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Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians In the early 1950s, both complex-analytic geometry and algebraic geometry began to undergo a radically new development, which finally led to the present shape of these two disciplines. It was basically the theory of sheaves and their cohomology which, under the influence of H. Cartan, K. Kodaira, J.-P. Serre, A. Grothendieck, and others, provided the appropriate framework for a subtantially new foundation of these two areas in mathematics and, simultaneously, for the great progress achieved since then.
One of the most spectacular and far-reaching results, obtained at the very beginning of this development, was due to the author. As early as in 1954, he succeeded in generalizing the classical Riemann-Roch theorem for algebraic curves and algebraic surfaces to general nonsingular complex projective varieties. His ingenious proof, which involved the entire power of the new sheaf- and cohomology-theoretic techniques, represented a milestone in transcendental algebraic geometry and made the ubiquitous significance (as well as the great perspectives) of the new methods strikingly evident. The author's epoch-making proof of a more general Riemann-Roch theorem was the decisive beginning of a further extensive development in this direction, culminating in more and more general Riemann-Roch-type theorems in the theory of algebraic schemes and, nowadays, even in arithmetical algebraic geometry.
The first edition of the author's book (under review) appeared in 1956; see Zbl 0070.16302. At that time, it was intended as being a research report, including an introduction to the theory of sheaves, vector bundles, and their cohomology machinery (cohomology groups, characteristic classes, cobordism theory, index theory, etc.) as used in the course of the treatise. This very layout made the book also into a textbook which, for the first time, offered a systematic and coherent account of the new techniques (and concepts) in algebraic geometry. Very quickly Hirzebruch's book became a standard text in transcendental algebraic geometry, and in the sequel it saw many new editions and translations.
The third edition of the book, which appeared in 1978 as a translation into English, was enriched and updated by two large appendices. These appendices, written by \textit{R. L. E. Schwarzenberger} and \textit{A. Borel}, explained the new developments in Riemann-Roch theory and index theory since the first edition of the book. -- The present edition is an unchanged reprint of the second, corrected printing of that third, enlarged edition. Over the years, the author's outstanding classic text has maintained its strong attractivity to any generation of mathematicians. The unique character of this book is based upon its high degree of clarity, rigor and comprehension, not to speak of the depth and beauty of its contents. There is certainly no doubt that the book under review is (and will remain) among the most important classics in transcendental algebraic geometry. fibre bundles; characteristic classes; holomorphic vector bundles; Atiyah-Singer index theorem; Riemann-Roch theorem; index theory F. Hirzebruch, Topological methods in algebraic geometry, in \textit{Classics in Mathematics} (Springer-Verlag, Berlin, 1995). Translated from the German and Appendix One by R. L. E. Schwarzenberger, With a preface to the third English edition by the author and Schwarzenberger, Appendix Two by A. Borel, Reprint of the 1978 edition. Topological properties in algebraic geometry, Research exposition (monographs, survey articles) pertaining to algebraic geometry, Research exposition (monographs, survey articles) pertaining to several complex variables and analytic spaces, Research exposition (monographs, survey articles) pertaining to algebraic topology Topological methods in algebraic geometry. Translation from the German and appendix one by R. L. E. Schwarzenberger. Appendix two by A. Borel. | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians The classical Riemann hypothesis states that all zeros of the Riemann zeta function in \(\mathbb C\) lie on the line \(\text{Re}\,s=1/2\). Due to the work of E. Artin, F. K. Schmidt and H. Hasse, it was shown that a zeta function can be associated to an arbitrary algebraic variety \(X\) over a finite field \(\mathbb F_ q\). These zeta functions turned out to be much simpler than the classical one, and since they are defined in terms of numbers of points of \(X\) in the finite algebraic extensions of \(\mathbb F_ q\), the analogue of Riemann hypothesis has immediate arithmetic implications allowing to give estimates for the number of solutions of algebraic equations in finite fields. In 1936 Hasse proved an analogue of the Riemann hypothesis for the zeta functions of elliptic curves. In the process of proving this conjecture for arbitrary curves, A. Weil developed the modern theory of Jacobian and abelian varieties which led him to the concept of abstract algebraic varieties over arbitrary domains. This work summarized in \textit{A. Weil}'s book ``Foundations of algebraic geometry.'' Providence, R.I.: American Mathematical Society (1962; Zbl 0168.18701)] marked the beginning of a new era in algebraic geometry.
By 1949 A. Weil proved the analogue of the Riemann hypothesis for a number of classes of algebraic varieties over finite fields, but he realized that in order to prove the conjecture in full generality it was necessary to think of a substitute for the classical topology which would be defined for varieties over arbitrary fields and would have the usual properties. - Assuming the existence of such a topology and a suitable cohomology theory and using the Lefschetz trace formula for the computation of the number of fixed points of the Frobenius automorphism, A. Weil was able to formulate (in 1949) his famous conjectures. In particular, the analogue of the Riemann hypothesis in the form suggested by Weil says that the eigenvalues of the representation of the Frobenius endomorphism in the (substitute of the) \(i\)-th cohomology space of \(X\) are integral algebraic numbers whose absolute values are equal to \(q^{i/2}\).
Weil suggested the Jacobian as a substitute for the cohomology of curves and the Chow ring of algebraic cycles as a substitute for the cohomology of certain rational varieties (e.g. Grassmannians), but it was quite unclear how to proceed further. This circle of problems, where arithmetic, analysis, algebra and geometry come together in such a beautiful way, attracted much attention, but only much later Grothendieck, following a suggestion of Serre, introduced the notion of étale topology and \(\ell\)-adic cohomology and succeeded in establishing some basic properties of this cohomology similar to properties of classical cohomology. From the existence of such cohomology theory it already follows that the zeta function is rational (this is the first of the Weil conjectures). Grothendieck also made the next step by formulating the so called ``standard conjectures'' about algebraic cycles [cf. \textit{S. Kleiman} in: Dix Exposés Cohomologie Schémas, Adv. Stud. Pure Math. 3, 359--386 (1968; Zbl 0198.25901)]. These standard conjectures imply the Weil conjectures and in fact go far beyond them, but at present there still is no idea how to prove some of the standard conjectures.
At last, in 1973 P. Deligne proved the remaining Weil conjectures (the independence of the coefficients of the characteristic polynomial of the Frobenius endomorphism \(F\) in \(H^*(X,\mathbb Q_{\ell})\) of the prime \(\ell\) and the fact that the absolute value of the eigenvalues of \(F\) in \(H^i(X,{\mathbb Q}_{\ell})\) is equal to \(q^{i/2})\).
The goal of the book under review is to present Deligne's proof of the Weil conjectures and the prerequisites from algebraic geometry, étale topology and monodromy theory used in this proof.
The book begins with a historical introduction (from Gauss to Deligne) by \textit{J. Dieudonné} [reprinted from Math. Intell. 10, 7--21 (1975; Zbl 0329.14012)].
The first chapter is devoted to Grothendieck's étale cohomology theory going as far as finiteness theorems and comparison theorems for complex varieties. The exposition here is based on \textit{A. Grothendieck}'s Sémin. géométrie algébrique 4 = SGA 4 (1963/64; see Lect. Notes Math. 269, 270, 305 (1972/73; Zbl 0234.00007; Zbl 0237.00012; Zbl 0245.00002)], but the authors use the language of derived categories.
In chapter 2 the authors prove Poincaré's duality theorem, the Lefschetz fixed points theorem, and the rationality of zeta [a more detailed exposition of this material can be found in SGA 7 (1967-69); see Lect. Notes Math. 288 (1972; Zbl 0237.00013) and 340 (1973; Zbl 0258.00005) and Exposé VIII (1971; Zbl 0221.14026)]. It should be noted that another textbook presentation of the contents of chapters 1 and 2 is given by \textit{J. S. Milne} [``Étale cohomology.'' Princeton, New Jersey: Princeton University Press (1980; Zbl 0433.14012)].
The third chapter is devoted to the monodromy theory of Lefschetz pencils which plays a major role in Deligne's proof. The principal reference here is SGA 7 II [Lect. Notes Math. 340 (1973; loc. cit.)].
Finally, in chapter 4 the authors present Deligne's proof of the Weil conjectures and discuss some generalizations and applications. There are three appendices devoted respectively to the fundamental group, derived categories, and descent.
The interest in Deligne's proof of Weil's conjectures obtained as a result of an interplay of ideas from diverse parts of mathematics is widespread among mathematicians, and since the proof requires rather complicated techniques, there certainly is a need for a self-contained exposition. Thus the book under review fills in an important gap in mathematical literature. On the other hand, the authors' presentation is still rather abstract and formal, and the book would have benefited from introducing examples, exercises and discussions. Riemann hypothesis; \(\ell\)-adic cohomology; Weil conjectures; algebraic cycles; standard conjectures; étale cohomology; monodromy theory of Lefschetz pencils Freitag, Eberhard; Kiehl, Reinhardt, Étale cohomology and the Weil conjecture, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)] 13, xviii+317 pp., (1988), Springer-Verlag, Berlin Étale and other Grothendieck topologies and (co)homologies, Varieties over finite and local fields, Finite ground fields in algebraic geometry, Research exposition (monographs, survey articles) pertaining to algebraic geometry, Parametrization (Chow and Hilbert schemes) Étale cohomology and the Weil conjecture. With a historical introduction by J. A. Dieudonné | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians In this paper it is shown that one of the aimed assumptions for constructing the moduli space of stable schemes does not hold. I.e., even for flat families of normal varieties the relative canonical sheaf is not compatible with base change. The proof of the main theorem (1.2) is constructive and gives a method to find flat families of \(S_j\) varieties (with \(\dim (X) > j \geq 2\)) and central fiber with just a singular point for which the base change does not hold. Furthermore, interesting results with several geometric consequences are given considering the intersection of the \(S_{n-1}\) locus and the components of the central fiber. relative canonical sheaf; dualizing complex; base change; depth; moduli of stable varieties; degenerations; Cohen-Macaulay Zsolt Patakfalvi, Base change behavior of the relative canonical sheaf related to higher dimensional moduli, Algebra Number Theory 7 (2013), no. 2, 353 -- 378. Families, moduli, classification: algebraic theory, Fibrations, degenerations in algebraic geometry, Differentials and other special sheaves; D-modules; Bernstein-Sato ideals and polynomials, Minimal model program (Mori theory, extremal rays), Families, moduli of curves (algebraic) Base change behavior of the relative canonical sheaf related to higher dimensional moduli | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians Let \(C\) be a smooth projective curve of genus \(g\geq 2\). The moduli space \({\mathcal U}(r,g-1)\) (respectively \({\mathcal S}{\mathcal U}(r,g-1))\) of stable vector bundles on \(C\) of rank \(r\) and degree \(g-1\) (respectively of rank \(r\), degree \(g-1\) and fixed determinant) admits the so called generalized theta divisor \(\Theta_ r\) (respectively \(\Theta_ r'\)) defined as \(\Theta_ r=\{E\in{\mathcal U}(r,g-1)\mid h^ 0(C,E)\geq 1\}\) (respectively \(\Theta_ r'\) is the restriction of \(\Theta _ r\) to \({\mathcal S}{\mathcal U}(r,g-1))\).
The main theorem of the paper is the following generalization of Riemann's singularity theorem: The multiplicity of \(\Theta_ r\) at \(E\) equals the dimension of the space of global sections of \(E\). If \(r\geq 2\) the analogous result holds for \(\Theta_ r'\). --- As a corollary it is shown that the divisor \(\Theta_ r'\) is a normal irreducible variety. For any \(E\in{\mathcal U}(r,g-1)\) denote by \(\Theta_ E\) the reduced subscheme of the Jacobian \(J(C)\) with support \(\{L\in J(C)\mid h^ 0(C,E\otimes L)\geq 1\}\). \(\Theta_ E\) is a divisor on \(J(C)\), if \(r=2\) and if \(r\geq 3\) for generic \(E\). It is shown that \(\Theta_ 2'\) admits a unique point \(E\) of multiplicity 3, and that its associated divisor \(\Theta_ E\) on \(J(C)\) is the image of the difference map \(C^ 2\to J(C)\), \((p,q)\mapsto{\mathcal O}_ C(p-q)\). Finally let \(E\in{\mathcal U}(2,g- 1)\) with \(\text{det} E=K_ C\). The following singularity theorem is proven for the divisor \(\Theta_ E\) on \(J(C)\):
\(\Theta_ E\) is singular at \(L\) if and only if either \(h^ 0(C,E\otimes L)\geq 2\) or \(h^ 0(C,E\otimes L)=1\), \(E\otimes L\) is an extension of \(K_ C(-D)\) with \(L^ 2(D)\) (where \(D\) is a nonzero section of \(E^*\otimes L^{-1}\otimes K_ C)\) and \(h^ 0(C,L^ 2(D))=1\). Riemann singularity theorem; stable vector bundle; generalized theta divisor Laszlo, Un théorème de Riemann pour les diviseurs thêta sur les espaces de modules de fibrés stables sur une courbe, Duke Math. J. 64 (2) pp 333-- (1991) Theta functions and curves; Schottky problem, Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20], Families, moduli of curves (algebraic), Vector bundles on curves and their moduli A Riemann theorem for the theta divisors on moduli spaces of stable fibre bundles over a curve | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians The paper is a survey dealing with the relation between Weil and Cartier divisors of a compact variety \(X^m\) on the one hand, and its \((2m-2)\)-th homology and second cohomology, respectively, on the other hand.
The main results are taken from two preprints written by the author and \textit{G. Barthel, K.-H. Fieseler}, and \textit{L. Kaup}: In case \(X\) is a compact toric variety, the canonical homomorphisms \(\text{Pic} (X)\to H^2(X)\) and \(\text{WeilDivCl} (X)\to H_{2m-2}(X)\) are isomorphisms. In particular, the natural inclusion \(\text{Pic} (X) \hookrightarrow \text{WeilDivCl} (X)\) corresponds to the Poincaré morphism.\ Moreover, the paper recalls the well known combinatorial description of all these groups for toric varieties.
At the end, the author mentions an interesting result obtained by the same authors concerning the intersection homology (middle perversity) of a toric variety. Sitting in between cohomology and homology, \(IH_{2m-2}(X)\) may be identified with those divisor classes on \(X\) that are represented by equivariant Weil divisors being Cartier in codimension two. Weil divisor; Cartier divisors; toric varieties; intersection homology Toric varieties, Newton polyhedra, Okounkov bodies, Divisors, linear systems, invertible sheaves, (Co)homology theory in algebraic geometry, Other algebro-geometric (co)homologies (e.g., intersection, equivariant, Lawson, Deligne (co)homologies) Poincaré morphism and divisors of compact toric varieties | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians Let \(f : {\mathcal X}\to M\) be a projective abelian scheme over an arithmetic quotient of a hermitian symmetric domain \(M = \Gamma \backslash {\mathcal D}\), constructed from a symplectic representation of the associated algebraic group \(G_ \mathbb{Q}\). Such an abelian scheme is called a Kuga fiber space of abelian varieties. The Mordell-Weil group \(\text{MW} ({\mathcal X}/{\mathcal M})\) of the Kuga fiber space \(f : {\mathcal X} \to M\) is defined to be the group of rational sections of \(f\).
In the paper, we show a finiteness theorem of the Mordell-Weil groups of Kuga fiber spaces of abelian varieties arising from standard \(\mathbb{Q}\)- primary symplectic representations. We remark that standard \(\mathbb{Q}\)- symplectic representations are defined and classified by Satake, and each of them can be decomposed into \(\mathbb{Q}\)-primary ones. The class of standard representations are very natural and it includes most of interesting and important cases such as elliptic modular cases, Siegel modular cases, Shimura's moduli families of abelian varieties and so on. Therefore our theorem is a generalization of Shioda's results on elliptic modular surfaces [\textit{T. Shioda}, J. Math. Soc. Japan 24, 20-59 (1972; Zbl 0226.14013)], and Silverberg's results on some important classes of Kuga fiber spaces [\textit{A. Silverberg}, Invent. Math. 81, 71-106 (1985; Zbl 0576.14020)].
The main idea of the proof is a generalization of Silverberg's cohomological criterion for finiteness of Mordell-Weil groups of Kuga fiber spaces [cf. Duke Math. J. 56, No. 1, 41-46 (1988; Zbl 0645.14019)]. Let \(\Gamma \to GL (W_ \mathbb{C})\) be the symplectic representation associated to a Kuga fiber space, and let \(H^ \bullet (\Gamma, W_ \mathbb{C})\) denote the Eilenberg-MacLane cohomology group. Then Silverberg's criterion says that if \(\dim M>1\) or \(M\) is compact and \(H^ q (\Gamma, W_ \mathbb{C}) = 0\) for \(q=0,1\), the Mordell-Weil group MW\(({\mathcal X}/M)\) is finite. Therefore vanishing theorems for \(H^ q (\Gamma, W_ \mathbb{C})\) imply the finiteness of the Mordell-Weil groups. In case when the \(\mathbb{R}\)-rank of \(G_ \mathbb{R}\) is greater than one, we can show that \(H^ q (\Gamma, W_ \mathbb{C}) = 0\) for \(q \leq 1\) by using the intersection cohomology group, Zucker conjecture proved by Looijenga and by Saper and Stern, and Borel-Wallach vanishing theorem.
In case when \(\text{rank}_ \mathbb{R} G_ \mathbb{R} = 1\) which includes the case of elliptic modular surfaces, we do have examples with \(H^ 1 (\Gamma, W_ \mathbb{C}) \neq 0\). Therefore we have to generalize the Silverberg criterion by using (mixed) Hodge theory on \(H^ 1 (\Gamma, W_ \mathbb{C})\). The generalized criterion says that the Mordell-Weil group is finite if and only if \(H^ 1 (\Gamma, W_ \mathbb{Q}) \cap H^{0,0} = 0\) where \(H^{0,0}\) is the Hodge component of \(H^ 1 (\Gamma, W_ \mathbb{C})\) of type (0,0). (In case when \(M\) is not compact, we will consider the intersection cohomology and its pure Hodge structure.) Then, though we have examples with \(H^{0,0} \neq 0\), we can show that \(H^ 1 (\Gamma, W_ \mathbb{Q}) \cap H^{0,0} = 0\), hence the Mordell-Weil group is finite. abelian scheme over an arithmetic quotient of a hermitian symmetric domain; symplectic representation of the algebraic group; standard representations; finiteness of Mordell-Weil groups; Kuga fiber spaces Saito, M.-H.: Finiteness of the Mordell-Weil groups of Kuga fiber spaces of abelian varieties. Publ. Res. Inst. Math. Sci. 29, 29--62 (1993) Algebraic theory of abelian varieties, Rational points, Arithmetic varieties and schemes; Arakelov theory; heights, Families, moduli, classification: algebraic theory, Homogeneous spaces and generalizations Finiteness of Mordell-Weil groups of Kuga fiber spaces of abelian varieties | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians Let \(X\) be an algebraic variety over a finite field \(\mathbb{F}_q\) of characteristic \(p> 0\). One of the famous Weil conjectures was that the zeta function \(\zeta(X,T)\) of \(X\), which is a priori defined as a certain power series, is in fact a rational function in \(T\). This conjecture was first proved by \textit{B. Dwork} [Am. J. Math. 82, 631--648 (1960; Zbl 0173.48501)] via \(p\)-adic analysis. Somewhat later, A. Grothendieck gave a different proof based on \(\ell\)-adic cohomology for \(\ell\neq p\).
In the sequel, B. Dwork began the study of the variation of the zeta function within an algebraic family of varieties over \(\mathbb{F}_q\), which led him to the investigation of certain other zeta functions, the so-called unit root zeta functions. These functions are analytic in nature, and generally no longer rational functions, which makes the understanding of their \(p\)-adic analytic properties especially important. In this context, B. Dwork conjectured that a unit root zeta function must be \(p\)-adically meromorphic, i.e., it should be expressible as a quotient of two power series which are convergent everywhere on the completion of \(\mathbb{Q}_p\). In the early 1970s, Dwork himself was able to prove this conjecture of his for special families of curves and surfaces, but a general proof of this hard conjecture had to wait until D. Wan's work under review was completed.
In the meantime, it was figured out that unit root zeta functions can be interpreted as the \(L\)-functions of \(F\)-crystals on the base space of the respective family of varieties, and this has finally proved to be the right approach to tackle Dwork's conjecture in full generality. The paper under review, which is the first of two consecutive articles, provides a complete proof of Dwork's conjecture in the so-called higher rank case, whereas the subsequent paper [cf.: \textit{D. Wan}, J. Am. Math. Soc. 13, No. 4, 853--908 (2000; Zbl 1086.11031)] accomplishes the remaining case, the so-called rank one case.
The author has outlined his ingenious approach to Dwork's conjecture in two foregoing articles entitled ``A quick introduction to Dwork's conjecture'' [Fried, Michael D. (ed.), Applications of curves over finite fields. 1997 AMS-IMS-SIAM joint summer research conference, July 27--31, 1997, Seattle, WA, USA. Providence, RI: American Mathematical Society. Contemp. Math. 245, 147--163 (1999; Zbl 0977.11028)] and ``Dwork's conjecture on unit root zeta functions'' [Ann. Math. (2) 150, No. 3, 867--927 (1999; Zbl 1013.11031)], respectively, in which he formulated this conjecture in a more general context, namely in terms of his newly established ``\(\sigma\)-modules'' and ``overconvergent power series rings over complete discrete valuation rings''. Wan's \(\sigma\)-modules are suitable generalizations of \(F\)-crystals, and the \(p\)-adic analytic properties of their associated \(L\)-functions form the crucial part of his subtle investigations, in the present paper.
The main theorem, whose complete proof is carried out in the two consecutive papers under review, is formulated in this generalized framework and reads as follows:
Theorem 1.1. Let \(X\) be a smooth affine variety defined over a finite field \(\mathbb{F}_q\) of characteristic \(p> 0\). Let \((M,\Phi)\) be a finite-rank overconvergent \(\sigma\)-module over \(X\). Then, for any rational number \(s\), the pure slope-\(s\) \(L\)-function \(L_s(\Phi,T)\) attached to \((M,\Phi)\) is \(p\)-adically meromorphic everywhere.
This theorem gives an affirmative answer to Dwork's long-standing conjecture, and it even shows that this conjecture holds in greater generality.
As to the contents of the present first paper, which is merely algebraic in nature, there are 10 sections discussing in full detail the following subjects: (1) Introduction, and main results; (2) \(\sigma\)-modules and their \(L\)-functions; (3) Monsky's trace formula; (4) Hodge-Newton decomposition of a \(\sigma\)-module and Dwork's conjecture; (5) An easier case of Dwork's conjecture; (6) The ordinary case of Dwork's conjecture; (7) The non-ordinary case of Dwork's conjecture; (8) The general case of Dwork's conjecture; (9) Reduction to the base scheme \(\mathbb{A}^n\); (10) Appendix: Proof of the extended Monsky trace formula.
Basically, the author reduces Dwork's conjecture from the higher rank case over any smooth affine variety to the rank one case over the affine space \(\mathbb{A}^n\). In the subsequent second part, the proof of the main theorem (Theorem 1.1) is finished by proving just that rank one case of Dwork's conjecture over \(\mathbb{A}^n\).
Altogether, this is a very important contribution toward the developments around Dwork's conjecture, which finally provides a long and fascinating story with a happy end. The paper is extremely rich of new ideas, concepts, constructions, and results, thereby utmost detailed, lucid and comprehensible. algebraic varieties over finite fields; \(L\)-functions; zeta functions; \(p\)-adic analysis; Dwork's conjecture Wan, D., Higher rank case of dwork\(###\)s conjecture, J. Amer. Math. Soc., 13, 807-852, (2000) \(L\)-functions of varieties over global fields; Birch-Swinnerton-Dyer conjecture, Zeta functions and \(L\)-functions, Other Dirichlet series and zeta functions, Varieties over finite and local fields, Complex multiplication and moduli of abelian varieties, Zeta functions and related questions in algebraic geometry (e.g., Birch-Swinnerton-Dyer conjecture) Higher rank case of Dwork's conjecture | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians The authors give completely algebro-geometric proofs of a theorem by \textit{T. Shiota} [Invent. Math. 83, 333--382 (1986; Zbl 0621.35097)], and of a theorem by \textit{I. Krichever} [Ann. Math. (2) 172, No. 1, 485--516 (2010; Zbl 1215.14031)], characterizing Jacobians of algebraic curves among all irreducible principally polarized abelian varieties. Shiota's characterization is given in terms of the Kadomtsev-Petviashvili (KP) equation. Krichever's characterization is given in terms of trisecant lines to the Kummer variety. Here, the authors treat the case of flexes and degenerate trisecants. The methods of Shiota and Krichever are completely analytic in nature. In the present article, the authors provide simple algebro-geometric proofs of Shiota's theorem (Novikov's conjecture) and of parts of Krichever's theorem (the trisecant conjecture). The basic tool they use is a theorem they prove asserting that the base locus of the linear system associated to an effective line bundle on an abelian variety is reduced. This result allows them to remove all the extra assumptions that were introduced in theorems of the first author et al. [Math. Res. Lett. 13, No. 1, 109--123 (2006; Zbl 1098.14020)], \textit{O. Debarre} [J. Algebr. Geom. 1, No. 1, 5--14 (1992; Zbl 0783.14015); Compos. Math. 107, No. 2, 177--186 (1997; Zbl 0890.14013)], and \textit{G. Marini} [Math. Ann. 309, No. 3, 483--490 (1997; Zbl 0885.14015); Compos. Math. 111, No. 3, 305--322 (1998; Zbl 0945.14017)], in order to achieve algebro-geometric proofs of the results above. In other words, their arguments build on the methods introduced by De Concini and the first author and on the subsequent developments mostly by Debarre and Marini, which all provided algebro-geometric proofs of the above theorems, but only under certain additional assumption This paper is organized as follows: Section 1 is an introduction to the subject and a description of the results. Section 2 is devoted to some base loci. The main result of this section is that the base locus of an effective line bundle on an abelian variety is always reduced. Section 3 deals with subschemes will play a central role in some arguments. Section 4 is devoted to Shiota's theorem. Section 5 deals with Kummer varieties with one flex come from Jacobians. Here the authors give a completely algebro-geometric proof of the Krichever's theorem. Section 6 deals with Kummer varieties with one degenerate trisecant line come from Jacobians. In this section the authors give a completely algebro-geometric proof of Krichever's theorem, in the case of a degenerate trisecant. K-P equations; principally polarized abelian variety; Jacobian variety; Novikov conjecture; theta functions; integrable hierarchies Theta functions and abelian varieties, Analytic theory of abelian varieties; abelian integrals and differentials, Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.) Characterizing Jacobians via the KP equation and via flexes and degenerate trisecants to the Kummer variety: an algebro-geometric approach | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians Let \(X\) be smooth separated connected scheme over an algebraically closed field of characteristic \(p \geq 0\), let \(f: Y \to X\) be a smooth proper morphism and \(x\) a geometric point of \(X\).
The authors prove the geometric variant with \(\mathbb{F}_\ell\)-coefficients of the Grothendieck-Serre semisimplicity conjecture:
Theorem 1.1. For \(\ell \gg 0\) (depending on \(f\)), the action of \(\pi_1(X,x)\) on \(H^*(Y_x,\mathbb{F}_\ell)\) is semisimple.
The analogous result with \(\mathbb{Q}_\ell\)-coefficients is a famous theorem of \textit{P. Deligne} [Publ. Math., Inst. Hautes Étud. Sci. 52, 137--252 (1980; Zbl 0456.14014)].
They prove that Theorem 1.1 is equivalent to:
Theorem 1.2. After replacing \(X\) by a connected etale cover and for \(\ell \gg 0\) (depending on \(f\)), the image of \(\pi_1(X,x)\) in the group of \(\mathbb{Q}_\ell\)-points of its Zariski closure in \(\mathrm{GL}(H^*(Y_x,\mathbb{Q}_\ell))\) is almost hyperspecial. étale cohomology; étale fundamental group; positive characteristic; tensor invariants; semisimplicity; algebraic groups; big monodromy; Tate conjecture Étale and other Grothendieck topologies and (co)homologies, Linear algebraic groups over local fields and their integers, Maximal subgroups Geometric monodromy -- semisimplicity and maximality | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians Chow's construction of the Jacobian \(J\) of an algebraic curve \(C\) as a quotient variety of \(C(n)\) -- the \(n\)-fold symmetric product of \(C\) -- ``fibered'' by the linear systems [cf. \textit{W.-L. Chow}, Am. J. Math. 76, 453--476 (1954; Zbl 0056.14404)] raises the question of whether \(C(n)\), for \(n> 2g-2\), is actually an algebraic projective bundle over \(J\). We have shown elsewhere [Ill. J. Math. 5, 550--564 (1961; Zbl 0107.14702)] that this is so; it is thus natural to ask what the Chern classes (to speak somewhat loosely) of this bundle are. One of the objectives of this paper is to exhibit these classes as elements of \(A(J)\), the rational equivalence ring of \(J\). Once this is done, one has the structure of \(A(C(n))\) explicitly as an extension of \(A(J)\). This gives for example in a natural form the structure of the homology rings of high symmetric products of the closed orientable topological surfaces, which have hitherto only been computed ``in principle'' by the use of Eilenberg-MacLane spaces. As a by-product of this determination of Chern classes, we get certain intersection relations among the subvarieties of \(J\).
Part II of this paper is devoted to the Chern classes. Part I is preliminary, and has connection with theorems of Chow and Andreotti. In it we prove that if \(C\) is a non-hyperelliptic curve, then any \(g-1\) points of a generic canonical divisor are algebraically independent (but as will be seen, only just!). In addition to being used in a critical argument of Part II, we also use it to squeeze out the dimension and irreducibility of certain subvarieties of \(C(n)\) which play an important role in Part II, as well as in the other applications alluded to above. Our emphasis throughout is on rational equivalence, not anything coarser. --------, Symmetric products and Jacobians , Amer. J. Math. 83 (1961), 189-206. JSTOR: Jacobians, Prym varieties, Vector bundles on curves and their moduli, Homology of classifying spaces and characteristic classes in algebraic topology, (Equivariant) Chow groups and rings; motives Symmetric products and Jacobians | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians In this paper we prove the following main theorem:
Let \(g\geq 3\), and let \(X\) be a generic abelian variety of dimension \(g\). Then \(X\) does not contain a hyperelliptic curve.
To be more precise: Consider the moduli space \({\mathcal A}_g\) of polarized abelian varieties of dimension \(g\). Let \(K\) be a field, and let \({\mathcal A}\) be an irreducible component of \({\mathcal A}_g \otimes K\). Let \(\eta\) be the generic point of \({\mathcal A}\), and let \(X\) be the abelian variety over \(L:=K({\mathcal A})^a\) corresponding with \(\overline\eta \in {\mathcal A}(L)\) (here the superscript \(a\) represents taking the algebraic closure). Let \(C\subset X\) be an irreducible algebraic curve. If \(g\geq 3\), then the normalization \(C^\sim\) is not a hyperelliptic curve.
This theorem in characteristic zero was proved by \textit{G. P. Pirola} [see Duke Math. J. 59, No. 3, 701-708 (1989; Zbl 0717.14021), section 2, theorem 2; see also: Abelian varieties, Proc. int. Conf., Egloffstein 1993, 237-249 (1995; Zbl 0837.14036), p. 238]. The proof by Pirola follows from the fact that hyperelliptic curves in an abelian variety in characteristic zero are ``rigid up to translation''. In positive characteristic this rigidity is no longer true, and we analyze the situation there [for surfaces, see \textit{C. Schoen}, J. Reine Angew. Math. 411, 196-204 (1990; Zbl 0702.14015), section 2]. These results by Pirola and by Schoen inspired us to give the proofs below. In Nieuw Arch. Wisk., IV. Ser. 13, No. 3, 427-434 (1995; Zbl 0859.14004), \textit{F. Oort} analyzed the proof by Pirola, and some details of that paper will be used and recorded here. hyperelliptic curves in abelian varieties Algebraic moduli of abelian varieties, classification, Elliptic curves Hyperelliptic curves in abelian varieties | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians The book contains an introduction to the theory of complex algebraic curves, which is well suited for a postgraduate course in algebraic geometry, or even for a graduate course of high level. The book is reasonably self-contained, for students with some background on the theory of polynomial rings. More advanced basis of commutative algebra are briefly collected in section 3.6. The point of view of the author is based on the classical work of \textit{A. Brill} and \textit{M. Nöther} [Math. Ann. 7, 269--316 (1874; JFM 06.0251.01)]. The author mainly studies the theory of plane curves with ordinary singularities. Since any projective curve is birationally equivalent to a plane curve with ordinary singularities, the exposition covers the basis of the theory of all projective curves.
After the first, introductory chapter on hypersurfaces, the author devotes the second chapter to the local theory of germs of plane curves. Branches are defined and studied in terms of the associated Puiseux series. The chapter contains the formal definition of intersection multiplicity of two branches. The third chapter is devoted to the study of intersections of plane curves. Combining the local theory with properties of the resultant of two polynomials, the author proves the Theorem of Bézout, and the AF+BG Theorem of M. Noether. The final chapter starts with an introduction to birational transformations and the proof that Cremona transormations of \(\mathbb P^2\) can turn any curve into a curve with ordinary singularities. Then, the author introduces divisors and general linear series on plane curves, starting with linear series cut by all curves of fixed degree. The main tool used through the chapter is the definition of adjoint linear series to curves with
ordinary singularities. The author defines the genus of a curve and describes its basic properties, including proofs of the Riemann-Roch Theorem, the Clifford Theorem, and the Hurwitz formula.
The analysis contained in the book, in addition to a good initial insight into algebraic geometry, can stimulate the reader to reconsider the classical Brill-Noether approach to the theory of curves, still suitable of application to problems on special linear series which remain open even now. algebraic curves Introductory exposition (textbooks, tutorial papers, etc.) pertaining to algebraic geometry, Curves in algebraic geometry, Intersection theory, characteristic classes, intersection multiplicities in algebraic geometry, Riemann-Roch theorems Algebraic curves, the Brill and Noether way | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians Several applications of model-theoretic methods have recently appeared in the theory of cohomology of varities, e.g. the construction of a new Weil cohomology theory with good properties by the author [Bull. Lond. Math. Soc. 36, No. 5, 663-670 (2004; Zbl 1070.14023)] and Brünjes and Serpé [Nonstandard Etale Cohomology, Preprint]. In such applications, one question is always: which model theoretic language does one use, and what does this language allow us to say about the algebraic objects one would like to consider? The present short note clarifies what one can say about étale sheaves in the pure ring language. More precisely, suppose that \(\mathcal{F}\) is a constructible sheaf on a variety \(X\) (either a torsion sheaf, or a \(\bar{\mathbb{Q}}_l\)-sheaf). Then for any field \(k\) and any \(x \in X(k)\), the stalk \(\mathcal{F}_x\) carries an action of the absolute Galois group of \(k\), well defined up to conjugacy.
The article describes a setting in which the isomorphism type of this action (as a function of \(k\) and \(x\)) can be described by a first order formula.
The author gives two example applications. The first one is an application to cohomology with compact support, with coefficients in a torsion sheaf. The second one is a generalization of the results of \textit{Z. Chatzidakis, L. van den Dries} and \textit{A. Macintyre} [J. Reine Angew. Math. 427, 107--135 (1992; Zbl 0759.11045)] concerning the number of points of a definable set over a finite field: instead of just counting points, one now sums over the traces of the Frobenius automorphism on the stalks of a (non-constant) sheaf, in the same way as one does in the Lefschetz trace formula.
The article contains a section recalling the basic notions concerning étale sheaves, fundamental groups and their relation. Moreover, the definitions of Galois stratifications and Galois formulae are recalled. étale sheaves; fundamental groups; Artin symbol; model theory of sheaves; pseudo-finite fields Étale and other Grothendieck topologies and (co)homologies, Model-theoretic algebra Constructible sheaves and definability | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians In this interesting paper, the author studies on the base of a generalization of Grothendieck's descent theorem in the framework of locally Krull schemes on which group schemes operate, relations among enriched principal bundles, class groups and Picard groups. In the previous work [J. Algebra 459, 76--108 (2016; Zbl 1348.13010)], the author introduced for a flat \(S\)-group scheme \(G\) and for a locally Krull \(G\)-scheme \(X\) a principal \(G\)-bundle \(\varphi : X \to Y\) and an equivariant class group \(Cl(G,X)\). It was shown, by the utilizing of Grothendieck's descent theorem, that \(Y\) is locally Krull and that `the inverse image functor \(\varphi^* \) induces an isomorphism \(Cl(Y) \to Cl(G,X)\)`. The present paper deals with an enriched version. Let \(f: G \to H\) be an fpqc homomorphism, \(N:= \ker f\), \(\varphi : X \to Y\) be a \(G\)-morphism which is also a principal \(N\)-bundle, \(Cl(G,X)\), \(Cl(H,Y)\) be equivariant class groups, \(\varphi^* : Cl(H,Y) \to Cl(G,X)\) be a morphism (really isomorphism). The main result presents and characterizes the enriched Grothendieck`s descent theorem which yields the isomorphism \(\varphi^*\) and a similar isomorphism \(\varphi^* : \mathrm{Pic}(H,Y) \to\mathrm{Pic}(G,X)\). The Grothendieck-Mumford approach consists in passing from commutative rings to corresponding affine schemes and analyzing their singularities by algebraic geometry methods. The theory of the Grothendieck descent is used for gluing schemes from affine pieces. Along with the definition of Krull rings, which the author uses, the Krull rings can be defined as corresponding rings for which there exists a theory of divisors by \textit{Z. I. Borevich} and \textit{I. R. Shafarevich} [Number theory. Translated by Newcomb Greenleaf. New York and London: Academic Press (1966; Zbl 0145.04902)]. After the introductory section the author of the paper under review gives in section 2 results on quasi-fpqc morphisms and enriched principal bundles. Section 3 deals with \(\kappa\)-schemes. Section 4 is on module sheaves over a ringed site. Big and interesting sections 5 and 6 on Grothendieck's descent and enriched Grothendieck's descent. The last short section contains observations on equivariant Picard groups and class groups. In this section, the results of the paper are applied to equivariant principal bundles, and, in particular, to the case of finite Galois extension of a field \(k\). In the last case let \(N_0\) be a finite étale \(k-\)group scheme, \(\varphi : X \to Y\) be a principal \(N_0\)-bundle and \(k^{`}/k\) be a finite Galois extension with Galois group \(H\), such that \(k^{`}\otimes N_0\) is a constant group scheme \(N\). Let \(G = N \rtimes H\). Then \(G\) and \(H\) are constant group schemes and \(\varphi\) is a principal \(N_0\)-bundle, and \(N_0\) need not be constant. group scheme; locally Krull scheme; class group; descent theory; Picard group; principal fiber bundle Group actions on varieties or schemes (quotients), Group schemes, Actions of groups on commutative rings; invariant theory Equivariant class group. II: Enriched descent theorem | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians One of the fundamental questions related to Grothendieck's program ``dessins d'enfants'' is to try to understand how the combinatorial data coming from the topological-combinatorial aspect of the dessins d'enfants can be translated into arithmetical data. The classical method to relate these two aspects makes use of the cartographic groups of the dessins d'enfants in order to produce étale coverings via the Galois theory. In other words, one has to study the arithmetic fundamental group of a curve defined over a numberfield. This approach involves therefore deep methods of algebraic and arithmetic geometry. The aim of the present paper is to present a very explicit and elementary method to understand how the combinatorial data completely determine the arithmetical data in the case of ``dessins d'enfants of genus one''. A special attention is payed to the action of the absolute fundamental group \(\text{Gal} (\overline{\mathbb Q}/\mathbb Q)\) on such structures. dessins d'enfants; action of absolute fundamental group; childrens' drawings; Galois group Enumerative problems (combinatorial problems) in algebraic geometry, Group actions on varieties or schemes (quotients) Childrens' drawings in genus 1 | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians \textit{V. G. Drinfel'd} [``Elliptic modules'', Math. USSR, Sb. 23 (1974), 561--592 (1976); translation from Mat. Sb., n. Ser. 94 (136), 594--627 (1974; Zbl 0321.14014)] introduced the analogues of Shimura varieties for \(\mathrm{GL}_d\) over a global field of positive characteristic. \textit{G. Laumon, M. Rapoport} and \textit{U. Stuhler} [``\({\mathcal D}\)-elliptic sheaves and the Langlands correspondence'', Invent. Math. 113, No. 2, 217--338 (1993; Zbl 0809.11032)] defined the corresponding varieties, which are the analogues of Shimura curves for \(d=2\). The higher dimensional analogues are moduli spaces of \(\mathcal{A}\)-elliptic sheaves. The author generalizes slightly the notion of \(\mathcal{A}\)-elliptic sheaves by using hereditary orders. Let \(X\) be a proper smooth curve over a finite field \(\mathbb{F}_q\) together with a distinguished point \(\infty\), and \(I\) an effective divisor on \(X\). Denote by \(F\) the function field of \(X\). Let \(A\) be a central simple \(F\)-algebra of dimension \(d^2\) and \(\mathcal{A}\) a locally principal hereditary \(\mathcal{O}_X\)-order in \(A\). It is proved that the moduli stack of \(\mathcal{A}\)-elliptic sheaves with \(I\)-level structure \(\mathcal{E}\!\ell\ell^\infty_{\mathcal{A},I}\) is a Deligne-Mumford stack which is locally of finite type and of dimension \(d-1\) over \(X-I\). The main result of this article is the following:
Let \(B\) be an other central simple \(F\)-algebra of dimension \(d^2\) such that there is a closed point \(\mathfrak{p} \in X- \{\infty\}\) such that the local invariants of \(B\) are given by
\[
\mathrm{inv}_\infty(B)=\mathrm{inv}_\infty(A)+\frac{1}{d},\;\; \mathrm{inv}_{\mathfrak{p}}(B)=\mathrm{inv}_{\mathfrak{p}(A)}-\frac{1}{d}
\]
and all other local invariants at points different than \(\mathfrak{p},\infty\) are equal. If \(\mathcal{B}\) is a locally principal hereditary \(\mathcal{O}_X\)-order in \(B\) with \(e_x(\mathcal{B})=e_x(\mathcal{A})\) for all \(x\), then the moduli stack \(\mathcal{E}\!\ell\ell^{\mathfrak{p}}_{\mathcal{B},I}\) is a twist of \(\mathcal{E}\!\ell\ell^\infty_{\mathcal{A},I}\). The positive integer \(e_x(\mathcal{A})\) is defined by \(\mathrm{Rad}(\mathcal{A}_x)^{e_x(\mathcal{A})}= \varpi_x \mathcal{A}_x\), where \(\varpi_x\) is the uniformizer at \(x\).
Using this result the author proves that the uniformization at \(\infty\) and the analogue of Cherednik-Drinfeld uniformization for the moduli spaces \(\mathcal{E}\!\ell\ell^\infty_{\mathcal{A},I}\) are equivalent. Drinfeld modular varieties; rigid analytic uniformization Spiess, M., Twists of Drinfeld-stuhler modular varieties, Doc. Math., 595-654, (2010), (Extra volume: Andrei A. Suslin sixtieth birthday) Drinfel'd modules; higher-dimensional motives, etc., Arithmetic aspects of modular and Shimura varieties, Rigid analytic geometry Twists of Drinfeld-Stuhler modular varieties | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians The starting point of this paper is \textit{R. Hartshorne}'s theory of reflexive sheaves and the Serre correspondence between rank 2 reflexive sheaves on \(\mathbb P^3\) and curves in \(\mathbb P^3\) [Math. Ann. 238, 229--280 (1978; Zbl 0411.14002); ibid. 254, 121--176 (1980; Zbl 0431.14004)]. In fact the author continues the study he began in his previous works [Pac. J. Math. 219, No. 2, 391--398 (2005; Zbl 1107.14032); J. Pure Appl. Algebra 211, no. 3, 622--632 (2007; Zbl 1128.14009); An effective bound for reflexive sheaves on canonically trivial 3-folds, to appear], investigating some cohomological properties of rank 2 reflexive sheaves on smooth projective threefolds (on which one can easily extend the Serre correspondence), with interesting consequences for the structure of the sheaves themselves. In particular, he finds conditions for these sheaves to be locally free and he studies a case in which the Riemann-Roch formula becomes quite simple. Finally the author applies these results to the relation between the moduli space of torsion free sheaves and the Hilbert scheme of elliptic curves on a particular class of threefolds (e.g. Fano threefolds). reflexive sheaf; Fano threefold; elliptic curve Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20], Algebraic moduli problems, moduli of vector bundles, Vector bundles on surfaces and higher-dimensional varieties, and their moduli, Fano varieties The Hilbert scheme of elliptic curves and reflexive sheaves on Fano 3-folds | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians The Jacobian \(J\) of a smooth curve \(C\) parameterizes degree zero line bundles on \(C\). Suppose that \(C\) degenerates to a possibly singular curve via a 1-parameter family. Algebraically, this is encoded by viewing \(C\) as the general fiber of a family of curves over a discrete valuation ring \(S\); the special fiber is the limit of the smooth curve \(C\), and the total space of this family is an arithmetic surface \(\widetilde{C}\). The Jacobian \(J\) extends to the special fiber via a construction called the Néron model, resulting in a scheme (also denoted \(J\)) over \(S\). A key issue, however, is that the Néron model \(J/S\) need not be proper, so one naturally studies its various compactifications.
On the one hand, a recent result of \textit{C. Pépin} [Math. Ann. 355, No. 1, 147--185 (2013; Zbl 1263.14046)] is that the Néron model \(J/S\) admits a ``semi-factorial'' model, meaning a compactification of \(J\), flat and projective over \(S\), such that every line bundle on the general fiber extends to a line bundle on the total space. On the other hand, assuming the curve \(C\) and its limit are sufficiently nice, there is a compactification \(\overline{J}\)/S, called the Altman-D'Souza-Kleiman family of compactified Jacobians parameterizing degree zero, rank one, torsion-free sheaves on \(\widetilde{C}/S\) (the locus corresponding to line-bundles is precisely \(J/S\)). We therefore have two natural compactifications of the Jacobican. The purpose of the present short note is to prove that, when the Picard rank of the Jacobian of \(C\) is one (i.e., the theta divisor generates the torsion-free part of the Néron-Severi group of \(J\)), then the compactified Jacobian \(\overline{J}\) enjoys Pépin's desirable property of being a semi-factorial model. Projectivity was already known, and flatness is a short lemma, so the main work (though still only one page) is to show that line bundles extend. Jacobian; compactification; arithmetic surface; Néron model Jacobians, Prym varieties, Abelian varieties of dimension \(> 1\), Arithmetic ground fields for curves, Picard schemes, higher Jacobians, Divisors, linear systems, invertible sheaves, Local ground fields in algebraic geometry An explicit semi-factorial compactification of the Néron model | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians The main result in this paper is a very nice generalization of a beautiful theorem of \textit{A. A. Rojtman} [Ann. Math., II. Ser. 111, 553- 569 (1980; Zbl 0504.14006)]. Rojtman had proved that for any smooth projective variety X over an algebraically closed field k of characteristic \(p\geq 0\), the natural map \(CH_ 0(X)\to Alb X\) is an isomorphism on torsion where \(CH_ 0(X)\) is the group of zero cycles of degree zero modulo rational equivalence. (Throughout this review I will slur over p-torsion points.) Here, the author generalizes this result as follows: Let X be a projective variety defined over an algebraically closed field k, smooth in codimension one. Define \(CH_ 0(X,X_{\sin g})\) as the group of zero cycles supported on \(X-X_{\sin g}\) of degree zero modulo rational equivalence on (complete) curves missing \(X_{\sin g}\). Then the natural map \(CH_ 0(X,X_{\sin g})\to Alb X\) is an isomorphism on torsion. In particular, if \(Y\to X\) is a desingularization, the natural map \(CH_ 0(X,X_{\sin g})\to CH_ 0(Y)\) is an isomorphism on torsion. - The proof is similar to that of Rojtman, reducing first to the case of a surface and then introducing a suitable notion of ''morphisms'' from varieties to the group of zero cycles, which enables one to study rational equivalences more closely.
Recently, \textit{A. Collino} (preprint) has generalized the above result and shown in particular that n-torsion in \(CH^ 2(X)\) is always finite, where X is any projective variety with isolated singular points. This is actually a generalization of a result of \textit{A. B. Merkur'ev} and \textit{A. A. Suslin} [Math. USSR, Izv. 21, 307-340 (1983); translation from Izv. Akad. Nauk SSSR, Ser. Mat. 46, No.5, 1011-1046 (1982; Zbl 0525.18008)]. Albanese variety; Chow group; zero cycles; rational equivalence; isomorphism on torsion; desingularization M. Levine, ''Torsion zero-cycles on singular varieties,'' Amer. J. Math., vol. 107, iss. 3, pp. 737-757, 1985. Algebraic cycles, (Equivariant) Chow groups and rings; motives, Singularities in algebraic geometry Torsion zero-cycles on singular varieties | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians For a smooth algebraic curve \(X\) over a field, applying \(\mathrm{H}_1\) to the Abel map \(X \to \mathrm{Pic} X / \partial X\) to the Picard scheme of \(X\) modulo its boundary realizes the Poincaré duality isomorphism
\[
\mathrm{H}_1(X, \mathbb{Z}/ \ell) \to \mathrm{H}^1(X/ \partial X, \mathbb{Z}/\ell(1)) \cong \mathrm{H}^1_c(X, \mathbb{Z}/\ell(1)).
\]
We show the analogous statement for the Abel map \(X/\partial X \to \overline{\mathrm{P}}\mathrm{ic}X/\partial X\) to the compactified Picard, or Jacobian, scheme, namely this map realizes the Poincaré duality isomorphism \(\mathrm{H}_1(X/ \partial X, \mathbb{Z}/\ell) \to \mathrm{H}^1(X, \mathbb{Z}/\ell(1))\). In particular, \(\mathrm{H}_1\) of this Abel map is an isomorphism. In proving this result, we prove some results about \(\overline{\mathrm{P}}\mathrm{ic}\) that are of independent interest. The singular curve \(X/\partial X\) has a unique singularity that is an ordinary fold point, and we describe the compactified Picard scheme of such a curve up to universal homeomorphism using a presentation scheme. We construct a Mayer-Vietoris sequence for certain pushouts of schemes, and an isomorphism of functors \(\pi_1^{\ell}\operatorname{Pic}^0(-) \cong \mathrm{H}^1(-,\mathbb{Z}_{\ell}(1))\). Abel map; compactified Picard scheme; compactified Jacobian; Poincaré duality J. L. Kass and K. Wickelgren, An arithmetic count of the lines on a smooth cubic surface, preprint, arXiv:1708.01175v1 [math.AG]. Homotopy theory and fundamental groups in algebraic geometry, Algebraic moduli problems, moduli of vector bundles, Étale and other Grothendieck topologies and (co)homologies, Picard groups An Abel map to the compactified Picard scheme realizes Poincaré duality | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians We prove the test function conjecture of Kottwitz and the first named author for local models of Shimura varieties with parahoric level structure attached to Weil-restricted groups, as defined by B. Levin. Our result covers the (modified) local models attached to \textit{all} connected reductive groups over \(p\)-adic local fields with \(p\geqslant 5\). In addition, we give a self-contained study of relative affine Grassmannians and loop groups formed using general relative effective Cartier divisors in a relative curve over an arbitrary Noetherian affine scheme. Shimura varieties; local models; nearby cycles Modular and Shimura varieties, Arithmetic aspects of modular and Shimura varieties, Zeta functions and related questions in algebraic geometry (e.g., Birch-Swinnerton-Dyer conjecture) The test function conjecture for local models of Weil-restricted groups | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians In [J. Am. Math. Soc. 5, No. 2, 373--444 (1992; Zbl 0796.14014)] \textit{R. E. Kottwitz} studied the cohomology of Shimura varieties with good reduction, using the fact that in this case the cohomology on the special and generic fibre agree, and in the simplest case the computation reduces to count the number of \(\mathbb{F}_{p^r}\) rational points.
This article (which generalizes results from [\textit{P. Scholze}, Invent. Math. 192, No. 3, 663--715 (2013; Zbl 1305.22025)]) is concerned with the case of bad reduction. Some of the arguments of Kottwitz pass without change for a suitable integral model with no level at \(p\) of the Shimura variety and give a description of the \(\mathbb{F}_{p^r}\)-rational points in the special fibre. Also for a proper model the cohomology of the generic fibre is related to the cohomology of the special fibre with coefficients in the nearby cycle sheaves which in turn depend on the formal completion of the Shimura variety at the given point and finally on the \(p\)-divisible group at the given point. Next, the case nontrivial level is reduced to the case of no level at \(p\). The results can be interpreted as relating the cohomology of Shimura varieties to the cohomology spaces of \(p\)-divisible groups. A formula of the trace is given, which involves certain weight factors in terms of the nearby cycle sheaves. Shimura Varieties; \(p\)-divisible groups; deformation theory; Langlands programme Scholze, P., The Langlands-Kottwitz method and deformation spaces of \textit{p}-divisible groups, J. Amer. Math. Soc., 26, 1, 227-259, (2013) Arithmetic aspects of modular and Shimura varieties, Rigid analytic geometry, Formal groups, \(p\)-divisible groups, Modular and Shimura varieties, Local deformation theory, Artin approximation, etc. The Langlands-Kottwitz method and deformation spaces of \(p\)-divisible groups | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians Let \(S\) be a locally noetherian scheme and consider two extensions \(G_1\) and \(G_2\) of abelian \(S\)-schemes by \(S\)-tori. In this note we prove that the \textit{fppf}-sheaf \(\mathbf{Corr}_S(G_1,G_2)\) of divisorial correspondences between \(G_1\) and \(G_2\) is representable. Moreover, using divisorial correspondences, we show that line bundles on an extension \(G\) of an abelian scheme by a torus define group homomorphisms between \(G\) and \(\mathbf{Pic}_{G/S}\). group schemes; divisorial correspondences Group schemes, Generalizations (algebraic spaces, stacks), Brauer groups of schemes, Separable algebras (e.g., quaternion algebras, Azumaya algebras, etc.) A note on divisorial correspondences of extensions of abelian schemes by tori | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians Let \(K\) be a number field, \(S\) a finite set of primes (i.e. discrete valuations) and \({\mathfrak O}_ S\) the ring of \(S\)-integers in \(K\). The authors consider complete, smooth irreducible curves of genus 2 over \(K\). Such a curve \(C\) has exactly six Weierstrass points, and these are precisely the ramification points of the associated canonical double covering \(C \to \mathbb{P}^ 1_ K\). A curve is said to have good reduction at a prime \(v \notin S\) if there exists a relatively proper, smooth scheme \({\mathcal C}\) of dimension one over \(\text{Spec} ({\mathfrak O}_ v)\) such that \({\mathcal C} \otimes_{{\mathfrak O}_ v} K \cong C\). It can be shown that if \(C\) has good reduction at \(v\) then so does its Jacobian variety \(J_ C\). We regard two curves as equivalent if their corresponding Jacobian varieties are isogenous. This may in principle be verified by computing the traces of Frobenius at all primes yielding good reduction for the two Jacobian varieties and checking agreement up to an effective bound. In general, though, this bound is too large for practical computation. A courser equivalence relation is introduced in this paper: two curves of genus 2 are \(N\)-equivalent if, for all rational prime \(p\) with \(2 \leq p \leq N\) and \(p \notin S\), the traces of Frobenius at \(p\) for the Jacobians of the two curves agree. In this paper the authors choose the convenient but somewhat arbitrary bound of 89. The main result of the paper is that there are ninety-five 89-equivalence classes of curves of genus 2 defined over \(\mathbb{Q}\) having a rational Weierstrass point and good reduction away from 2. A consequence is that there are at least 95 isogeny classes of Jacobian varieties of such curves. isogenous Jacobian varieties; equivalence classes of curves; Weierstrass points; good reduction J.R. Merriman and N.P. Smart, Curves of genus 2 with good reduction away from 2 with a rational Weierstrass point, Math. Proc. Cambridge Philos. Soc., 114 (1993), 203-214. MR 94h:14031 Riemann surfaces; Weierstrass points; gap sequences, Special algebraic curves and curves of low genus, Jacobians, Prym varieties, Isogeny Curves of genus 2 with good reduction away from 2 with a rational Weierstrass point | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians The authors prove the following theorem: Let \(J\) be a general Jacobian variety of dimension \(g\geq 4\) defined over \({\mathbb{C}}\). If \(\chi: J'\to J\) is an isogeny and J' is a Jacobian then there is an isomorphism of principally polarized abelian varieties \(\mu: J\to J'\) and \(\chi\circ \mu\) is the multiplication by an integer.
The proof is based on a \((3g-3)\)-variables degeneration of \(J\) to a generalized Jacobian of an irreducible stable curve with two nodes, on the analysis of the extension classes coming from each node and on the comparison of the Gauss maps associated to the two surfaces of extension classes coming from the corresponding degenerations of \(J\) and of \(J'\) respectively.
If \(J\) is the Jacobian \(J(C)\) of a smooth curve \(C\), \(\Phi: C\to J\) is some fixed Abel Jacobi embedding, and \(\Phi _ n: C\to J\) is the map \(n\circ \Phi\) the theorem above has the following corollary: The irreducible curves of geometric genus \(g\) contained in a general Jacobian \(J(C)\) of dimension \(g\geq 4\) are the curves \(\Phi _ n(C)\) and all their translates. In particular they are all birationally equivalent. general Jacobian variety of dimension \(g\geq 4\); generalized Jacobian of an irreducible stable curve with two nodes; isogeny; geometric genus Bardelli, F., Pirola, G.P.: Curves of genusg lying on ag-dimensional Jacobian variety, Invent. Math.95, 263--276 (1989) Picard schemes, higher Jacobians, Jacobians, Prym varieties Curves of genus \(g\) lying on a \(g\)-dimensional Jacobian variety | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians For an algebraic variety \(X\) over a field \(k\) the étale fundamental group \(\pi_1(X)\) is an extension of the Galois-group \(\text{Gal}(k)\) by the geometric fundamental group \(\pi_1(\overline X)\). Rational points in \(X(k)\) induce \(\pi_1(\overline X)\)-conjugacy classes of sections of this extension. The section conjecture (due to Grothendieck) states that for suitable \(X\) and \(k\) this induces a bijection. ``Suitable'' of course implies that \(\pi_1(\overline X)\) and \(\text{Gal}(k)\) are sufficiently big, for example \(X\) could be a hyperbolic curve and \(k\) a number field. Except in cases where there are no sections the truth of conjecture is unknown.
In the present book the author gives an overview of mathematics around the conjecture. For example it is shown that many aspects of rational points generalise to sections. étale fundamental groups; rational points [6] J. Stix, \(Rational points and arithmetic of fundamental groups: Evidence for the section conjecture\), Springer Lecture Notes in Mathematics, 2054, Springer, Heidelberg, (2013). &MR 29 | &Zbl 1272. Research exposition (monographs, survey articles) pertaining to algebraic geometry, Homotopy theory and fundamental groups in algebraic geometry, Rational points Rational points and arithmetic of fundamental groups. Evidence for the section conjecture | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians Let \(J=J_0(N)\) denote the Jacobian of the modular curve \(X_0(N)\) of Hecke type, where \(N\) is a rational prime. \textit{B. Mazur} in groundbreaking work [Publ. Math., Inst. Hautes Étud. Sci. 47, 33--186 (1977; Zbl 0394.14008)] figured out a quotient \(\widetilde{J}\) of \(J\) (the Eisenstein quotient) with remarkable properties. Investigation of \(\widetilde{J}\) finally led him to his famous theorem on the uniform boundedness of \(E(\mathbb Q)_{\text{tor}}\) for elliptic curves \(E\) over the rationals.
Already in the above paper, Mazur proposed to consider the analogous quotients in the context of Drinfel'd's theory, where \(K = \mathbb Q\) has to be replaced by a global function field \(K\) (e.g., \(K=\mathbb{F}_q(T)\)), \(X_0(N)\) by some Drinfel'd modular curve \(X_0({\mathfrak n})\) with a divisor \({\mathfrak n}\) of \(K\), etc.
An important step in this program is made in the present article, where, among others things, the analogue of \(\widetilde{J}\) is defined and properties similar to those of \(\widetilde{J}\) are proved (although the results are weaker than Mazur's, due to problems specific to positive characteristic).
In detail: Let \(K=\mathbb F_q(T)\) with its ring \(A=\mathbb F_q[ T ]\) of integers, and let \({\mathfrak n}\) be a nonzero ideal in \(A\). Then the Drinfel'd modular curve of Hecke type \(X_0({\mathfrak n})\) and its Jacobian \(J_0({\mathfrak n})\) are defined and have properties similar to their classical counterparts [\textit{E.-U. Gekeler}, Compos. Math. 57, 219--236 (1986; Zbl 0599.14032)]. If \({\mathfrak n}\) is prime (which we suppose from now on), \(J=J_0({\mathfrak n})\) agrees with its new part. Define \(T\) to be the \(\mathbb Z\)-algebra generated by the Hecke operators \(T_{{\mathfrak p}}\) (\({\mathfrak n} \not= {\mathfrak p}\) prime) and the canonical involution \(w_{{\mathfrak n}}\). The splitting of the semisimple commutative ring \(T \otimes \mathbb Q\) into fields corresponds to the splitting (up to isogeny) of \(J\) into \(K\)-irreducible abelian varieties. (In fact, in the appendix it is shown, using ideas of Ribet, that \(T\otimes \mathbb Q = \text{End}_{\overline{K}}(J) \otimes \mathbb Q\).) The cuspidal divisor class group \(C\) of \(J\) is a finite cyclic group of order prime to \(q\), and the Eisenstein ideal \(I\) is defined as its annihilator in \(T\).
The Eisenstein quotient is then \(\widetilde{J} = J/{\mathfrak b}_I J\), where \({\mathfrak b}_I = \bigcap_{r \in \mathbb{N}} I^r\). Similarly, Eisenstein prime numbers \(l\) in \(\mathbb{N}\) (the divisors of \(\#C\)), Eisenstein prime ideals in \(T\) and the corresponding quotients \(\widetilde{J}^{(l)}\) of \(\widetilde{J}\) are defined. It is easy to see that \(l=2\) is an Eisenstein prime if and only if \(q\) is odd and \(d := \deg {\mathfrak n} \equiv 0\bmod 4\). If this is not the case then (Proposition 4.12) \(\widetilde{J}= \widetilde{J}^-\) (i.e., \(w_{{\mathfrak n}}\) acts as \(-1\) on \(\widetilde{J}\) as in the classical situation). Most likely this assertion remains true in the general case, but it cannot be proved with the present methods, due to the lack (?) of a reasonable theory of automorphic forms mod \(p\) (\(p = \text{char}\,K\)). We should point out that it is much more difficult here to work out significant numerical examples than in the classical case.
The central result of the paper is Theorem 5.7, which states that (i) \(\widetilde{J}^-(K)\) and (ii) Ш\((\widetilde{J}^-/K) \otimes \mathbb{Z}[q^{-1}]\) are finite (Ш = Tate-Shafarevich group). By Proposition 4.14, \(\widetilde{J}^- \not= 0\) whenever \(J \not= 0\), so we can hope that \(\widetilde{J}^-\) will play a similar role in the arithmetic of \(X_0({\mathfrak n})\) as does \(\widetilde{J}\) in the number field situation.
The proof of Theorem 5.7 relies on the study of \(H^1_{\text{ét}}(S,{\mathcal B}[l^n])\), where \(S=\mathbb P^1/\mathbb F_q\), \({\mathcal B}\) is the global Néron model of \(B=J/{\mathfrak p}J\) (\({\mathfrak p} =\) minimal prime ideal of \(T\) contained in some \(l\)-Eisenstein prime ideal), \({\mathcal B}[l^n]= l^n\)-division points, and on work of \textit{P. Schneider} [Math. Ann. 260, 495--510 (1982; Zbl 0509.14022)]. The necessary prerequisites are given in the first three sections, where the author collects material which is more or less known but difficult to find elsewhere.
Besides its important result, the article is highly laudable for its lucidity of exposition and intelligent choice of notation. Tamagawa A., The Eisenstein quotient of the Jacobian variety of a Drinfel'd modular curve, Publ. Res. Inst. Math. Sci. 31 (1995), no. 2, 203-246. Drinfel'd modules; higher-dimensional motives, etc., \(L\)-functions of varieties over global fields; Birch-Swinnerton-Dyer conjecture, Arithmetic ground fields for abelian varieties The Eisenstein quotient of the Jacobian variety of a Drin'feld modular curve. | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians The main object of the paper under review is a smooth projective variety \(X\), defined over a number field, which admits a structure of fibration \(\pi: X\to C\) over a smooth projective curve (\(X\), \(C\) and the generic fibre of \(\pi\) are assumed geometrically integral). The author considers the conjecture raised by \textit{J.-L.~Colliot-Thélène} [Proc. Symp. Pure Math. 67, 1--12 (1999; Zbl 0981.14003)] which asserts that the Brauer-Manin obstruction to the Hasse principle for zero-cycles of degree 1 is the only one. The main result of the paper establishes this conjecture under the assumptions that all fibres of \(\pi\) are geometrically integral, the Tate-Shafarevich group of the Jacobian of \(C\) is finite and all fibres over some ``generalized Hilbertian'' subset of \(C\) satisfy the Hasse principle. Some parallel results are obtained for the Brauer--Manin obstruction to weak approximation. This is applied to the example of \textit{B.~Poonen} [Ann. Math. (2) 171, No. 3, 2157--2169 (2010; Zbl 1284.11096)] where there is no Brauer-Manin obstruction for points but, according to \textit{J.-L.~Colliot-Thélène} [Bull. Soc. Math. Fr. 138, No. 2, 249--257 (2010; Zbl 1205.11075)], there is a zero-cycle of degree 1. In particular, the author shows that there are many such cycles (in the sense of weak approximation).
The author mentions some parallel results obtained by O.~Wittenberg under different assumptions. Brauer-Manin obstruction; Hasse principle; weak approximation Liang, Y., Principe local-global pour LES zéro-cycles sur certaines fibrations au-dessus d'une courbe: I, Math. Ann., 353, 1377-1398, (2012) Global ground fields in algebraic geometry, Varieties over global fields, Arithmetic ground fields (finite, local, global) and families or fibrations The local-global principle for zero cycles on certain fibrations over a curve. I. | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians Let \(f\) be a newform of weight two and trivial character for \(\Gamma_ 0(N)\), with rational coefficients, then there is an elliptic curve over \(\mathbb Q\), quotient of the Jacobian variety of \(X_ 0(N)\), whose \(L\)-function is that of \(f\). Conversely, Weil has conjectured that every elliptic curve over \({\mathbb Q}\) is isogenous to such a curve. In the present paper it is investigated, mostly numerically the case of a complex quadratic field \(K\) as a base field instead of \(\mathbb Q\). If \(\mathfrak a\) is an ideal of \(K\) and \(\Gamma_ 0(\mathfrak a)\) the associated congruence subgroup of \(\mathrm{SL}(2,\mathfrak G_ K)\), then cusp forms on \(\Gamma_ 0({\mathfrak a})\) can be seen as dual to the space \(V(\mathfrak a)=H_ 1(\Gamma_ 0({\mathfrak a})\backslash X,\mathbb C)\) where \(X\) is the completion of the hyperbolic 3-space \(H_ 3\) by cusps; then \(V(\mathfrak a)\) can be computed via modular symbols, using a tessellation of \(H_ 3\) relative to \(\Gamma_ 0(\mathfrak a)\). In case \(K\) is euclidean, these tessellations are studied in \S2 and a generator and relation description is given; the action of Hecke operator is also expressed (\S3). This results in extensive tables (see also work of Grunewald, Elstrodt, Mennicke when \(\mathfrak a=\mathfrak p\) is a prime) giving much evidence for the existence of a correspondence between rational newforms in \(V({\mathfrak a})^+\) (where \(\pm\) means \(\pm 1\) eigenspace for \(\left( \begin{matrix} \varepsilon\\ 0\end{matrix} \begin{matrix} 0\\ 1\end{matrix} \right)\) where \(\varepsilon\) is a generator for roots of unity in \(K\)), and isogeny classes of elliptic curves defined over \(K\), but with no complex multiplication by \(K\). However, as remarked in a forthcoming paper of the author [``Abelian varieties with extra twists and cusp forms for imaginary quadratic fields''] rational newforms in \(V(\mathfrak a)^+\) not corresponding to elliptic curves can be obtained from two-dimensional abelian varieties over \(\mathbb Q\), not splitting over \(K\), and with extra twist by the quadratic character defining \(K\).
The final \S4 examines twisting by quadratic characters and its action on the decompositions \(V(\mathfrak a)=V(\mathfrak a)^+\otimes V(\mathfrak a)^-\). modular symbols; hyperbolic tessellations; elliptic curve; complex quadratic field; twisting by quadratic characters Cremona, [Cremona 84] J. E., Hyperbolic tessellations, modular symbols, and elliptic curves over complex quadratic fields, \textit{Compos. Math.}, 51, 3, 275-324, (1984) Arithmetic ground fields for abelian varieties, Quadratic extensions, Special values of automorphic \(L\)-series, periods of automorphic forms, cohomology, modular symbols, Global ground fields in algebraic geometry, Holomorphic modular forms of integral weight Hyperbolic tessellations, modular symbols, and elliptic curves over complex quadratic fields | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians This paper relates two different approaches to extending families of Jacobian varieties. If \(X_{0}\) is a smooth projective curve of genus \(g,\) then the associated Jacobian variety is a \(g-\)dimensional smooth projective variety \(J_{0}\) that can be described in two different ways: as the Albanese variety and as the Picard variety. If \(X_{U} \rightarrow U\) is a family of smooth, projective curves, then the Jacobians of the fibers fit together to form a family \(J_{U} \rightarrow U.\) Let \(U\) will be an open subset of a smooth curve \(B\) (or a Dedekind scheme). The author investigates extending \(J_{U}\) to a family over \(B.\)
Viewing the Jacobian as the Picard variety is to extend \(J_{U} \rightarrow U\) as family of moduli spaces of sheaves (the approach of Mayer and Mumford). One first extends \(X_{U} \rightarrow U\) to a family of curves \(X \rightarrow B\) and then extends \(J_{U}\) to a family \(\overline{J} \rightarrow B\) with the property that the fiber over a point \(b \in B\) is a moduli space of sheaves on \(X_{b}\) parametrizing certain line bundles, together with their degenerations. In this paper, the author showed that the line bundle locus \(J\) in \(\overline{J}\) is canonically isomorphic to the Néron model for some schemes \(\overline{J}.\) The main result is
Theorem 1. Fix a Dedekind scheme \(B.\) Let \(f: X \rightarrow B\) be a family of geometrically reduced curves with regular total space \(X\) and smooth generic fiber \(X_{\eta}.\) Let \(J \subset \overline{J}\) the locus of line bundles in one of the following moduli spaces:
the Esteves compactified Jacobian \(\overline{J}^{\sigma}_{{E}};\)
the Simpson compactified Jacobian \(\overline{J}^{0}_{{L}}\) associated to an \(f\)-ample line bundle \({L}\) such that slope semistability coincides with slope stability.
The \(J\) is the Néron model of its generic fiber. projective curve; Jacobian varieties; Albanese variety and Picard variety; Néron model Chiodo, A.: Néron models of \(Pic^0\) via \(Pic^0\). arXiv:1509.06483 (Preprint) Jacobians, Prym varieties, Group schemes, Fine and coarse moduli spaces, Families, moduli of curves (algebraic) Two ways to degenerate the Jacobian are the same | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians There have been several generalizations of the concept of a fundamental group for algebraic varieties. Grothendieck defined the étale fundamental group by considering finite étale covers of the variety. In the complex case, under suitable hypotheses, this turns out to be the profinite completion of the topological fundamental group. Later, Nori defined the fundamental group scheme of a scheme \(X\) over a perfect field \(k\) as the affine group scheme associated to the neutral Tannaka category of essentially finite vector bundles over \(X\). If \(k\) is algebraically closed, this has the étale fundamental group as its quotient, and is in fact equal to it if \(k\) has characteristic 0. In addition to these two fundamental groups, Simpson defined the universal complex pro-algebraic group as the inverse limit of the directed system of representations \(\rho: \pi_1(X,x) \to G\) for complex algebraic groups \(G\), such that the image of \(\rho\) is Zariski-dense in \(G\). This group can also be described as being associated to the neutral Tannaka category of semistable Higgs bundles with vanishing rational Chern classes. \textit{I. Biswas}, \textit{A. J. Parameswaran} and \textit{S. Subramanian} [Duke Math. J. 132, No. 1, 1--48 (2006; Zbl 1106.14032)] defined the S-fundamental group scheme of a smooth, projective curve \(X\) over an algebraically closed field as the affine group scheme associated to the neutral Tannaka category of strongly semistable vector bundles of degree zero over \(X\).
In the paper under review, the author seeks to study the S-fundamental group scheme for a general complete, connected, reduced \(k\)-scheme \(X\). This is defined as the affine group scheme associated to the neutral Tannaka category of strongly semistable vector bundles with vanishing Chern classes over \(X\), directly generalizing the definition given in Biswas. We note that, for a smooth and projective variety \(X\), this category can also be described as the category of numerically flat vector bundles. Here, a vector bundle \(E\) on \(X\) is called numerically flat if both \(E\) and \(E^*\) are nef. For \(G\) a connected reductive \(k\)-group and \(E_G\) a numerically flat principal \(G\)-bundle on \(X\), the monodromy group scheme of \(E_G\) is defined and it is shown that this is the smallest subgroup scheme of \(G\) to which \(E_G\) has a reduction of structure group. Next, the author discusses basic properties of the S-fundamental group scheme. Among these, we have that the S-fundamental group scheme of the projective space is trivial, and the S-fundamental group scheme is well-behaved under blow-ups and base change. The author also conjectures that the S-fundamental group scheme of a product of complete \(k\)-varieties is the product of the corresponding S-fundamental group schemes. (The author proves this result in a sequel to this paper.) After two vanishing theorems for the cohomology of strongly semistable sheaves with vanishing Chern classes, the author proves Lefschetz-type theorems for the S-fundamental group scheme, with similar statements for Nori's and étale fundamental groups deriving as a corollary. fundamental group; positive characteristic; numerically flat bundles; Lefschetz type theorems Langer, A., On the \(S\)-fundamental group scheme, Ann. Inst. Fourier, 61, 2077-2119, (2011) Homotopy theory and fundamental groups in algebraic geometry, Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20], Vector bundles on surfaces and higher-dimensional varieties, and their moduli, Group schemes On the S-fundamental group scheme | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians Let \(C\) be a reduced, irreducible projective curve with normalization \(\widetilde C\). The generalized Jacobian \(JC\) of \(C\) is an extension of \(J\widetilde C\) by an affine commutative group. \(J(C)\) is an open subset of the compactified Jacobian \(\overline J C\) of \(C\), the points of which correspond to isomorphism classes of rank one torsion free sheaves \(F\) of degree zero. In the paper under review the Euler number of \(\overline J C\) is computed in the case \(C\) is rational and admits only planar singularities. generalized Jacobian; compactified Jacobian; Euler number Fantechi, B.; Göttsche, L.; van Straten, D., Euler number of the compactified Jacobian and multiplicity of rational curves, J. algebraic geom., 8, 1, 115-133, (1999) Jacobians, Prym varieties, Topological properties in algebraic geometry Euler number of the compactified Jacobian and multiplicity of rational curves | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians The Jacobian \(J\) of a curve defined over a field \(K\) is a projective group variety also defined over \(K\). In general it is hard to give an explicit description of the Jacobian. In the case where the curves are hyperelliptic there is a relatively explicit description of the Jacobian as an abstract variety defined over \(K\). On the other hand, if \(K\) is a subfield of \(\mathbb{C}\), then \(J(\mathbb{C})\) carries the structure of an abelian torus \(\mathbb{C}^g/\Lambda\), and this torus can be embedded into the projective space by theta functions. For a hyperelliptic curve \(y^2= f(x)\) the equations defining this variety in projective space are given by the Frobenius identities for half-integer characteristic theta functions. This description can be very useful because it is so explicit, but in general it is not isomorphic to the Jacobian over \(K\) (only over \(\mathbb{C}\)) and we therefore lose any arithmetic information. We can now ask whether we might be able to modify this description in such a way that the resulting variety would be isomorphic to the Jacobian over a smaller field. This paper gives one answer to this question.
The main result is theorem 9, which gives a variety defined by theta functions which is isomorphic to the Jacobian of a hyperelliptic curve over the splitting field of \(f\). The description is a modification of the usual embedding of the complex points of the Jacobian into projective space using theta functions. It describes the Jacobian as an explicit projective variety and not just as an abstract variety. Jacobian; hyperelliptic curve; theta constant; Thomae's identity; abelian torus; half-integer characteristic theta functions van Wamelen, Paul, Equations for the Jacobian of a hyperelliptic curve, Trans. Amer. Math. Soc., 350, 8, 3083-3106, (1998) Jacobians, Prym varieties, Theta functions and curves; Schottky problem, Theta functions and abelian varieties Equations for the Jacobian of a hyperelliptic curve | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians This paper contains applications to abelian schemes of previous differential algebraic work by the author. The first application applies to an abelian scheme \(A\) of relative dimension \(g\) over the ring of integers \(R\) of a finite unramified extension \(K\) of the \(p\)-adic field (odd \(p\)). Let \(k\) denote the residue field of \(R\) and assume \(|k|=p^\nu\); let \(A_0\) be the closed fibre of \(A\), which is assumed ordinary, and let \(p\) be the characteristic polynomial of the \(\nu^{\text{th}}\) power of the Frobenius on \(A_0\). Let \(R_{ur}\) be the maximal unramified extension of \(R\) and let \(p^\infty A(R_{ur})\) be all the infinitely \(p\) divisible points of \(R\) in \(R_{ur}\). Then under an appropriate condition on the matrix of Serre-Tate parameters of \(A\) (which is automatically satisfied if \(g=1\) and also in the most degenerate and most generic cases for the parameter matrix) and the assumption that \(P\) has only simple roots, the author proves that all points of \(p^\infty A(R_{ur})\) are torsion.
The second application concerns abelian schemes \(A\) of relative dimension \(1\) (not necessarily with ordinary reduction) over an absolutely unramified DVR \(R\) with quotient field \(K\) and algebraically closed residue field \(k\) of odd prime characteristic \(p\). Let the automorphism \(\phi: R \to R\) lift the Frobenius. In previous work, the author defined the operator \(\delta: R \to R\) by \(\delta x=(\phi(x)-x^p)/p\) and \(p\) derivatives of \(x\) of order \(i\) to be \(\delta^i x\). A \(\delta\) character of order \(n\) is a group homomorphism \(\psi: A(R) \to R\) which is Zariski locally representable as a \(p\)-adic limit of \(R\) polynomials in the affine coordinates and their \(p\) derivatives of order \(\leq n\). Let \(\psi\) be the uniquely determined (up to \(K\) scalar multiple) \(\delta\) character of minimal order. Then the author proves that if the closed fibre \(A_0\) is supersingular then \(p^\infty A(R)= \text{Ker } \psi\). abelian scheme; \(p\)-adic field; Frobenius; derivatives; differential algebra Buium A.: Differential characters and characteristic polynomial of Frobenius. J. Reine Angew. Math. 485, 209--219 (1997) Arithmetic ground fields for abelian varieties, Modules of differentials, Local ground fields in algebraic geometry, Characteristic \(p\) methods (Frobenius endomorphism) and reduction to characteristic \(p\); tight closure Differential characters and characteristic polynomial of Frobenius | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians By a well-known lemma of Serre, if a vector bundle \(E\) of rank \(r\) on a variety \(X\) is spanned by global sections, then there is a subspace \(W\) of \(H^ 0(X,E)\) with \(\dim W\leq r+\dim X\) such that \(W\) spans \(E\). The author studies here the case \(\dim W=4\), \(r=2\), \(X\) an integral projective surface. Let \(i\) be the morphism from \(X\) to the Grassmannian \(G(2,4)\subset \mathbb{P}^ 5\) associating to \(x\) in \(X\) the quotient \(E_ x\) of \(W\), \(E_ x=\) fibre of \(E\) at \(x\). Assume that
(1) \(i\) is an embedding, \(i(X)\) is ordinary (i.e. reflexive with dual a hypersurface);
(2) there is \(s\in W\) with zero scheme \(s_ 0\) union of \(C_ 2(E)-2\) smooth points of \(X\) (with reduced structure) and a length 2 scheme supported by another smooth point of \(X\);
(3) all general \(s_ 0\) are reduced and projectively equivalent.
The main result in the first part of the paper says that under these assumptions \(C_ 2(E)\leq 28\).
The second part of the paper is devoted to plane curves \(C\) with all general tangent lines to \(C\) intersecting \(C\) in projectively equivalent subsets (under \(\Aut \mathbb{P}^ 1\)). It is shown that if \(\text{Card}(C\cap\ell)_{\text{red}})\geq 3\) for every line \(\ell\), then varying line \(\ell\) in \((\mathbb{P}^ 2)^*\) the projective equivalence classes of sets \(C\cap\ell\) vary in a two dimensional family. vector bundles; global sections; zero locus; Chern class Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20], Characteristic classes and numbers in differential topology, Vector bundles on surfaces and higher-dimensional varieties, and their moduli On vector bundles whose general sections have all projectively equivalent zero-loci | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians Let \(\mathcal{M}_g\) denote the moduli space of smooth curves of genus \(g\), and \(p:\mathcal{C}_g\rightarrow\mathcal{M}_g\) the universal curve. The tautological ring \(\mathcal{R}(\mathcal{M}_g)\) on the curve side is defined to be the \(\mathbb{Q}\)-subalgebra of the Chow ring \(\text{CH}(\mathcal{M}_g)\) (with \(\mathbb{Q}\)-coefficients) generated by \(\kappa_i=p_*(K^{i+1})\), with \(K\) the relative canonical divisor. On the Jacobian side, the tautological ring \(\mathcal{T}(J)\) for the Jacobian of a smooth pointed curve \((C,x_0)\) is defined to be the smallest \(\mathbb{Q}\)-subspace of \(\text{CH}(J)\) containing the curve class \([C]\) and stable under both ring structures as well as the multiplication by \(N\in \mathbb{Z}\).
The first main result of the present paper extends the Jacobian side to the relative setting. Denote by \(\mathcal{L}_{g,1}\) the universal Jacobian over \(\mathcal{M}_{g,1}\), the moduli of smooth pointed curves. The tautological ring \(\mathcal{T}(\mathcal{J}_{g,1})\) is defined similarly. Then the ring \(\mathcal{T}(\mathcal{J}_{g,1})\) is proved to have an explicit finite set of generators (with respect to the intersection product), and the action on \(\mathcal{T}(\mathcal{J}_{g,1})\) can also be described explicitly in terms of the generators.
The second main result connects the tautological rings on both sides. Let \(\mathcal{C}_{g,1}\) denote the universal curve over \(\mathcal{M}_{g,1}\), and \(\mathcal{C}_{g,1}^n\) (respectively, \(\mathcal{C}_{g,1}^{[n]}\)) its \(n\)-th power (respectively, symmetric power). Passing to the limit \(\mathcal{C}_{g,1}^{[\infty]}=\varinjlim\mathcal{C}_{g,1}^{[n]}\), one gets the tautological ring \(\mathcal{R}(\mathcal{C}_{g,1}^{[\infty]})\). Then the ring \(\mathcal{R}(\mathcal{C}_{g,1}^{[\infty]})\) is shown to be a polynomial algebra over \(\mathcal{T}(\mathcal{J}_{g,1})\). In particular, the ring \(\mathcal{R}(\mathcal{M}_{g,1})\) is a \(\mathbb{Q}\)-subalgebra of \(\mathcal{T}(\mathcal{J}_{g,1})\). algebraic curves; Jacobian varieties; Chow ring Yin, Q., \textit{cycles on curves and Jacobians: a tale of two tautological rings}, Algebraic Geom., 3, 179-210, (2016) Families, moduli of curves (algebraic), Jacobians, Prym varieties Cycles on curves and Jacobians: a tale of two tautological rings | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians The Narasimhan-Seshadri theorem gave a bijective correspondence between irreducible unitary representations of the (topological) fundamental group of a compact Riemann surface and stable vector bundles of degree \(0\) on the surface. Hitchin, Simpson and Corlette established a one to one correspondence between semisimple representations of the fundamental group of a complex Kähler manifold and Higgs bundles on the manifold.
In this interesting paper Faltings gives a \(p\)-adic analogue of these famous results. Using almost étale coverings and \(p\)-adic Hodge theory, he defines generalized representaions of the étale fundamental group of a curve over a \(p\)-adic field, these include usual representaions as a full subcategory. He establishes an equivalence between the category of generalized representations and the category of Higgs bundles on the curve. Under this equivalence, the Higgs bundles associated to usual representations are semistable of degree \(0\). It is not known if the converse is true, the converse is true for line bundles on curves over \(p\)-adic local fields. The equivalence is not canonical, it depends on certain choices, including a choice of an exponential function of the multiplicative group. Most of the constructions are in fact done and work in the more general set up of schemes with toroidal singularities (with some restrictions). Related interesting results are due to \textit{C. Deninger} and \textit{A. Werner} [Ann. Sci. Éc. Norm. Supér Sér. 38, No. 4, 553--597 (2005; Zbl 1087.14026); in: Number fields and function fields -- two parallel worlds. Progr. Math. 239, 101--131 (2005; Zbl 1100.11019)]. Higgs bundles; almost étale coverings; generalized representations G. Faltings, A \textit{p}-adic Simpson correspondence, Adv. Math. 198 (2005), 847-862. Vector bundles on curves and their moduli, Local ground fields in algebraic geometry, Algebraic moduli problems, moduli of vector bundles A \(p\)-adic Simpson correspondence | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians The present paper provides a point counting algorithm for some plane curves defined over a finite field, in fact the class of curves with affine equation \(C:y^r=f(x)\) (including therefore elliptic and hyperelliptic curves) over \(\mathbb{F}_q,\, q=p^n,\, p\nmid r\).
The proposed algorithm generalizes the algorithm of \textit{P. Gaudry} and \textit{N. Gürel} [Lect. Notes Comput. Sci. 2248, 480--494 (2001; Zbl 1064.11080)] for superelliptic curves (curves with \(\mathrm{gcd}(r, d=d^{o}f)=1)\). The author points out that ``our algorithm has essentially the same complexity as the Gaudry-Gürel algorithm'' (in the best case \(\tilde{O}(pn^3d^4r^3)\) elementary operations), but he enumerates three main simplifications and improvements.
Section 2 recalls the concept of cyclic cover of the projective line and Section 3 studies the action of the Frobenius automorphism on the first Monsky-Washnitzer cohomology group \(H^1_{MW}(C,\mathbb{Q}_q)\). Then Section 4 presents the point counting algorithm for \(C\) (Algorithm 1). Theorem 4.1 computes the Weil polynomial of \(C\)\, and Theorem 4.3 (whose proof is postponed to Section 6) gives the precision \(p\)-adic bounds needed to recover the Weil polynomial over \(\mathbb{Z}_q\). Then the paper details the steps of Algorithm 1 and Section 8 shows four numerical examples using Magma 2.18. algebraic geometry; number theory; point counting; cyclic cover; \(p\)-adic precision Rational points, Curves over finite and local fields, Finite ground fields in algebraic geometry, Analysis of algorithms and problem complexity, Zeta functions and related questions in algebraic geometry (e.g., Birch-Swinnerton-Dyer conjecture), Number-theoretic algorithms; complexity A point counting algorithm for cyclic covers of the projective line | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians The authors prove that the Hilbert functor \(\text{Hilb}_{X/S}\) of flat families of closed subschemes of a projective scheme \(X \to S\) over a locally noetherian base is representable by a scheme that is a disjoint union of locally projective schemes. The existence of the Hilbert scheme has been established by several others [\textit{A. Grothendieck}, ``Techniques de construction et théorèmes d'existence en géométrie algébique. IV: Les schéma de Hilbert'', Sém. Bourbaki 13 (1960/61), No. 221 (1961; Zbl 0236.14002); \textit{D. Mumford}, ``Lectures on curves on an algebraic surface.'' (1966; Zbl 0187.42701); \textit{E. Sernesi}, ``Topics on families of projective schemes'', Queen's Pap. Pure Appl. Math. 73 (1986); \textit{E. A. Strømme}, in: Parameter spaces: enumerative geometry, algebra and combinatorics. Proc. Banach Center conf., Warsaw, Poland, February 1994. Banach Cent. Publ. 36, 179--198 (1996; Zbl 0877.14002); \textit{A. B. Altman} and \textit{S. L. Kleiman}, Adv. Math. 35, 50--112 (1980; Zbl 0427.14015); \textit{M. Haiman} and \textit{B. Sturmfels}, J. Algebr. Geom. 13, No. 4, 725--769 (2004; Zbl 1072.14007)]).
The approach in the paper under review is the following. For a finitely generated \(\mathbb{N}\)-graded \(A\)-algebra \(R\), the authors show that the functor \(\text{HomHilb}_R\) of homogeneous ideals in \(R\) having \(A\)-flat quotients is representable by a scheme. To prove this representability result it suffices to consider the functor \(\text{HomHilb}^\varphi_R\) of flat and homogeneous quotients of the polynomial ring \(R=A[x_0, \ldots, x_m]\) having a fixed Samuel function \(\varphi\).
The Homogeneous Hilbert functor is a special case of the Multigraded Hilbert functor introduced by Haiman and Sturmfels [loc. cit.], and the approach given in the paper under review is comparable with the one of Haiman and Sturmfels. However the proof of the representability of the Homogeneous Hilbert functor is appealingly simple:
Let \(G(B^n)\) be the closed subscheme of the Grassmannian of the truncated polynomial ring \(B^n=A[x_0, \ldots, x_m]/(x_0, \ldots, x_m)^n\) that represents \(\text{HomHilb}^{\varphi}_{B^n}\). The inverse limit \(G(B^{n+1})\to G(B^n)\) is the functor \(\text{HomHilb}^{\varphi}_R\). Let \(C_n\subseteq G(B^n)\) denote the intersection of the schematic images of \(G(B^{n+i})\to G(B^n)\). It follows by noetherian induction that \(C_n\subseteq G(B^n)\) is a closed subscheme. There exists, furthermore, an integer \(r\) such that any ideal \(I\) corresponding to a point of \(\text{HomHilb}^{\varphi}_R\) is generated in degrees \(\leq r\), and consequently the maps \(C_{n+r+1}\to C_{n+r}\) are injective. By Zariski's Main Theorem one obtains that \(C_{r+n+1}=C_{r+n}\), which proves representability, and projectivity, of \(\text{HomHilb}^{\varphi}_{R}\). The authors also supply a second proof that uses the theory of Fitting ideals to realize the representing object as a locally closed subscheme inside a product of suitable Grassmannians.
Representability of the Hilbert functor is then obtained in the following way. A flat family \(\text{Proj}(R/I)\to \text{Spec}(A)\) with Hilbert polynomial \(p(n)\) on each fiber is equivalent with the existence of some integer \(r\) such that the \(A\)-module of degree \(n\) elements \((R/I)_n\) is locally free of rank \(p(n)\), for \(n \geq r\). In particular, by truncating the first \(r\) degrees and forming the ideal \(I'=\bigoplus_{n\geq r} I\) one gets that the homogeneous quotient \(R/I'\) is \(A\)-flat. Furthermore, as the integer \(r\) uniformly depends on the Hilbert polynomial only, there is a map \(\text{Hilb}_{\mathbb{P}}^m\to\text{HomHilb}_R\) which is a section of the natural map \(\text{HomHilb}_R \to \text{Hilb}_{\mathbb{P}}^m\) -- giving representability of the Hilbert functor. homogeneous ideals; representable functors; Hilbert scheme A. Álvarez, F. Sancho, P. Sancho, Homogeneous Hilbert scheme. \textit{Proc. Amer. Math. Soc}. \textbf{136} (2008), 781-790. MR2361849 Zbl 1131.14008 Parametrization (Chow and Hilbert schemes) Homogeneous Hilbert scheme | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians Let \(K\) be a number field and \(\bar{K}\) the algebraic closure of \(K\). Let \(\phi : \tilde{V}\rightarrow V\) be a finite étale covering of normal projective varieties, defined over \(K\). The Chevalley-Weil theorem asserts that there exists an integer \(T> 0\) such that for any \(P\in V(K)\) and \(\tilde{P} \in \tilde{V}(\bar{K})\) with \(\phi(\tilde{P}) = P\), the relative discriminant of \(K(\tilde{P})/K\) divides \(T\). This theorem is useful in diophantine analysis since it provides a reduction of a diophantine problem on the variety \(V\) to that on the variety \(\tilde{V}\) which could be easier. Furthermore, it is used implicitly in the proofs of some finiteness theorems of Mordell-Weil. In this paper, a quantitative version of this theorem in dimension \(1\) is presented which improves and generalizes previous results of the first author [Ph. D. Thesis, Beer Sheva (1993)] and of \textit{K. Draziotis} and \textit{D. Poulakis} [Rocky Mt. J. Math. 39, No. 1, 49--70 (2009; Zbl 1222.14063)] and [Houston J. Math. 38, No. 1, 29--39 (2012; Zbl 1239.14018)]. The method of the proof is different of that of Draziotis and Poulakis and goes back to the first author's thesis. Chevalley-Weil theorem; integral point; rational point; ramification; Puiseux series Strambi, Yu. B. M.; Surroca, A., Quantitative Chevalley-Weil theorem for curves, Monatshefte für Mathematik, 171, 1-32, (2013) Curves of arbitrary genus or genus \(\ne 1\) over global fields, Rational points Quantitative Chevalley-Weil theorem for curves | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians The authors exhibit a large subgroup of the Grothendieck-Teichmüller group which has the following property: it acts on the profinite mapping class groups \(\Gamma^m_{g,n}\) for all \(g,n,m\geq 0\) (recall that these groups, also called Teichmüller modular groups in an algebraic context, are isomorphic to fundamental groups of moduli spaces of curves with a given finite number of marked points and boundary components). It is emphasized the fact that this action, which extends the known action in the genus 0 case, respects certain fundamental homomorphisms between the mapping class groups, namely those naturally induced by inclusions of the associated subsurfaces.
A Teichmüller tower consists of a collection of mapping class groups linked by certain natural homomorphisms coming from corresponding homomorphisms between the associated moduli spaces. The geometric (outer) automorphism group of such a Teichmüller tower is the collection of tuples of automorphisms of each of the mapping class groups which commute, up to inner automorphism, with the homomorphisms of the tower. Until now, only Teichmüller towers consisting of mapping class groups in genus zero (or braid groups) have been studied.
Let \(GT^1\) be the subgroup of the Grothendieck-Teichmüller group having \(\lambda\)-component equal to 1. Let us define a subgroup \(\Lambda\) of \(GT^1\) by adding one additional defining relation to the definition of \(GT^1\), and show that \(\Lambda\) acts on the tower of profinite mapping class groups \(\Gamma^m_{g,n}\) for all \(g,n,m\geq 0\), respecting all the natural arrows \(\Gamma^{\prime m}_{\prime g,\prime n}\to\Gamma^m_{g,n}\) coming from cutting out a topological surface of genus \(\prime g\) with \(\prime n\) punctures and \(\prime m\) boundary components inside one of genus \(g\) with \(n\) punctures and \(m\) boundary components. The proof that these homomorphisms are respected is an easy consequence of a certain local inertia conjugation property of the action of \(\Lambda\). Teichmüller-Grothendieck groups; Teichmüller towers of mapping class groups; profinite mapping class groups; fundamental groups; moduli spaces; actions Hatcher, A.; Lochak, P.; Schneps, L., On the Teichmüller tower of mapping class groups, J. Reine Angew. Math., 521, (2000) Automorphism groups of groups, Fundamental groups and their automorphisms (group-theoretic aspects), Geometric group theory, Topological properties of groups of homeomorphisms or diffeomorphisms, Fundamental group, presentations, free differential calculus, Coverings of curves, fundamental group, Families, moduli of curves (algebraic) On the Teichmüller tower of mapping class groups | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians For any variety \(X\) over a perfect field the authors prove the existence of a natural isomorphism between the Grothendieck residue complex [\textit{R. Hartshorne}, ``Residues and duality'', Lect. Notes Math. 20 (1966; Zbl 0212.261)] and a residue complex described by \textit{A. Yekutieli} [``An explicit construction of the Grothendieck residue complex'', Astérisque 208 (1992; Zbl 0788.14011)]. As a result they get that the trace map \(\widetilde \vartheta_X\) defined by \textit{J. Lipman} [``Dualizing sheaves, differentials and residues on algebraic varieties'', Astérisque 117 (1984; Zbl 0562.14003)] and the trace \(\vartheta_X\) defined by \textit{A. Yekutieli} (loc. cit.) agree up to sign. -- Then formulas for residues of local cohomology classes of differential forms are written down explicitly. Thus, a clear relation between local cohomology residues and the Parshin residues [see \textit{A. Yekutieli} (loc. cit.)] is established. It should be remarked that for Cohen-Macaulay varieties similar results were obtained by \textit{R. Hübl} [Math. Ann. 300, No. 4, 605-628 (1994; Zbl 0814.14022)]. Grothendieck residue complex; trace map; local cohomology classes of differential forms; cohomology residues; Parshin residues Pramathanath Sastry and Amnon Yekutieli, On residue complexes, dualizing sheaves and local cohomology modules, Israel J. Math. 90 (1995), no. 1-3, 325 -- 348. Local cohomology and algebraic geometry, Residues for several complex variables, Complexes, Local cohomology and commutative rings, Étale and other Grothendieck topologies and (co)homologies, Differentials and other special sheaves; D-modules; Bernstein-Sato ideals and polynomials On residue complexes, dualizing sheaves and local cohomology modules | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians The present book grew out of a series of lectures delivered by the two authors at the Summer School 1995 of the Graduiertenkolleg ``Geometry and nonlinear analysis'' at Humboldt University in Berlin. While the original lectures were designed to discuss some of the recent results on the geometry of moduli spaces of (semi-)stable coherent sheaves on an algebraic surface, the text at hand is a considerably elaborated and extended version of the initial notes. The outcome of the authors' rewarding and admirable effort at completing their lecture notes is now a book that serves several purposes at the same time. On the one hand, and in regard of its first part, it provides a textbook-like introduction to the theory of (semi-)stable coherent sheaves over arbitrary algebraic varieties and to their moduli spaces. On the other hand, mainly in view of its second part, the text has the character of both a research monograph and a comprehensive survey on some very recent results on those moduli spaces of (semi-)stable sheaves over (special) algebraic surfaces.
In both aspects, this book is rather unique in the existing literature on the classification theory of sheaves and vector bundles. Namely, for the first time in this central current area of research in algebraic geometry, a successful attempt has been undertaken to develop both the general, conceptual and methodical framework and the present state of knowledge in one of the most important special cases in a systematic, detailed, nearly complete and didactically processed presentation.
The text is divided into two major parts. After a careful introduction, which provides several motivations for studying sheaves on algebraic surfaces, in particular with a view to their significance in the differential geometry of four-dimensional manifolds and in gauge field theory (e.g., via Donaldson polynomials and Seiberg-Witten invariants), part I is devoted to the general theory of semi-stable sheaves and their moduli spaces. Chapter 1 introduces the basic concept of semi-stability for coherent sheaves over algebraic varieties, in the sense of D. Gieseker as well as in the (original) version of Mumford-Takemoto, and the fundamental material on Harder-Narasimhan filtrations, Jordan-Hölder filtrations, \(S\)-equivalence for semi-stable sheaves, and boundedness conditions. Flat families of sheaves, Grothendieck's Quot-scheme, the deformation theory of flags of sheaves, and Maruyama's openness-of-stability theorem are discussed in chapter 2, while chapter 3 deals with the most general form of the so-called Grauert-Mülich theorem and its application in establishing the boundedness of the set of semi-stable sheaves.
Moduli spaces for semi-stable sheaves, in their local and global aspects, is the subject of chapter 4. The authors discuss in detail C. Simpson's more recent approach to the construction of these moduli spaces, together with the related general facts from geometric invariant theory, and sketch the original construction by D. Gieseker and M. Maruyama likewise in an appendix. Furthermore, deformation theory is used to analyze the local structure of these moduli spaces, including dimension bounds and estimates for the expected dimension in the case of an algebraic surface. In another appendix the authors give an outlook to their own research contributions, in that they briefly describe moduli for ``decorated sheaves'' [cf. \textit{D. Huybrechts} and \textit{M. Lehn}, Int. J. Math. No. 2, 297-324 (1995; Zbl 0865.14004)]. This topic, though not systematically treated in the text, has recently found spectacular applications in conformal quantum field theory (e.g., in M. Thaddeus's proof of the famous Verlinde formula) and in non-abelian Seiberg-Witten theory.
The second part of the book, starting with chapter 5, mainly focuses on moduli spaces of semi-stable sheaves on algebraic surfaces. At first, the authors present various construction methods for stable vector bundle on surfaces, including Serre's correspondence between rank-2 vector bundles and codimension-2 subschemes, Maruyama's method of elementary transformations, and some illustrating examples. The geometry of moduli spaces of semi-stable sheaves on K3 surfaces is thoroughly explained in chapter 6, where in particular some very recent results by S. Mukai, A. Beauville, L. Göttsche-D. Huybrechts, K. O'Grady, J. Li, G. Ellingsrud-M. Lehn, and others are systematically compiled. Chapter 7 deals with the restriction of sheaves on surfaces to curves, focusing on the related work of H. Flenner, F. Bogomolov, and V. Mehta-A. Ramanathan in the 1980's. In chapter 8, the authors turn the attention to line bundles on moduli spaces and their Picard groups. The construction of determinantal line bundles and ampleness results for special line bundles on moduli spaces are presented by essentially following the approaches of J. Le Potier (1989) and J. Li (1993). As an application, the authors provide a profound comparison between the (algebraic) Gieseker-Maruyama moduli spaces of semi-stable vector bundles and the (analytic) Donaldson-Uhlenbeck compactification of the moduli spaces of Mumford-stable bundles. Chapter 9 is almost entirely devoted to K. O'Grady's recent work on the irreducibility and generic smoothness of moduli spaces for vector bundles on projective surfaces [cf. \textit{K. O'Grady}, Invent. Math. 123, No, 1, 141-207 (1996; Zbl 0869.14005)] and the related results by \textit{D. Gieseker} and \textit{J. Lie} [J. Am. Math. Soc. 9, 107-151 (1996; Zbl 0864.14005)].
Chapter 10, entitled ``Symplectic structures'', turns to differential forms on moduli spaces of stable sheaves on surfaces. After a lucid survey of the technical background material such as Atiyah classes, trace maps, cup products, the Kodaira-Spencer map, etc., the authors describe the tangent bundle of the smooth part of a moduli space by means of the universal family of vector bundles. Then, via the explicit construction of closed differential forms on moduli spaces, Mukai's theorem on the existence of a non-degenerate symplectic structure on the moduli space of stable sheaves on a K3 surface is derived.
The concluding chapter 11 deals with the birational properties of moduli spaces of semi-stable sheaves on surfaces. The main result presented here is a simplified proof of \textit{J. Li}'s recent theorem [Invent. Math. 115, No. 1, 1-40 (1994; Zbl 0799.14015)] stating that moduli spaces of semi-stable sheaves on surfaces of general type are also of general type. Other results on the birational type of such moduli spaces are surveyed in a brief sub-section, and the treatise concludes with two instructive examples showing how the Serre correspondence can be used to obtain information about the birational structure of moduli spaces of sheaves on a K3 surface. Actually, both examples are variations on two recent theorems due to T. Nakashima (1993) and K. O'Grady (1995), respectively, and their discussion is based upon an elegant combination of the results from chapter 8 and 10 in the book.
Altogether, the present text fascinates by comprehensiveness, rigor, profundity, up-to-dateness and methodical mastery. The bibliography comprises 263 references, most of which are really referred to in the course of the text. Each chapter comes with its own specific introduction and, always at the end, with a list of extra comments, hints to the original literature, and remarks on related topics, further developments and current research problems. The authors have successfully tried to keep the presentation of this highly advanced material as self-contained as possible, so that the text should be accessible for readers with a solid background in algebraic geometry. Active researchers in the field will appreciate this book as a valuable source and reference for their work. vector bundles on projective surfaces; stable coherent sheaves; moduli spaces; gauge field theory; Donaldson polynomials; Seiberg-Witten invariants; Grauert-Mülich theorem; semi-stable sheaves; geometric invariant theory; conformal quantum field theory; Verlinde formula; Seiberg-Witten theory; Picard groups; determinantal line bundles; Gieseker-Maruyama moduli spaces; Donaldson-Uhlenbeck compactification; differential forms on moduli spaces of stable sheaves; birational properties Hu D.~Huybrechts and M.~Lehn. \newblock \em Geometry of moduli spaces of sheaves, Vol. E31 of \em Aspects in Mathematics. \newblock Vieweg, 1997. Algebraic moduli problems, moduli of vector bundles, Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20], Research exposition (monographs, survey articles) pertaining to algebraic geometry, Vector bundles on surfaces and higher-dimensional varieties, and their moduli The geometry of moduli spaces of sheaves | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians The author extends his joint work with Ponza [\textit{L.\ Gatto} and \textit{F.\ Ponza}, Trans. Am. Math. Soc. 351, 2233-2255 (1999; Zbl 0917.14016)] on ``taking derivatives of the relative Wronskian'' from flat families of smooth curves of genus \(g\) to certain flat families of stable curves of genus \(g\). This technique allows relatively simple calculation of certain classes in \(\text{Pic}(\overline{\mathcal M}_g)\otimes {\mathbb{Q}}\), the rational Picard group of the stack of genus-\(g\) stable curves. Most notable among these classes is the class \(wt(2)\), the closure of the locus of smooth curves with Weierstrass point of weight at least \(2\).
The author also gives a proof of a widely believed, but previously unpublished, formula expressing \(wt(2)\) as a sum of \(\overline{D_{g-1}}\), the closure of the loci of curves with a Weierstrass point whose first non-gap is \(g-1\), and \(\overline{E(1)}\), the closure of the locus of curves with a Weierstrass point of type \(g+1\). Using this formula and the computation of \textit{S.\ Diaz} [``Exceptional Weierstrass points and the divisor on moduli space that they define'', Mem. Am. Math. Soc. 327 (1985; Zbl 0581.14018)] for the class of \(\overline{D_{g-1}}\) in \(\text{Pic}(\overline{\mathcal M}_g)\otimes {\mathbb{Q}}\) the author gives a new proof of a formula of \textit{F.\ Cukierman} [Duke Math. J. 58, 317-346 (1989; Zbl 0687.14026)] for \(\overline{E(1)}\). derivatives of relative Wronskians; families of special Weierstrass points; moduli spaces of curves; divisor classes; families of smooth curves L. Gatto, On the closure in \(\overline{M}_{g}\) of smooth curves having a special Weierstrass point, Math. Scand., to appear. Families, moduli of curves (algebraic), Families, moduli of curves (analytic), Riemann surfaces; Weierstrass points; gap sequences, Curves in algebraic geometry On the closure in \(\overline M_g\) of smooth curves having a special Weierstrass point | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians Let \(f:X\to S\) be a morphism of smooth and connected schemes over an algebraically closed field \(k\). What can be said about the relation between the three fundamental group schemes in sight: \(\Pi(X)\), \(\Pi(S)\), and \(\Pi(\mathrm{Fibre})\)? This work addresses this question under the extra assumption that f is smooth, projective, and geometrically connected. Moreover, it wishes to reach a natural answer by employing a method of proof analogous to the one presented by its precursor, Exposé X of [\textit{A. Grothendieck} (ed.), Séminaire de géométrie algébrique du Bois Marie 1960--61. Revêtements étales et groupe fondamental (SGA 1). Un séminaire dirigé par Alexander Grothendieck. Augmenté de deux exposés de M. Raynaud. Édition recomposée et annotée du original publié en 1971 par Springer. Paris: Société Mathématique de France (2003; Zbl 1039.14001)]. Our approach is then based on a new Tannakian criterion for studying exact sequences, and on infinitesimal equivalence relations. The latter objects are inspired by Ehresmann's image of connections on general fibre spaces. The proofs make no distinction between the cases \(\mathrm{char}(k)=0\) and \(\mathrm{char}(k)>0\). J.~P. Dos~Santos, \emph{The homotopy exact sequence for the fundamental group scheme and infinitesimal equivalence relations}, Algebraic Geometry \textbf{2} (2015), no.~5, 535--590. DOI 10.14231/AG-2015-024; zbl 1336.14017; MR3421782 Homotopy theory and fundamental groups in algebraic geometry, Group schemes The homotopy exact sequence for the fundamental group scheme and infinitesimal equivalence relations | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians The following theorem is proved: Assume that \(\mathrm{Lie }G_k\) is of dimension \(\leq 1\) and that \(Y_k\) does not arise as the push-forward of a torsor over \(X_k\) under a proper subgroup scheme of \(G_k\). Then, there exist a smooth formal curve \(X\) over \(R\) and a \(G\)-torsor \(Y \to X\) whose special fiber is the \(G_k\)-torsor \(Y_k \to X_k\).
Here \(R\) is a complete local ring with residue field \(k\) of positive characteristic \(p >0\), \(G\) is a finite, flat and of finite presentation, commutative group scheme over \(R\) and \(X_k\) is a smooth curve over \(k\).
This article extends some earlier works of the authors [J. Algebra 318, No. 2, 1057--1067 (2007; Zbl 1135.14036)].
In the acknowledgements, the authors indicate that they `would like to thank M. Raynaud who suggested the problem to us and for his encouragement.'
The interesting feature of the article is the use of the equivariant cotangent complex by \textit{L. Illusie} [Complexe cotangent et déformations. II. Berlin-Heidelberg-New York: Springer-Verlag (1972; Zbl 0238.13017)], results on algebraic spaces and schemes by \textit{M. Raynaud} and \textit{L. Gruson} [Invent. Math. 13, 1--89 (1971; Zbl 0227.14010)], the analogue of the stack by \textit{D. Abramovich} et al. [J. Algebr. Geom. 20, No. 3, 399--477 (2011; Zbl 1225.14020); corrigendum ibid. 24, No. 2, 399--400 (2015)] and moduli of Galois \(p\)-covers by \textit{D. Abramovich} and \textit{M. Romagny} [Algebra Number Theory 6, No. 4, 757--780 (2012; Zbl 1271.14032)].
In the last section of the paper under review, the authors show that these techniques can also be applied to the theory moduli of \(p\)-covering of curves. Several interesting examples are given, in particular on relation with Jacobians. lifting of torsors; finite flat group scheme; algebraic curve; cotangent complex Local ground fields in algebraic geometry, Curves over finite and local fields, Coverings of curves, fundamental group, (Equivariant) Chow groups and rings; motives Deformation of torsors under monogenic group schemes | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians According to the author's abstract, this paper is intended to be a survey on tangent bundles \(T_{X/Y}\) of (relative) schemes \(f:X\to Y\) drawing on previous results of algebraic nature about symmetric algebras as applied to the case of the module of relative differentials \(\Omega_{X/Y}\). As it turns out, the paper serves an interesting additional purpose by conveying a neat introduction to (relative) vector fields of schemes. After these preliminaries, the author illustrates in the context of tangent bundles of schemes the formulas for the dimension of a symmetric algebra. Pushing further these connections, he deduces other formulas for the fibres of the structural projection of the bundle and for the fibres of \(f\). As might be expected, in this generality, the formulas regarding \(\Omega_{X/Y}\) turn out to be a bit more complicated, involving the inseparability degree and the regularity defect of a local morphism \(R\to S\) of local rings which is essentially of finite type.
A special section is devoted to the case in which \(Y=\text{Spec}(K)\), where \(K\) is a field of arbitrary characteristic. Here the author is at his best by developing a notion of admissible fields that is suited to safely take the place of perfect fields. He then derives results on equidimensionality and criteria of regularity known to hold when the base field is perfect.
The author also devotes a section to obtaining a condition, back in the general relative case (with sufficient conditions on the scheme \(X)\), for equidimensionality to trigger irreducibility for the relative tangent bundle. This question had been essentially distilled by other authors before in the case of an abstract module, but here the author brings into the picture a different inequality by unveiling the role of the inseparable degree and the (local) regularity defect. He applies this in the case in which \(X\) is a one-dimensional scheme over a field \(K\), showing that the torsionfreeness of the symmetric algebra of \(\Omega_{X/K_0}\) \((K_0\) standing for an admissible subfield of \(K)\) triggers the regularity of \(X\) -- this is still a few yards from proving the Berger conjecture, which requires that only \(\Omega_{X/K_0}\) itself be torsionfree.
The last part of the paper deals with local (relative) complete intersections \(X\to Y\) and a criterion for the tangent bundle \(T_{X/Y}\) to be a local complete intersection over \(Y\). module of relative differentials; vector fields of schemes; tangent bundles of schemes; Berger conjecture Modules of differentials, Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20], Special types (Cohen-Macaulay, Gorenstein, Buchsbaum, etc.) On the tangent bundle of a scheme | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians This voluminous book is the long-awaited second part of the authors' comprehensive monograph on the geometry of algebraic curves. The first volume [Geometry of algebraic curves. Volume I. Grundlehren der mathematischen Wissenschaften, 267. New York etc.: Springer-Verlag (1985; Zbl 0559.14017)] was published more than 25 years ago, at that time with the main goal to provide the first unified and systematic presentation of the basic old and new ideas, methods, and results in the theory of algebraic curves. In particular, the central topic of the first volume was the study of linear series on a fixed curve, together with the then very recent developments in the theory of special divisors, varieties of special linear series on a curve, and Brill-Noether theory.
Also, the authors announced the second volume as a forthcoming book, which was to contain an exposition of the fundamentals of deformation theory and of the main properties of the moduli spaces of curves, some of the further results of Brill-Noether theory, a presentation of the basic properties of the varieties of special linear series on a moving curve, and a proof of the (then new) theorem that the moduli space of curves of sufficiently high genus is of general type.
However, since the 1980s, the study of the moduli spaces of curves has undergone a rapid, nearly explosive development and still continues to do so, especially in view of the increasingly growing interrelation with mathematical physics. As the authors point out, the subject has developed in ways that are incredibly richer than could have been imagined at the time of writing Volume I in 1985. Taking these developments into account and trying to keep up with them, the authors have radically changed the disposition of the second volume, apparently revised the contents continuously, and finally presented the somewhat different, now realy existing second volume of their monograph ``Geometry of Algebraic Curves''.
The main purpose of this book is to provide comprehensive and detailed foundations for the theory of moduli of complex algebraic curves, and that from multiple perspectives and various points of view. In fact, this book treats for the first time the different aspects of moduli theory in a coherent manner, thereby instructively combining algebro-geometric, complex-analytic, topological, and combinatorial methods. The originally envisioned centerpiece of the second volume, namely the further study of linear series on a general or variable curve, culminating in a proof of the Petri conjecture, is still an important part of the book, but it is not the central aspect anymore.
As for the contents of the current text, recall that the first volume contained the first eight chapters, while the present book comprises the following thirteen chapters, starting with Chapter IX and ending with Chapter XXI.
Chapter IX provides a self-contained introduction to the Hilbert scheme, thereby stressing the significance of the concept of flatness and the special case of curves. Chapter X presents the fundamental results for the construction of the moduli space \(M_{g,n}\) of stable \(n\)-pointed curves of genus \(g\), including the so-called stable reduction theorem and such basic constructions as lutching, projection, and stabilization. The central objects of study in this chapter are nodal curves. Chapter XI deals with the deformation theory of (stable) nodal curves, including the Kodaira-Spencer approach, Kuranishi families, the Hilbert scheme of \(n\)-canonical curves, the period map and the local Torelli theorem, Hodge bundles and their curvature, deformations of symmetric products, and other related topics. Chapter XII is then devoted to the construction of the moduli space \(\overline M_{g,n}\) in different ways.
This space is first exhibited as an analytic space, then as an algebraic space, and finally as an orbifold and as a Deligne-Mumford stack. Along the way, the first properties of the boundary strata of these moduli spaces are studied, and an essentially self-contained introduction to the theory of stacks is given as well.
Chapter XIII discusses the theory of line bundles on moduli stacks of stable curves. The reader gets acquainted with the necessary theory of descent, the Hodge bundle, the tangent bundle to the stack, the canonical bundle, and the normal bundles to the various boundary strata. Furthermore, the determinant of cohomology, the Deligne pairing, Mumford's class, and various notions of Picard groups of moduli stacks are analyzed in great detail.
Chapter XIV gives a proof of the projectivity of the moduli space of stable curves. The authors' proof uses an interesting combination of two techniques, namely geometric invariant theory and the Hilbert-Mumford criterion of stability, on the one hand, and certain numerical inequalities among cycles in moduli spaces and positivity results, on the other hand. Chapter XV provides a self-contained introduction to the Teichmüller point of view in the theory of complex curves. Actually, Teichmüller theory is needed in the authors' subsequent discussion of smooth Galois covers of moduli spaces, and therefore this chapter explains the Teichmüller space and the mapping class group, quadratic differentials and Teichmüller deformations, the proof of Teichmüller's uniqueness theorem, the boundary of Teichmüller space, and the simple connectedness of the moduli stack of stable curves. These techniques are used in Chapter XVI, where smooth Galois covers of moduli spaces are described. More precisely, the authors construct moduli spaces of stable pointed curves as quotients of smooth varieties by finite groups, thereby presenting a variation of E. Looijenga's construction due to Abramovich, Corti, and Vistoli. The central subject of Chapter XVII is the theory of cycles in the moduli space \(\overline M_{g,n}\), including the necessaryintersection theory of certain stacks, tautological classes on moduli spaces of curves, tautological relations and the tautological ring, Mumford's relations for the Hodge classes, and a description of the Chow ring of the moduli space \(\overline M_{0,P}\).
Chapter XVIII turns to the combinatorial study of the moduli space \(M_{g,n}\). To this end, the authors introduce a number of simplicial complexes associated to a pointed oriented surface, and these complexes are used to define certain cell decompositions of the Teichmüller space and of its bordification. These decompositions descend to orbicell-decompositions of the moduli space \(M_{g,P}\) and of suitable compactifications of it.
The basic tools for the combinatorial description of the associated Teichmüller spaces are certain graphs, the so-called ribbon graphs, Jenkins-Strebel differentials, uniformization theory, and the hyperbolic geometry of Riemann surfaces.
Chapter XIX discusses first consequences of the cellular decomposition of moduli spaces as studied in the foregoing chapter. The authors use the cellular decomposition to compute the rational cohomology of the space \(\overline M_{g,n}\) in degrees one and two, touch upon Harer's stability theorem, the Madsen-Weiss theorem and the Tillmann theorem on the stable rational cohomology of \(M_{g,n}\) and give then a proof of Harer's theorem on the second homology of \(M_{g,n}\) by applying Deligne's spectral sequence for the complement of a divisor with normal crossings. At the end of this chapter, further uses of the cellular decomposition are presented by deriving some combinatorial formulae of M. Kontsevich in this context.
Chapter XX presents a nearly self-contained version of Kontsevich's proof of Witten's conjecture on the intersection numbers of the cohomological (so-called) \(\psi\)-classes.This chapter is titled ``Intersection Theory of Tautological Classes'' and explains Witten's generating series, Virasoro operators and the KdV hierarchy, Feynman diagrams and matrix models, equivariant cohomology and the virtual Euler-Poincaré characteristic of \(M_{g,n}\), Gromov-Witten invariants, and other related material.
Chapter XXI returns to one of the central themes of the first volume: Brill-Noether theory. Here the authors study the Brill-Noether theory for smooth curves ``moving with modul''. As the authors point out, this chapter is based on a draft version from the 1980s, when it was planned as a major part of the second volume. Thus it is meant here to give a snapshot of what Brill-Noether theory looked some twenty-five ago, both in content and style, and does not completely reflect the present state of the theory. Among the topics included in this final chapter are the following: the relative Picard variety, Brill-Noether varieties on moving curves, Looijenga's vanishing theorem for the tautological ring of \(M_g\), Lazarsfeld's proof of Petri's conjecture, Horikawa theory, the Hurwitz scheme and its irreducibility, the unirationality of \(M_g\) for genus \(g\leq 10\) and other meanwhile classical results.
Each chapter comes with two extra sections at the end. One of them provides bibliographical notes and hints for further reading concerning the respective chapter, and the other gives a wealth of related exercises, many of which were contributed by J. Harris.
As in the first volume, the exercises are arranged in well-structured series, most of which usually cover an additional topic in the context of the respective section.
The bibliography at the end of the book is extremely rich and very up-to-date. There are 695 references listed in this bibliography, which must be seen as a highly valuable service to the reader, too.
Altogether, the present second volume of ``Geometry of Algebraic Curves'' has appeared with some delay, but it was worth waiting for it. The authors have presented a wealth of topical material from the moduli theory of curves, and that in a unique blend of different aspects, old and new viewpoints, relations to mathematical physics, and in great versatility. The presentation is utmost lucid, detailed, inspiring and instructive, written in a very motivating and user-friendly style.
The current book is an excellent research monograph and reference book in the theory of complex algebraic curves and their moduli, which is very likely to become an indispensable source for researchers and graduate students in both complex geometry and mathematical physics. monograph (algebraic geometry); algebraic curves; stable curves; moduli spaces; stacks; Hilbert schemes; Brill-Noether theory; intersection theory; Teichmüller theory E.~Arbarello, M.~Cornalba, P.A.~Griffiths: {\em Geometry of Algebraic Curves}, Vol. II, Grundlehren der math. Wiss., 268, Berlin: Springer (2011). DOI 10.1007/978-3-540-69392-5; zbl 1235.14002; MR2807457 Research exposition (monographs, survey articles) pertaining to algebraic geometry, Families, moduli of curves (algebraic), Families, moduli of curves (analytic), Theta functions and curves; Schottky problem, Teichmüller theory for Riemann surfaces, Complex-analytic moduli problems, Special divisors on curves (gonality, Brill-Noether theory), Generalizations (algebraic spaces, stacks), Relationships between algebraic curves and physics Geometry of algebraic curves. Volume II. With a contribution by Joseph Daniel Harris | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians The main object of the paper under review is an \(X\)-torsor \(Y\) under \(G\), where \(X\) is a smooth variety over a local field \(K\) (the field of fractions of a Henselian discrete valuation ring \(R\)) and \(G\) is an algebraic \(K\)-torus. The residue field \(k\) of \(K\) is assumed perfect. The author's goal is to show that, under certain conditions, the evaluation map \(X(K)\to H^1(K,G)\), which associates to each point \(P\) the isomorphism class of the fibre \(Y_P\), factors through reduction to the special fibre, i.e., for a suitable \(R\)-model \(\mathcal X\) of \(X\), the evaluation map comes from \(\mathcal X_s(k)\to H^1(k,\mathcal G_s)\), where \(\mathcal G\) is the Néron--Raynaud model of \(G\) and the subscript \(s\) refers to the special fibres of \(\mathcal X\) and \(\mathcal G\). This is proved when the torus \(G\) is split by a tamely ramified extension of \(K\). The author presents an example showing that one should not expect such a result in wildly ramified cases. Some intermediate results on extending torsors are formulated for more general algebraic groups \(G\) and are interesting by their own right. Finally, the author discusses some applications to the case where \(X\) is a rational surface. torsor; Néron model Local ground fields in algebraic geometry, Rational points, Étale and other Grothendieck topologies and (co)homologies, Varieties over finite and local fields Torsors under tori and Néron models | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians Let \(X\) be a projective variety defined over a finite field, and let \(K\) denote the function field of \(X\). Assuming \(X\) to be non-singular in codimension one, the author introduces and studies the power series \(Z(t)= \sum_{d=0}^ \infty A_ d(n) t^ d\), where \(A_ d(n)\) stands for the number of points in \(\mathbb{P}^ n (K)\) whose Neron height is equal to \(d\). He proves that \(Z(t)\) is a rational function if \(X\) is a curve, and that \(Z(t)\) is \(p\)-adically meromorphic if \(X\) is a smooth complete intersection of dimension \(\geq 3\). In general, the author conjectures that if the monoid of effective divisor classes of \(X\) is finitely generated, then \(Z(t)\) is a \(p\)-adic meromorphic function. As an application of his results and technique, the author improves the error term in the Serre asymptotic formula for \(A_ d(n)\) when \(X\) is a curve [cf. \textit{J.-P. Serre}, Lectures on the Mordell-Weil theorem (Vieweg, 1989; Zbl 0676.14005), p. 19]. varieties over finite fields; projective variety; power series; Neron height; \(p\)-adic meromorphic function Da Qing Wan, Heights and zeta functions in function fields, The arithmetic of function fields (Columbus, OH, 1991) Ohio State Univ. Math. Res. Inst. Publ., vol. 2, de Gruyter, Berlin, 1992, pp. 455-463. Varieties over finite and local fields, Curves over finite and local fields, Finite ground fields in algebraic geometry, Other analytic theory (analogues of beta and gamma functions, \(p\)-adic integration, etc.) Heights and zeta functions in function fields | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians Let \(X_k\) be a proper, smooth and geomerically connected curve over a global field \(k\), and let \(A\) be the Jacobian variety of \(X_k\). Let \(X\) be a 2-dimensional proper, regular model of \(X_k\). After the work of A. Grothendieck, J. Milne gave a connection between the Tate-Shafarevich group of \(A\) and the Brauer group of \(X\) under the assumption that the index \(\delta_v\) of \(X_{k_v}\) equals 1 for all prime \(v\) of \(\pi\). In this paper, the author generalizes Milne's formula when the \(\delta_v\)'s are no longer equal to 1 and thereby answers partially a question posed by Grothendieck. For the proof, the author generalizes Milne's methods and employs the Albanese-Picard pairing of Poonen and Stoll instead of the Cassels-Tate pairing used by Milne. In an appendix, the compatibility of the Cassels-Tate pairing with the Albanese-Picard pairing of Poonen and Stoll is verified. Brauer groups; Tate-Shafarevich groups; Jacobian variety; index and period of a curve; Cassels-Tate pairing González-Avilés C.: Brauer groups and Tate-Shafarevich groups. J. Math. Sci. Univ. Tokyo 10, 391--419 (2003) Varieties over global fields, Arithmetic ground fields for abelian varieties, Global ground fields in algebraic geometry Brauer groups and Tate-Shafarevich groups | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians This article serves as a very readable introduction to several articles in these Proceedings and moreover, ``surveys a few of the highlights in the arithmetic of curves: the proof of the Mordell Conjecture, and the more detailed theory that has developed around the classes of curves most studied until now by number theorists: modular curves, Fermat curves, and elliptic curves''. We cite from the text:
``Section 1 recalls some preliminary results that are used heavily in later sections: the main finiteness results of algebraic number theory, and the method of descent based on unramified coverings and the Chevalley-Weil theorem. \textit{H. Chapdelaine}'s article [Clay Math. Proc. 8, 55--69 (2009; Zbl 1223.11035)] in these proceedings further develops these themes by describing a relatively elementary application of Faltings' theorem to a Diophantine equation -- the generalised Fermat equation \(x^p + y^q + z^r = 0\) -- that appears to fall somewhat beyond the scope of the study of algebraic curves, but to which, it turns out, the ``fundamental trichotomy'' described in the introduction can still be
applied.
The main goal of Section 2 is to give a survey of Faltings' proof of the Mordell Conjecture. In many ways, this section forms the heart of these notes. The ideas in Section 2 are used to motivate the startlingly diverse array of techniques that arise in the Diophantine study of curves. These techniques are deployed in subsequent sections to study several important and illustrative classes of algebraic curves -- specifically, modular curves, Fermat curves, and elliptic curves.
Section 3 focuses on what may appear at first glance to be a rather special collection of algebraic curves, the so-called modular curves over \(\mathbb Q\) classifying isomorphism classes of elliptic curves with extra level structure. Singling out modular curves for careful study can be justified on (at least) two grounds.
(1) They are the simplest examples of moduli spaces. Classifying the rational points on modular curves translates into ``uniform boundedness'' statements for the size of torsion subgroups of elliptic curves over \(\mathbb Q\), and therefore leads to nontrivial results concerning rational points on curves of genus one.
(2) Modular curves are also the simplest examples of Shimura varieties, and their Jacobians and \(\ell\)-adic cohomology are closely tied to spaces of modular forms. (It is from this connection that they derive their name.) This makes it feasible to address finer questions about the rational points on modular curves, following a line of attack that was initiated by \textit{B. Mazur} [Publ. Math., Inst. Hautes Étud. Sci. 47, 33--186 (1977; Zbl 0394.14008)] in his landmark paper on the Eisenstein ideal.
Section 3 attempts to convey some of the flavour of Mazur's approach by describing a simple but illustrative special case of his general results: namely, his proof of the conjecture, originally due to Ogg, that the size of the torsion subgroup of elliptic curves over \(\mathbb Q\) is uniformly bounded, by 14. The approach we describe incorporates an important strengthening due to Merel exploiting progress on the Birch and Swinnerton-Dyer conjecture that grew out of later work of Gross--Zagier
and Kolyvagin--Logachev. \textit{Marusia Rebolledo}'s article [Clay Math. Proc. 8, 71-82 (2009; Zbl 1250.11059)] in these proceedings takes this development one step further by describing Merel's proof of the strong uniform boundedness conjecture over number fields: given \(d\geq 1\), the modular curves \(Y_1(p)\) contain no points of degree \(d\) when \(p\) is large enough (relative to \(d\)).
Section 4 describes the approach initiated by Frey, Serre, and Ribet for reducing Fermat's Last Theorem to deep questions about the relationship between elliptic curves and modular forms. This subject is only lightly touched upon in these notes. \textit{P. Charollois}'s article in this volume [Clay Math. Proc. 8, 83--89 (2009; Zbl 1250.11032)] describes a technique of Halberstadt and Kraus that strengthens the ``modular approach'' to prove a result on the generalised Fermat equation \(ax^p + by^p + cz^p = 0\) that is notable for its generality. This result also suggests that it might be profitable to view the modular
approach as part of a general method, rather than just a serendipitous ``trick'' for proving Fermat's Last Theorem.
Section 5 gives a rapid summary of the author's second week of lectures at the Göttingen summer school, devoted largely to curves of genus 1, particularly elliptic curves. This section is less detailed than the others, partly because it covers topics that have already been treated elsewhere, notably in the author's booklet [Rational points on modular elliptic curves. CBMS Regional Conference Series in Mathematics 101. Providence, RI: American Mathematical Society (AMS) (2004; Zbl 1057.11034)]. The main topics that are touched upon (albeit briefly) in Section 5 are:
(1) The collection of Heegner points on a modular elliptic curve, and Kolyvagin's use of them to prove essentially all of the Birch and Swinnerton-Dyer conjecture for elliptic curves with analytic rank \(\leq 1\). Kolyvagin's techniques also supply a crucial ingredient in Merel's proof of the uniform boundedness conjecture, further justifying its inclusion as a topic in the present notes. The article by \textit{S. Dasgupta} and \textit{J. Voight} [Clay Math. Proc. 8, 91--102 (2009; Zbl 1250.11057)] in these proceedings describes an application of the theory of Heegner points to Sylvester's conjecture on the primes that can be expressed as a sum of two rational cubes.
(2) Variants of the modular parametrisation which can be used to produce
more general systems of algebraic points on elliptic curves over \(\mathbb Q\). Such systems are likely to continue to play an important role in further progress on the Birch and Swinnerton-Dyer conjecture. A key example is the fact that many elliptic curves defined over totally real fields are expected to occur as factors of the Jacobians of Shimura curves attached to certain quaternion algebras. The articles by \textit{J. Voight} [Clay Math. Proc. 8, 103--113 (2009; Zbl 1250.11063)] and \textit{M. Greenberg} [Clay Math. Proc. 8, 115--124 (2009; Zbl 1213.11118)] in these proceedings discuss the problem of calculating with Shimura curves and their associated parametrisations from two different angles: from the point of view of producing explicit equations in Voight's article, and relying on the Cherednik-Drinfeld \(p\)-adic uniformisation in Greenberg's.
(3) The theory of Stark-Heegner points, which is meant to generalise classical Heegner points. \textit{M. Greenberg}'s second article [Clay Math. Proc. 8, 125--135 (2009; Zbl 1213.11118)] in these proceedings discusses Stark-Heegner points attached to elliptic curves over imaginary quadratic fields. Proving the existence and basic algebraicity properties of the points that Greenberg describes how to calculate numerically would lead to significant progress on the Birch and Swinnerton-Dyer conjecture -- at present, there is no elliptic curve that is ``genuinely'' defined over a quadratic imaginary field for which this conjecture is proved in even its weakest form.'' Faltings' theorem; modular curves and Mazur's theorem; Fermat curves; elliptic curves Curves of arbitrary genus or genus \(\ne 1\) over global fields, Elliptic curves over global fields, Arithmetic aspects of modular and Shimura varieties, \(L\)-functions of varieties over global fields; Birch-Swinnerton-Dyer conjecture, Rational points, Modular and Shimura varieties Rational points on curves | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians Let \(K_0(\text{Var}_k)\) denote the Grothendieck ring of algebraic \(k\)-schemes, with addition and multiplication given by disjoint sums and products, respectively. This paper computes the subring generated by the smooth conics \(C\subset\mathbb P^2\), which are the 1-dimensional Severi--Brauer varieties. There is a technical assumption on the ground field \(k\), but number fields, functions fields of complex surfaces, and more generally \(C_2\)-fields are allowed.
To describe the result, let \(G\) be a finite subgroup inside the 2-torsion of the Brauer group \(\text{Br}(k)\). Choose a basis \(C_1,\ldots,C_n\in G\) consisting of smooth conics, where \(G\) is regarded as vector space over the field with two elements, and let \(C(G)=[C_1\times\ldots\times C_n]\) be the class in the Grothendieck ring. Then the subring of the Grothendieck ring generated by smooth conics is the free abelian group generated by elements of the form \(C(G) \cdot [\mathbb P^1]^m\), for varying \(G\) and \(m\). There is also an explicit formula for multiplication.
The computation depends on another result of the paper, which characterizes when two schemes of the form \(C_1\times\ldots\times C_n\) and \(C'_1\times\ldots\times C'_{n'}\), where all factors are smooth conics, have the same class in the Grothendieck ring. It turns out that this holds if and only if the following equivalent conditions hold: The two schemes are (1) birational, (2) stably birational, or (3) we have \(n=n'\) and the subgroups of the Brauer groups generated by the factors \(C_1,\ldots,C_n\) and \(C_1',\ldots, C'_n\) are the same.
The proofs are based on work of \textit{M. Larsen} and \textit{V. A. Lunts} [Mosc. Math. J. 3, 85--95 (2003; Zbl 1056.14015)], and a nice geometric description of the Brauer product of two smooth conics \(C_1,C_2\) as a subscheme of the Hilbert scheme of divisors on \(C_1\times C_2\) of bidegree (1,1). Grothendieck ring; conics; Severi-Brauer variety János Kollár, ``Conics in the Grothendieck ring'', Adv. Math.198 (2005) no. 1, p. 27-35 Brauer groups of schemes, Other nonalgebraically closed ground fields in algebraic geometry Conics in the Grothendieck ring | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians Let \(C\) be a curve of genus \(g\geq 2\) defined over the rational function field \(\mathbb{Q} (t)\), let \(J\) be the Jacobian of \(C\), and let \(r(C)\) denote the rank of the Mordell-Weil group \(J(\mathbb{Q} (t))\). \textit{A. Néron} [in Proc. Int. Congr. Math., 1954, Amsterdam, Vol. III, 481-488 (1956; Zbl 0074.15901] proved for each \(g\) the existence of a curve \(C\) with \(r(C)\geq 3g+7\). In this paper the author combines ideas of \textit{J.-F. Mestre} [C. R. Acad. Sci. Paris Sér. I Math. 313, 139-142 (1991; Zbl 0745.14103)] for the \(g=1\) case with his own work on Mordell-Weil lattices to construct an explicit example of a curve \(C/\mathbb{Q} (t)\) of genus 2 whose rank satisfies \(r(C)\geq 15\). This curve is constructed so that its Jacobian splits (up to isogeny) as \(E_1\times E_2\) for two distinct elliptic curves \(E_i/\mathbb{Q} (t)\), whose ranks the author computes via (geometric) height computations. As a final remark, he observes that his construction shows the existence of infinitely many (hyperelliptic) curves \(\Gamma/\mathbb{Q} \) satisfying \(\#\Gamma(\mathbb{Q})\geq 8g+16\), which improves the lower bound of \(8g+12\) obtained by \textit{L. Caporaso, J. Harris} and \textit{B. Mazur} [The moduli space of curves (Texel Island, 1994), Progr. Math. 129, 13-31 (1995; Zbl 0862.14012]. rank; Mordell-Weil group T. Shioda, Genus two curves over Q(t) with high rank, Comment. Math. Univ. St. Pauli 46 (1997), 15--21. Curves of arbitrary genus or genus \(\ne 1\) over global fields, Abelian varieties of dimension \(> 1\), Jacobians, Prym varieties Genus two curves over \(\mathbb{Q}(t)\) with high rank | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians Let \(X\) be a connected scheme, smooth and separated over an algebraically closed field \(k\) and let \(\rho_\ell :\pi_1(X)\rightarrow \mathrm{GL}_{r_\ell}(\mathbb{F}_\ell), \ell\in L\) be a family (indexed by an infinite set \(L\) of primes) of continuous \(\mathbb{F}_\ell\)-linear representations of the étale fundamental group of \(X\) of bounded degree \(r_\ell\leq r\). The most important examples of such families are those arising from the étale cohomology with \(\mathbb{F}_\ell\)-coefficients of the geometric generic fiber of a smooth proper scheme over \(X\). The main result of this article asserts that, under a mild finiteness assumption, the image \(G_\ell\) of \(\rho_\ell:\pi_1(X)\rightarrow\mathrm{GL}_{r_\ell}(\mathbb{F}_\ell), \ell\in L\) is ``almost algebraic'' for \(\ell\gg 0\). This is the analog for \(\mathbb{F}_\ell\)-coefficients of Grothendieck's unipotency theorem -- a crucial step in the proof of Deligne's semisimplicity theorem in Weil II. Just as for \(\mathbb{Q}_\ell\)-coefficients, our result is a crucial step to establish the analog of Deligne's semisimplicity theorem for \(\mathbb{F}_\ell\)-coefficients (proved by Chun Yin Hui and the authors in a subsequent article). Our result also has a wide range of other applications -- in particular to the variation of invariants in one-dimensional families of varieties and the existence of closed Galois-generic points for motivic representations. We give a first simple example of such applications in the final section of this article. Étale and other Grothendieck topologies and (co)homologies, Fundamental groups and their automorphisms (group-theoretic aspects) On the geometric image of \(\mathbb{F}_\ell\)-linear representations of étale fundamental groups | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians Let \(\ell\) be a prime, and \(H\) a curve of genus 2 over a field \(k\) of characteristic not 2 or \(\ell\). If \(S\) is a maximal Weil-isotropic subgroup of \(\mathrm{Jac}(H)[\ell]\), then \(\mathrm{Jac}(H)/S\) is isomorphic to the Jacobian of some (possibly reducible) curve \(X\). We investigate the Dolgachev-Lehavi method for constructing the curve \(X\), simplifying their approach and making it more explicit. The result, at least for \(\ell=3\), is an efficient and easily programmable algorithm suitable for number-theoretic calculations. Dolgachev-Lehavi method; low-degree isogeny Smith, B, Computing low-degree isogenies in genus 2 with the dolgachev-lehavi method, Arith. Geom. Cryptogr. Coding Theory, 574, 159-170, (2012) Computational number theory, Computational aspects of algebraic curves, Elliptic curves over global fields, Special algebraic curves and curves of low genus Computing low-degree isogenies in genus 2 with the Dolgachev-Lehavi method | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians A classical result of Gabriel implies that two algebraic varieties are isomorphic if they have equivalent categories of coherent sheaves. How much can be said if the varieties are no longer algebraic?
The paper under review gives a precise answer to this question for generic \(K3\) surfaces and generic compact complex tori. The main result of this article says, in stark contrast to Gabriel's result, that for such manifolds, the category of coherent sheaves does not depend on the complex structure. A complex structure on a \(K3\) surface or on a compact torus of dimension \(n\geq 2\) is said to be generic if it does not have non-trivial integral \((p,p)\)-classes for \(0<p<2n\). In particular, such manifolds are not algebraic.
For the proofs, the author makes essential use of the fact that the manifolds considered are holomorphically symplectic, hence carry a hyperkähler structure. The corresponding twistor space comes as a \(\mathbb{P}^1\)-family of complex structures on the underlying manifold. The generic complex structures form a dense subset of this \(\mathbb{P}^1\) and have a countable complement. A result of the author [Geom. Funct. Anal. 6, No. 4, 601--611 (1996; Zbl 0861.53069)] says that any two points in the same connected component of the moduli space of complex structures on a compact hyperkähler manifold can be connected by a chain of hyperkähler \(\mathbb{P}^1\)'s, such that their intersection points represent generic complex structures. This reduces the proof of the main result to showing the required equivalence for any two generic complex structures in the same hyperkähler family.
The proof of the main result uses the twistor correspondence which associates a holomorphic vector bundle on the twistor space to a vector bundle with a connection with \(\text{SU}(2)\)-invariant curvature on the hyperkähler manifold. In a first step, he proves that a certain subcategory of the category of reflexive sheaves on the twistor space is equivalent to the category of reflexive sheaves (i.e.\ bundles) on the given manifold, equipped with any generic complex structure from the hyperkähler family. In the final step of the proof, singularities are incorporated into the above picture: a certain subcategory of the category of coherent sheaves on the twistor space is shown to be equivalent (via restriction) to the category of coherent sheaves for any generic complex structure in the hyperkähler family. An important ingredient into the proofs is the result that on generic \(K3\) surfaces and tori, reflexive sheaves are automatically locally free and coherent sheaves have isolated singularities only. category of coherent sheaves; reflexive sheaf; hyperkähler manifold; holomorphic symplectic manifold; Hermite-Einstein bundle; twistor space; twistor correspondence; reflexive sheaves on Cohomology of compact hyperkähler manifolds and its applications Verbitsky, M., Coherent sheaves on general \textit{K}3 surfaces and tori, Pure Appl. Math. Q., 4, 3, part 2, 651-714, (2008) Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20], Vector bundles on surfaces and higher-dimensional varieties, and their moduli, \(K3\) surfaces and Enriques surfaces, Calabi-Yau manifolds (algebro-geometric aspects), Twistor theory, double fibrations (complex-analytic aspects), Analytic sheaves and cohomology groups Coherent sheaves on general \(K3\) surfaces and tori | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians Recent research has demonstrated a connection between Weil-étale cohomology and special values of zeta functions. In particular, \textit{S. Lichtenbaum} [Compos. Math. 141, No. 3, 689--702 (2005; Zbl 1073.14024)] has shown that the order of vanishing and leading coefficient of the zeta function of a smooth, projective variety over a finite field has a Weil-étale cohomological interpretation in terms of certain secondary Euler characteristics. These results rely on a duality theorem stated in terms of cup-product in Weil-étale cohomology.
We define Weil-étale cohomology for varieties over \(p\)-adic fields, and prove a duality theorem for the cohomology of \(\mathbb{G}_m\) on a smooth, proper, geometrically connected curve of index 1. This duality theorem is a \(p\)-adic analogue of Lichtenbaum's Weil-étale duality theorem for curves over finite fields, as well as a Weil-étale analogue of his classical duality theorem for curves over \(p\)-adic fields. Finally, we show that our duality theorem implies this latter classical duality theorem for index 1 curves. Galois cohomology; étale cohomology; duality Galois cohomology, Étale and other Grothendieck topologies and (co)homologies, Duality in applied homological algebra and category theory (aspects of algebraic topology) Weil-étale cohomology of curves over \(p\)-adic fields | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians Let \({\mathcal A}_{g,N}/\mathbb{F}_p\) be the moduli scheme of principally polarized abelian varieties of a fixed dimension \(g\geq 1\) in characteristic \(p\) with level \(N\)-structure where \(N\geq 3\) is some integer prime to \(p\), and let \(\chi\to {\mathcal A}_{g,N}\) be the universal family. It is a classical result that the ordinary locus in \({\mathcal A}_{g,N}\) is open and dense, where the ordinary locus consists of those points \(s\in {\mathcal A}_{g,N}\) such that the \(p\)-divisible group of \(\chi_{\overline s}\) has only slopes in \(\{0,1\}\). The fact that the ordinary locus is open results from Grothendieck's specialization theorem for crystals. The density of the ordinary locus can be proved in three different ways: (a) There is a proof by Koblitz who investigates by deformation theoretical methods the stratification of \({\mathcal A}_{g,N}\) by the \(p\)-rank of \(\chi\). (b) A second proof is obtained by explicitly constructing deformations by using Cartier theory which raise the \(p\)-rank. (c) A third method to prove the density is the construction of a smooth compactification of the moduli stack \({\mathcal A}_g\) over \(\mathbb{Z}\) of principally polarized abelian varieties and applying Zariski's connectedness theorem to show that \({\mathcal A}_g\) is irreducible.
The aim of this work is to generalize the above statement to good reductions of Shimura varieties of PEL-type. For a Shimura variety of PEL-type the reduction can be considered as a moduli space of abelian varieties with additional structures. We will use a variant of method (b) above. The naive generalization, namely the density of the locus where the underlying abelian variety is ordinary, turns out to be false in general. \(p\)-divisible group; principally polarized abelian varieties; density of the ordinary locus; good reductions of Shimura varieties of PEL-type [We] T. Wedhorn, Ordinariness in good reductions of Shimura varieties of PEL-type, preprint, KoEln 1998, Ann. Sci. EAc. Norm. Sup., to appear. Algebraic moduli of abelian varieties, classification, Modular and Shimura varieties, Finite ground fields in algebraic geometry Ordinariness in good reductions of Shimura varieties of PEL-type | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians In this fundamental paper the author, after a long period of working, proved the Grothendieck conjecture, a very deep and difficult result concerning the converse of the Grothendieck specialization theorem for \(p\)-divisible groups -- the Newton polygon goes up under specialization in a family of \(p\)-divisible groups [\textit{A. Grothendieck}, Groupes de Barsotti-Tate et cristaux de Dieudonné (Les Presses de l'Universite de Montreal) (1974; Zbl 0331.14021)]. The analogous statement for the reduction mod \(p\) of Siegel moduli spaces is also proved. The proof relies on two rather different aspects of deformation theory of \(p\)-divisible groups. The first one is on deformations of simple \(p\)-divisible groups keeping the Newton polygon constant. The author uses the methods and results derived from ``Purity'' as obtained in [\textit{A. J. de Jong} and \textit{F. Oort}, J. Am. Math. Soc. 13, 209--241 (2000; Zbl 0954.14007)] . The other one is an explicit deformation theory with invariant \(a\)-number one [\textit{F. Oort}, Ann. Math. (2) 152, 183--206 (2000; Zbl 0991.14016)]. This is an effective method of reading the Newton polygon of a subvariety in a local deformation space. In this paper a combination of these two methods gives the desired result. \(p\)-divisible groups; Newton polygons; Grothendieck conjecture F. Oort: Newton polygon strata in the moduli space of abelian varieties, Moduli of abelian varieties (Texel Island, 1999), 417--440, Progr. Math. 195, Birkhäuser, Basel, 2001. Algebraic moduli of abelian varieties, classification, Formal groups, \(p\)-divisible groups Newton polygon strata in the moduli space of abelian varieties | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians In the seminal work [Seminaire de géométrie algébrique Du Bois-Marie 1967-1969. Groupes de monodromie en géométrie algébrique (SGA 7 I). Lecture Notes in Mathematics. 288. Berlin-Heidelberg-New York: Springer-Verlag (1972; Zbl 0237.00013)], \textit{A. Grothendieck} introduced the concept of biextension of a pair of groups \((P,Q)\) by another group \(G\) and applied it to the study of Néron models of abelian varieties over DVR's. In particular, he defined a pairing controlling the obstructions for a biextension over the generic fiber to come from a biextension over the whole DVR. The work under review studies these objects in a logarithmic setting, showing that no obstruction occurs to extend logarithmic biextensions of \((A,A^t)\) by \(\mathbb{G}_m\) to logarithmic biextensions over the DVR when \(A\) is an abelian variety over the fraction field of the DVR and \(A^t\) is its dual abelian variety.
Fix a scheme \(S\) and three commutative \(S\)-groups \(P,Q,G\): extensions of \(P\) by \(G\) (resp. of \(Q\) by \(G\)) over \(S\) are identified with \(S\)-torsors under \(G_P:=G\times_S P\) (resp. \(G_Q:=G\times_S Q\)), and these are in turn classified by classes in \(H^1_{\text{top}}(P,G_P)\) (resp. classes in \(H^1_{\text{top}}(Q,G_Q)\)); here any ``reasonably nice'' topology on the category of \(S\)-schemes (inducing that on \(P\)- or \(Q\)-schemes) can be considered in order to compute these cohomology groups. Similarly, a biextension of the pair \((P,Q)\) by \(G\) is defined to be a torsor \(E\) under the base-change \(G_{P\times Q}\) with additional and compatible structures of an extension of \(Q_P\) by \(G_P\) and one of \(P_Q\) by \(G_Q\): then Grothendieck showed in [loc. cit.] that these biextensions are classified by a certain group \(\mathrm{Biext}^1_{\text{top}}(P,Q;G)\) which has a purely homological definition in terms of the topos attached to the site of \(S\)-schemes endowed with any ``nice'' topology \(\text{top}\), as before.
Consider now the case when \(S\) is the spectrum of a discrete valuation ring and endow it with its canonical log structure. The starting point of the paper under review is to follow Grothendieck's construction in the topos attached to the site of fine saturated \(S\) log schemes endowed with the (Kummer) logarithmic flat topology [see \textit{K. Kato}, ``Logarithmic structures of Fontaine-Illusie 2, Logarithmic flat topology'', Preprint (1991)] -- that the author denotes by \(\text{kpl}\) -- and to study the corresponding group \(\mathrm{Biext}^1_{\text{kpl}}(P,Q;G)\). The main result of the paper is (see Théorème 4.1.1) that, if \(P\) and \(Q\) are smooth of finite type over \(S\), then restricting to the generic fiber \(\eta\) gives isomorphisms
\[
\text{Ext}^1_{\text{kpl}}(P,\mathbb{G}_m)\overset{\cong}{\longrightarrow}\text{Ext}^1_{\text{pl}}(P_\eta,\mathbb{G}_{m,\eta})\tag{P}
\]
\[
\text{Biext}^1_{\text{kpl}}(P,Q;\mathbb{G}_m)\overset{\cong}{\longrightarrow}\text{Biext}^1_{\text{pl}}(P_\eta,Q_\eta;\mathbb{G}_{m,\eta})\tag{PeQ}
\]
where we denote by \(\text{pl}\) the fppf topology on \(Sch/S\). This should be compared with the classical situation, where these maps fail to be isomorphisms in general: indeed, in [loc. cit., exposé VIII, théorème 7.1], \textit{A. Grothendieck} showed that the vanishing of a certain map (resp. a certain pairing) from the group of connected components of the special fiber of \(P\) (resp. between the groups of connected components of the special fibers of \(P\) and \(Q\)) is a necessary and sufficient condition for (P) (resp. for (PeQ)) to be an isomorphism. In the logarithmic world this obstruction disappears but, as the author says, the price to pay is to switch from schemes to log-schemes. The main application of the above result is when \(P=\mathcal{A}\) is the Néron model of an abelian variety \(A\) over \(\eta\) and \(Q=\mathcal{A}^t\) is the Néron model of the dual abelian variety \(A^t\). In this setting, a canonical biextension (``of Weil'') of \((A,A^t\)) by \(\mathbb{G}_m\) was constructed in [Grothendieck, loc. cit.], and (PeQ) above shows the existence, in the logarithmic world, of a unique biextension \(\mathcal{W}^{\text{log}}\) extending the Weil biextension over the whole of \(S\).
The paper falls into four sections: the first is a clear and neat introduction, hinting at the future work [\textit{J. Gillibert}, ``Cohomologie log-plate, actions modérées et structures galoisiennes'', \url{arXiv:0905.1902}] of the author where the extension \(\mathcal{W}^{\text{log}}\) will play a role. In Section 2 the needed logarithmic technology is introduced (Proposition 2.2.6 is the key result) and Section 3 studies logarithmic \(\mathbb{G}_m\)-torsors in terms of ``logarithmic divisors'' (see Définition 3.1.2). In Section 4 the author first introduces logarithmic biextensions and proves the main theorem (Théorème 4.1.1); later, he leaves the setting of a DVR to study more general monodromy pairings à la Mazur-Tate over noetherian regular schemes of dimension \(1\) (Definition 4.2.2) as well as cubic torsors in the very last subsection. log schemes; Néron models; biextensions Étale and other Grothendieck topologies and (co)homologies, Abelian varieties and schemes, Local ground fields in algebraic geometry Extension of biextensions and pairings in the log flat cohomology | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians Let \(\mathbb{C}\), the complex field, be the ground field. The renowned Jacobian conjecture is stated as follows:
If a polynomial mapping \(\varphi=(f_1,\dots,f_n): \mathbb{A}^n\to\mathbb{A}^n\) has a nonzero constant as the Jacobian determinant \(|\partial(f_1,\dots, f_n)/ \partial(x_1,\dots, x_n)|\), then \(\varphi\) is an automorphism.
Recalling that the affine space \(\mathbb{A}^n\) is topologically simply connected, we can formulate this conjecture as follows:
Jacobian conjecture. If \(\varphi:\mathbb{A}^n \to\mathbb{A}^n\) is an étale endomorphism, then \(\varphi\) is a finite morphism.
Thus formulated, the Jacobian conjecture can be generalized in the following way:
Generalized Jacobian conjecture. Let \(X\) be a smooth algebraic variety defined over \(\mathbb{C}\) and let \(\varphi:X\to X\) be an étale endomorphism. Then \(\varphi\) is an finite étale morphism except when \(X\) has essentially logarithmic Kodaira dimension zero.
The authors discuss the second conjecture. Their strategy is to use the geometric structures of a given variety to look into an étale endomorphism. The main focus here is on affine and quasi-affine varieties. étale endomorphism; finite morphism Miyanishi, M.; Masuda, K., Generalized Jacobian conjecture and related topics, (Algebra, Arithmetic and Geometry, Part I, II, Mumbai, 2000, Tata Inst. Fund. Res. Stud. Math., vol. 16, (2002), Tata Inst. Fund. Res Bombay), 427-466 Jacobian problem Generalized Jacobian conjecture and related topics | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians To every abelian scheme \(A/S\) we associate in a natural way an arithmetic group which acts projectively on \(D^b(\text{Coh}(A))\), the derived category of coherent sheaves on \(A\), following a construction of \textit{A. Weil} [Acta Math. 111, 143--211 (1964; Zbl 0203.03305)]. We show that the corresponding central extension of the arithmetic group by \(\mathbb Z \times \text{Pic}(S)\) embeds into the group of autoequivalences of \(D^b(\text{Coh}(A))\) and is related to determinantal line bundles for \(A/S\). The main construction is based on the theory of Schrödinger representations of Heisenberg group schemes. Our contribution to this theory is the theorem of existence of a representation of minimal possible rank for symmetric Heisenberg groups schemes of odd order.
This paper generalizes previous works by the author [``Biextensions, Weil representations on derived categories and theta-functions'', Ph.D. thesis, Harvard Univ., Cambridge, MA, 1996] and \textit{D. O. Orlov} [Izv. Ross. Akad. Nauk, Ser. Mat. 66, No. 3, 131--158 (2002; Zbl 1031.18007)]. derived category of coherent sheaves; Heisenberg group schemes; homological mirror conjecture DOI: 10.1515/crll.2002.016 Abelian varieties of dimension \(> 1\), Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20], Derived categories, triangulated categories, Linear algebraic groups over global fields and their integers Analogue of Weil representation for abelian schemes | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians The authors construct a Riemann-Hilbert correspondence for smooth schemes over a field of characteristic \(p> 0\). It is between constructable perverse étale \(F_p\)-sheaves and quasicoherent \({\mathcal O}_X\)-sheaves which are isomorphic to their Frobenius-transform. As both categories do not admit a reasonable duality-theory the usual definitions have to be somehow modified. On the étale side this has been done by \textit{O. Gabber} [in: Geometric aspects of Dwork theory. Vol. I, II. Berlin: Walter de Gruyter. 711--734 (2004; Zbl 1074.14018)]. On the quasicoherent side a local theory had been proposed by \textit{G. Lyubeznik} [J. Reine Angew. Math. 491, 65--130 (1997; Zbl 0904.13003)], and it is globalised here using derived categories. One difficulty is that usual direct images do not preserve the condition ``Frobenius is an isomorphism'', and one has to compose with a limit process to remedy this. The naive approach would lead to projective limits which are difficult to handle. The authors use instead some implicit duality to define an analogue of direct images with proper support, and an inductive limit process. The price to pay is that sheaves are only quasicoherent (however, still finitely generated over the twisted polynomials \({\mathcal O}_X[\text{Frob}]\)). For example for a closed immersion \(Y\subset X\) one typically obtains local cohomology with support in \(Y\). Once this machinery has been set up the Riemann-Hilbert correspondence amounts to the usual Artin-Schreier theory combined with devissage.
In the last chapter the theory is extended to schemes over \(\mathbb{Z}/p^n\mathbb{Z}\). To define Frobenius-pullbacks one needs sheaves with an integrable connection. If they are isomorphic to their Frobenius-transform they automatically admit an action by all differential operators. Thus the extended theory relates étale \(\mathbb{Z}/p^n\mathbb{Z}\)-sheaves and \({\mathcal D}_s[\text{Frob}]\)-modules. It relies an previous work by \textit{P. Berthelot} [Ann. Sci. Éc. Norm. Supér., IV. Sér. 29, 185--272 (1996; Zbl 0886.14004) and Mém. Soc. Math. Fr., Nouv. Sér. 81 (2000; Zbl 0948.14017)]. Riemann-Hilbert correspondence; Frobenius; perverse étale sheaves Emerton, M.; Kisin, M., The Riemann-Hilbert correspondence for unit \textit{F}-crystals, Asterisque, 293, (2004) \(p\)-adic cohomology, crystalline cohomology The Riemann-Hilbert correspondence for unit \(F\)-crystals | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians This is the first of the famous series of ``Séminaire de géométrie algébrique'' of Alexander Grothendieck and his collaborators, started in 1960-61, in which Grothendieck constructs (and studies the main properties of) the algebraic fundamental group \(\pi^{\text{alg}}_1(X)\) of a scheme \(X\). If \(X\) is a complex algebraic variety then the profinite completion of the topological fundamental group \(\pi_1(X)\) coincides with \(\pi^{\text{alg}}_1(X)\). This fundamental construction is extremely important when for example \(X\) is an algebraic scheme over a finite field extension \(K\) of \(\mathbb Q\) because the knowledge of \(\pi^{\text{alg}}_1(X)\) contains already a lot of information about \(X\) (in some cases, enough to reconstruct \(X\) itself). To quote Serre (from his talk in the Bourbaki Seminar of the fall of 1991), this seminar was the first great success of the theory of schemes.
Due to these facts it certainly deserved to be republished by the Société Mathématique de France as a volume of ``Documents Mathématiques''. The present volume is a (slightly) corrected and updated version of the previous edition [Lect. Notes Math. 224 (1971; Zbl 0234.14002)]. Some updated remarks have been added by M. Raynaud, which are bounded by brackets [ ] and indicated by the symbol (MR). étale covering; algebraic fundamental group; descent theory; smooth morphisms Raynaud, M., Catégories fibrée et descente, Revêtements étales et groupe fondamental (SGA 1), 3, (2003) Research exposition (monographs, survey articles) pertaining to algebraic geometry, Local structure of morphisms in algebraic geometry: étale, flat, etc., Coverings in algebraic geometry, Étale and other Grothendieck topologies and (co)homologies, Homotopy theory and fundamental groups in algebraic geometry Seminar on algebraic geometry at Bois Marie 1960-61. Étale coverings and fundamental group (SGA 1). A seminar directed by Alexander Grothendieck. Enlarged by two reports of M. Raynaud | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians The authors work on an older paper of \textit{S. Lubkin} [Ann. Math. (2) 87, 105--255 (1968; Zbl 0188.53004)] which introduced a Weil cohomology for the category of complete smooth liftable varieties over a field \(k\) of nonzero characteristic \(p\). In that paper Lubkin uses this \(p\)-adic cohomology theory to prove the first two Weil conjectures, the Lefschetz theorem and the functional equation. Let \({\mathfrak O}\) be a complete discrete valuation ring with quotient field \(K\) of characteristic zero and with residue field \(k\) of positive characteristic \(p\). A lifting of a complete smooth algebraic variety \(X\) over \(k\) is a proper smooth \({\mathfrak O}\)-scheme \({\mathcal X}\) such that \({\mathcal X}\otimes_{\mathfrak O}k=X\). In the paper of Lubkin the notion of \({\mathfrak O}\)-spaces was introduced; in particular \({\mathcal X}\) is an \({\mathfrak O}\)-space. In the paper, under review the \(p\)-adic cohomology and the relative \(p\)-adic cohomology for the category of \({\mathfrak O}\)-spaces are studied. The main result of this paper is the proof of a Lefschetz duality theorem for the \(p\)-adic cohomology of \({\mathfrak O}\)-spaces. \(p\)-adic cohomology; Weil cohomology; liftable varieties; Weil conjectures; Lefschetz theorem; Lefschetz duality theorem \(p\)-adic cohomology, crystalline cohomology, Duality theorems for analytic spaces A Lefschetz duality theorem in \(p\)-adic cohomology | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians The hyperbolicity statements for subvarieties and complement of hypersurfaces in abelian varieties admit arithmetic analogues, due to \textit{G. Faltings} [Ann. Math. (2) 133, No. 3, 549--576 (1991; Zbl 0734.14007)] (and for the semiabelian case [\textit{P. Vojta}, Am. J. Math. 121, No. 2, 283--313 (1999; Zbl 1018.11027)]). In [Atti Accad. Naz. Lincei, Cl. Sci. Fis. Mat. Nat., IX. Ser., Rend. Lincei, Mat. Appl. 29, No. 3, 401--411 (2018; Zbl 1439.11180)], by the second author, an analogy between the analytic and arithmetic theories was shown to hold also at proof level, namely in a proof of Raynaud's theorem (Manin-Mumford Conjecture). The first aim of this paper is to extend to the relative setting the above mentioned hyperbolicity results. We shall be concerned with analytic sections of a relative (semi-) abelian scheme \(\mathscr{A}\to B\) over an affine algebraic curve \(B\). These sections form a group; while the group of the rational sections (the Mordell-Weil group) has been widely studied, little investigation has been pursued so far on the group of the analytic sections. We take the opportunity of developing some basic structure of this apparently new theory, defining a notion of height or order functions for the analytic sections, by means of Nevanlinna theory. Legendre elliptic; semi-abelian scheme; Diophantine geometry; Nevanlinna theory Value distribution theory in higher dimensions, Abelian varieties and schemes, Elliptic surfaces, elliptic or Calabi-Yau fibrations, Hyperbolic and Kobayashi hyperbolic manifolds, Picard-type theorems and generalizations for several complex variables Analytic and rational sections of relative semi-abelian varieties | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians The paper under review fits into a general program initiated by Wildeshaus which aims at associating motives to certain automorphic forms for all PEL Shimura varieties. For classical modular forms, this was achieved by \textit{A. J. Scholl} [Invent. Math. 100, No. 2, 419--430 (1990; Zbl 0760.14002)], building on \textit{P. Deligne}'s realization of Galois representations in [in: Sémin. Bourbaki 1968/69, No. 355, 139--172 (1971; Zbl 0206.49901)]. They came with the aditional feature of purity, one of the central ingredients of many further findings.
Presently, the author investigates Picard modular surfaces, i.e., moduli spaces of abelian threefolds endowed with an action by an order in some imaginary quadratic field plus some additional structure. These moduli spaces are smooth, but non-compact, so the author considers the Baily-Borel compactification. This allows him to study variations of Hodge structures over the Picard modular surfaces. More precisely, he computes in thorough detail (and quite explicitly and concretely!) the weights and the types of degenerations at the cusps of the compactification.
As the main application, the results form a key ingredient for \textit{J. Wildeshaus}' construction of pure motives associated with Picard automorphic forms in [Manuscr. Math. 148, No. 3--4, 351--377 (2015; Zbl 1349.14106)]. Picard surface; Shimura variety; Hodge structure; motive; modular form; automorphic form; Baily-Borel compactification; weight; degeneration Other groups and their modular and automorphic forms (several variables), Arithmetic aspects of modular and Shimura varieties, Cohomology of arithmetic groups, Complex multiplication and moduli of abelian varieties, Variation of Hodge structures (algebro-geometric aspects), Families, moduli, classification: algebraic theory Degeneration of Hodge structures over Picard modular surfaces | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians Let \(\overline {\mathcal M}_{g,n}\) denote the moduli space of \(n\)-pointed stable curves of genus \(g\). There are combinatorical cycles on \(\overline {\mathcal M}_{g,n}\) which are defined using Strebel's theory of quadratic differentials. On the other hand there are algebro-geometric cohomology classes of \(\overline {\mathcal M}_{g,n}\), the so-called Mumford-Morita-Miller classes, defined via the relative dualizing sheaf of the universal curve over \(\overline {\mathcal M}_{g,n}\). It was first conjectured by Witten that the classes of the combinatorical cycles can be expressed in terms of the Mumford-Morita-Miller classes. The intersection numbers of the combinatorical cycles are best organized as coefficients of an infinite series \(F\) in infinitely many \(s\)- and \(t\)-variables. \textit{P. Di Francesco}, \textit{C. Itzykon}, and \textit{J.-B. Zuber} proved in Commun. Math. Phys. 151, No. 1, 193-219 (1993; Zbl 0831.14010) that the \(s\)-derivatives of \(F\) are linear combinations of the \(t\)-derivatives of \(F\) evaluated at the same point. The authors conjecture that this result interpreted geometrically should provide the link between the combinatorical and algebro-geometric classes on \(\overline {\mathcal M}_{g,n}\). The main result of the paper is the proof of this conjecture in codimension 1. Moreover in some cases the authors give an explicit expression for the Di Francesco-Itzykson-Zuber-correspondence. In these cases the correspondence gives an explicit expression for the intersection numbers of the combinatorial cycles in terms of the algebro-geometric classes. Mumford classes; moduli space of \(n\)-pointed stable curves; combinatorical cycles; quadratic differentials E. Arbarello and M. Cornalba, Combinatorial and algebro-geometric cohomology classes on the moduli spaces of curves, J. Algebraic Geom. 5 (1996), 705-749. Families, moduli of curves (algebraic), Families, moduli of curves (analytic) Combinatorial and algebro-geometric cohomology classes on the moduli spaces of curves | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians Let $X$ be a smooth quasi-projective scheme over a perfect field $k$ and $D\subset X$ an effective Cartier divisor. The main result of this paper relates the Chow group of zero-cycles with modulus $\mathrm{CH}_0(X|D)$ in the sense of \textit{M. Kerz} and \textit{S. Saito} [Duke Math. J. 165, No. 15, 2811--2897 (2016; Zbl 1401.14148)] to the Levine-Weibel Chow group [\textit{M. Levine} and \textit{C. Weibel}, J. Reine Angew. Math. 359, 106--120 (1985; Zbl 0555.14004)] of the double $S(X,D)$ of $X$ along $D$. Here $S(X,D)$ is defined as the amalgamated sum of two copies of $X$ along the inclusion of $D$ in both copies. The precise result is that the Levine-Weibel Chow group of $S(X,D)$ is a split extension of the usual Chow group of zero-cycles $\mathrm{CH}_0(X)$ by the Chow group with modulus $\mathrm{CH}_0(X|D)$. \par The proof of this remarkable decomposition theorem is long and technical. Once it is established, the authors can deduce results on $\mathrm{CH}_0(X|D)$ from known results for the Levine-Weibel Chow group. For instance, using the theory of the generalized Albanese variety of \textit{H. Esnault} et al. [Invent. Math. 135, No. 3, 595--664 (1999; Zbl 0954.14003)], the authors establish the existence of a generalized Albanese variety for the degree zero part of $\mathrm{CH}_0(X|D)$ and prove a version of Roitman's theorem for it. They also show that Bloch's conjecture on the Albanese map for surfaces implies a more general version with modulus. Finally, they study a cycle map from $\mathrm{CH}_0(X|D)$ to the relative $K$-group $K_0(X,D)$. In the case when $X$ is a surface and $k$ is algebraically closed, they prove it is injective and identify its cokernel with a Picard group with modulus. zero-cycles with modulus; Albanese variety; cycle map Algebraic cycles, Applications of methods of algebraic \(K\)-theory in algebraic geometry, \(K\)-theory of schemes Zero cycles with modulus and zero cycles on singular varieties | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians Let \(X\) be a smooth proper variety over a perfect field \(k\) and let \(D\) be an effective divisor on \(X\) with multiplicity. In this paper the author introduces an Albanese variety \(\text{Alb}^{(1)}(X,D)\) associated to the pair \((X,D)\) as a higher-dimensional analog of the generalized Jacobian variety with modulus introduced by Rosenlicht and Serre for smooth proper curves. If \(P\) is a torsor under a commutative algebraic \(k\)-group \(G\) and \(\varphi: X\dashrightarrow P\) is a rational map, the author defines a certain effective divisor \(\text{mod}(\varphi)\) on \(X\), called the modulus of \(\varphi\), which coincides with the classical definition in the curve case. Then \(\text{Alb}^{(1)}(X,D)\) and the corresponding Albanese map \(\text{alb}_{X,D}^{(1)}: X\dashrightarrow \text{Alb}^{(1)}(X,D)\) are defined by the following universal property: for every torsor \(P\) under a commutative algebraic \(k\)-group \(G\) and every rational map \(\varphi\) from \(X\) to \(P\) of modulus \(\leq D\), there exists a unique homomorphism of torsors \(h: \text{Alb}^{(1)}(X,D)\to P\) such that \(\varphi=h\circ \text{alb}_{X,D}^{(1)}\). To establish the existence of \(\text{alb}_{X,D}^{(1)}\), the author works with a broader notion of generalized Albanese varieties defined by a universal mapping property for categories of rational maps from \(X\) to torsors for commutative algebraic groups. To construct the latter, the author develops a notion of duality for smooth connected commutative algebraic groups over a perfect field via 1-motives with unipotent part, which generalize both Deligne's and Laumon's 1-motives. A 1-motive with unipotent part is roughly a homomorphism \((\mathcal F\to G)\), where \(G\) is an extension of an abelian variety \(A\) by a commutative linear group \(L\) and \(\mathcal F\) is a dual-algebraic commutative formal group, i.e., the Cartier dual of \(\mathcal F\) is algebraic. These 1-motives admit duality, e.g., the dual of \((0\to A)\) is \((L^{\vee}\to A^{\vee})\), where \(L^{\vee}\) is the Cartier dual of \(L\) and \(A^{\vee}\) is the abelian variety dual to \(A\). Using these 1-motives, the author obtains explicit and functorial descriptions of these generalized Albanese varieties and their dual functors. The following theorem, from which the existence of \(\text{Alb}^{(1)}(X,D)\) can be deduced, is one of the main results of the paper:
Theorem. Let \(\underline{\text{Div}}^{0,\text{red}}_{X}\) be the (sheaf) pullback of the Picard variety \(\text{Pic}^{0,\text{red}}_{X}\) of \(X\) under the cycle class map and let \(\mathcal F\) be a dual-algebraic formal \(k\)-subgroup of \(\underline{\text{Div}}^{0,\text{red}}_{X}\). By base change to an algebraic closure of \(k\), a rational map \(\varphi: X\dashrightarrow P\) induces a rational map \(\varphi: \overline{X}\dashrightarrow \overline{G}\) which in turn induces a natural transformation \(\tau_{\overline{\varphi}}: \overline{L}^{\vee}\to \underline{\text{Div}}^{0,\text{red}}_{\overline{X}}\). Let \(M_{\mathcal F}\) be the category of rational maps \(\varphi: X\dashrightarrow P\) such that the image of \(\tau_{\overline{\varphi}}\) lies in \(\overline{\mathcal F}\). Then \(M_{\mathcal F}\) admits a universal object \(\text{alb}_{\mathcal F}^{(1)}: X\dashrightarrow \text{Alb}^{(1)}_{_{\mathcal F}}(X)\), where \(\text{Alb}^{(1)}_{_{\mathcal F}}(X)\) is a torsor for an algebraic group \(\text{Alb}^{(0)}_{_{\mathcal F}}(X)\) which is an extension of the classical Albanese variety \(\text{Alb}(X)\) by \(\mathcal F^{\vee}\) and is dual to the 1-motive \((\mathcal F\to \text{Pic}^{0,\text{red}}_{X})\).
The author further defines a relative Chow group of zero cycles \(\text{CH}_{0}(X,D)\) of modulus \(D\) and shows that \(\text{Alb}^{(1)}(X,D)\) can be viewed as a universal quotient of \(\text{CH}_{0}(X,D)^{\text{deg}\,0}\). Finally, an application is given which rephrases Lang's class field theory of function fields of varieties over finite fields in explicit terms. Albanese with modulus; relative Chow group with modulus; geometric class field theory Russell, H., \textit{Albanese varieties with modulus over a perfect field}, Algebra Number Theory, 7, 853-892, (2013) Group varieties, Geometric class field theory, (Equivariant) Chow groups and rings; motives Albanese varieties with modulus over a perfect field | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians In this very readable and carefully written paper, the author studies the descent sequence associated to an isogeny between two abelian varieties defined over a number field. The techniques are Galois cohomological in nature.
Let \(\varphi: A\to A'\) be an isogeny of abelian varieties defined over a number field \(K\). Let \(L\) be the extension of \(K\) obtained by adjoining the points of the kernel \(A[\varphi]\) of \(\varphi\) to \(K\). Under the, frequently valid, assumption that \(A[\varphi]\) is a cohomologically trivial \(\text{Gal} (L/K)\)-module, the \(\varphi\)-Selmer group \(S^\varphi (K,A)\) can be viewed as a subgroup of \(\Hom (\text{Gal} (\overline{L}/L), A[\varphi])\). The author compares the Selmer group with the subgroup of homomorphism \(\text{Gal} (\overline{L}/L)\to A [\varphi]\) that are everywhere unramified. By class field theory, he obtains in this way relations between \(S^\varphi (K,A)\) and the ideal class group of \(K\).
The results are very explicit when \(A\) and \(A'\) are elliptic curves or when \(A=A'\) and \(\varphi=2\). In these cases much work has been done previously and it is enlightening to see this work from the point of view of this paper. The author gives several explicit applications of his methods: he exhibits a cubic number field whose class group has 2-rank at least 13 and he exhibits a curve of genus 2 whose Jacobian has a Mordell Weil group of rank 7. unramified homomorphisms; descent sequence; isogeny; abelian varieties defined over a number field; Galois cohomological; Selmer group; class field theory; ideal class group Schaefer E.\ F., Class groups and Selmer groups, J. Number Theory 56 (1996), no. 1, 79-114. Abelian varieties of dimension \(> 1\), Isogeny, Class numbers, class groups, discriminants, Galois cohomology, Varieties over global fields Class groups and Selmer groups | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians If \(C\) is a smooth curve of genus \(g\), the Brill-Noether locus \(W^r_d(C)\) parametrizes the line bundles \(L\) on \(C\) of degree \(d\) with \(h^0(L)\geq r+1\). It has a natural structure of determinantal variety and its ``expected'' dimension is the Brill-Noether number \(\rho(g,r,d)=g-(r+1)(g-d+r)\). It has been proved by Fulton-Lazarsfeld that \(W^r_d(C)\) is connected if \(\rho(g,r,d)>0\).
In the paper under review, the author extends this result to a class of singular curves, precisely to any irreducible curve \(C\) lying on a smooth regular surface \(S\) such that the anticanonical divisor \(-K_S\) is generated by its global sections. Now \(W^r_d(C)\) is replaced by the generalized Brill-Noether locus \(\overline W^r_d(C)\), parametrizing rank one torsion-free sheaves \(A\) on \(C\), of degree \(d\) with at least \(r+1\) independent sections. The rough idea of the proof is to deform \(C\) to a smooth curve and then use the connectedness theorem of Fulton-Lazarsfeld.
As an application of this result, the author gives a new proof of a theorem of \textit{K. G. O'Grady} [J. Algebr. Geom. 6, 599-644 (1997; Zbl 0916.14018)] concerning the irreducibility of certain moduli spaces of rank 2 Gieseker-semistable torsion-free sheaves with low \(c_2\) on a \(K3\) surface \(S\). Precisely, the moduli space considered are of the form \({\mathfrak M}_H(L,c_2)\), where \(L\) is a primitive nef and big line bundle, \(c_2\leq {1\over 2}L^2+3\) and \(H\) is an \((L,c_2)\)-generic polarization. Brill-Noether number; torsion-free sheaves; K3 surfaces; singular curves; generalized Brill-Noether locus; moduli spaces Gómez, Tomás L., Brill-Noether theory on singular curves and torsion-free sheaves on surfaces, Comm. Anal. Geom., 9, 4, 725-756, (2001) Special divisors on curves (gonality, Brill-Noether theory), Singularities of curves, local rings, Vector bundles on surfaces and higher-dimensional varieties, and their moduli, \(K3\) surfaces and Enriques surfaces Brill-Noether theory on singular curves and torsion-free sheaves on surfaces | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians The Grothendieck-Riemann-Roch theorem gives a formula that relates direct images and characteristic classes. In general this formula is not valid for characteristic forms. The singular Bott-Chern classes measure the failure of an exact Grothendieck-Riemann-Roch theorem for closed immersions at the level of characteristic forms. Similarly, the analytic torsion forms measure the failure of an exact Grothendieck-Riemann-Roch theorem for submersions. Among other things, the existence of analytic torsion, makes the arithmetic Riemann-Roch Theorem hard to use. A Lefschetz fixed point formula was developed by \textit{K. I. Köhler} and \textit{D. Roessler} [C. R. Acad. Sci., Paris, Sér. I, Math. 326, No. 6, 719--722 (1998; Zbl 0934.14017)] in representing an equivariant version of the Riemann Roch theorem. The authors consider arithmetic varieties endowed with an action of the group scheme of \(n\)-th roots of unity and defined equivariant arithmetic \(K_0\)-theory for these varieties. Still the fixed point theorem can only be used for smooth morphism over the complex. The work of \textit{J~. I. Burgos-Gil}, \textit{R.~ Litcanu} and \textit{G. Freixas} [Doc. Math., J. DMV 15, 73--176 (2010; Zbl 1192.14019); J. Math. Pures Appl. (9) 97, No. 5, 424--459 (2012; Zbl 1248.18011); in: ``Alexandru Myller'' Mathematical Seminar AIP Conf. Proc. 1329, Amer. Inst. Phys. (2010; \url{doi:10.1063/1.3546075})] seeks to extend the Riemann-Roch Theorem to arbitrary proper morphisms. The present work of Tang combines the equivariant Lefschetz type theorem with the extension of Riemann-Roch to arbitrary proper morphisms. The first step is to generalize the analytic torsion to closed immersions. The strategy of work is as follows: by deforming a resolution to a more treatable one, the author proved that the theory of equivariant singular Bott-Chern classes is determined by its effects on Koszul resolutions. By adding suitable axioms to the usual axiomatic of the equivariant Bott-Chern secondary classes, it is shown how Bismut's construction provides a uniqueness property for the Bott-Chern characteristic classes. As a consequence of this uniqueness an analytic statement is presented in the last section with represents a concentration formula for equivariant classes, extending the concentration theorem in algebraic \(K\)-theory to Arakelov Geometry. singular Bott-Chern forms; Riemann-Roch; equivariant secondary classes; axiomatic for Bott-Chern forms Arithmetic varieties and schemes; Arakelov theory; heights, Currents, Riemann-Roch theorems, Applications of methods of algebraic \(K\)-theory in algebraic geometry, Relations of \(K\)-theory with cohomology theories Uniqueness of equivariant singular Bott-Chern classes | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians The author studies the geometric and arithmetic properties of the theta divisor \(\Theta\) associated to the vector bundle of locally exact differential forms of a smooth, projective, connected curve \(X\) over an algebraically closed field \(k\) of characteristic \(p>0\). Let \(X_1\) be the inverse image of \(X\) with respect to the Frobenius automorphism of \(k\), \(J_1\) the Jacobian of \(X_1\), \(B\) the sheaf defined by the exact sequence \(0 \to \mathcal{O}_{X_1} \to F_*\mathcal{O}_X \to B \to 0\), where \(F : X \to X_1\) is the relative Frobenius homomorphism. The determinant of the complex \(Rf_*(B \otimes \mathcal{P})\), where \(f : X_1 \times J_1 \to J_1\) is the projection, is called the divisor \(\Theta\).
Section 1 of this long and important paper contains the definitions and the proofs of the basic properties of the divisor \(\Theta\). In particular, if \(X/S\) is a smooth curve, \(B\) is endowed with an alternating pairing with values in \(\Omega^1_{X/S}\), \(B\) is self-dual with respect to Serre's duality, and \(\Theta\) is totally symmetric in Mumford's sense. Using the Fourier-Mukai transformation, some relationships between \(B\) and \(R^1f_*(B \otimes P)\) are clarified in order to prove important properties of \(\Theta\) for generic curves. Among other things, it is proven that \(\Theta\) possesses the Dirac property, and a new and more direct proof of \textit{K. Joshi}'s result [C. R., Math., Acad. Sci. Paris 338, No. 11, 869--872 (2004; Zbl 1051.14043)] on the stability of \(B\) for a curve of genus \(\geq 2\) is given.
Section 2 is devoted to the differential study of \(\Theta\) and the Hilbert schema \(\mathcal{H}\) of invertible degree 0 sheaves in \(B\). Using the results of \textit{Y. Laszlo} [Duke Math. J. 64, No. 2, 333--347 (1991; Zbl 0753.14023)] the author investigates the multiplicity of \(\Theta\), elucidates the role of the Cartier forms, and examines the points \(x \in \mathcal{H}\) in the case where \(\mathcal{H}\) is smooth of dimension \(g-1\).
The main result of Section 3 asserts that for the generic curve of genus \(\geq 2\) in the space of modules, the divisor \(\Theta\) is geometrically integer and normal. The proof is technical and uses the reducible degenerate curves, the action of monodromy, the deformations of double ordinary points, the information on the Neron-Severi group of the Jacobian of the generic curve, and a formal GAGA property for a schema which is not necessarily proper.
Section 4 deals with the case where \(g\) is small or \(p=3\). If \(p=3\), it is shown that \(\Theta\) is reduced and contains no irreducible components -- translates of abelian subvarieties. If, in addition, \(g=2\) then \(\Theta\) is integer.
Finally, some applications of the divisor \(\Theta\) to the study of variation of the fundamental group of an algebraic curve are presented in Section 5. In particular, the author refines the recent result of \textit{A. Tamagawa} [J. Algebraic Geom. 13, No. 4, 675--724 (2004; Zbl 1100.14021)] on the specialization homomorphism between fundamental groups at least when the special fiber is supersingular. vector bundles; theta divisor; differential forms; algebraic curve; fundamental group Vector bundles on curves and their moduli, Coverings of curves, fundamental group, Picard schemes, higher Jacobians Theta divisors and differential forms | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians The Jacobian variety of a smooth plane cubic \(f=0\) over a field can be given in the form \(f^*=0\) with \(f^*\) an explicit Weierstrass form whose coefficients are homogeneous polynomials with integral coefficients in the coefficients of \(f\). \textit{A. Weil} treated the case of fields of characteristic not \(2\) and \(3\) [Arch. Math. 5, 197--202 (1954; Zbl 0056.03402)].
In the present paper this is done in the wider context of a family \(X\) of cubics in the projective plane \(\mathbb P^2\) over a base scheme \(S\). The authors give an explicit Weierstrass cubic equation \(f^*=0\) such that the smooth locus of the associated scheme represents \(\text{Pic}^0_{X/S}\). The proof uses that \(\text{Pic}^0_{X/S}\) satisfies a certain characterization of algebraic spaces that are commutative groups given by a Weierstrass equation. cubic curve; Jacobian; relative Picard scheme M. Artin, F. Rodriguez-Villegas and J. Tate, \textit{On the Jacobians of Plane Cubics}, \textit{Adv. Math.}\textbf{198} (2005) 366. Picard schemes, higher Jacobians, Jacobians, Prym varieties, Elliptic curves over global fields, Elliptic curves On the Jacobians of plane cubics | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians The following result is established: Every isomorphism between the Prym varieties of two admissible [in the sense of \textit{A. Beauville}, Invent. Math. 41, 149-196 (1977; Zbl 0333.14013)] coverings of stable plane curves of the same degree \(= 7\) (one of which satisfies a mild technical condition) is induced by an isomorphism between these coverings. - It follows that if two complete intersections of three quadrics of odd dimension are given, then any isomorphism between their generalized jacobians is induced, modulo sign, by an isomorphism between them. - This provides a definitive completion of the results of \textit{R. Friedman} and \textit{R. Smith} [Invent. Math. 85, 615-635 (1986; Zbl 0619.14027)] valid generically. The author's methods are constructive, and make use of an idea of Welters introduced for the classical Torelli theorem, and later used for Prym varieties [\textit{G. E. Welters}, Am. J. Math. 109, 165-182 (1987; Zbl 0639.14026)]. Prym varieties; coverings of stable plane curves; complete intersections of three quadrics; Torelli theorem Debarre, O, Le théorème de Torelli pour LES intersections de trois quadriques, Invent. Math., 95, 507-528, (1989) Jacobians, Prym varieties, Picard schemes, higher Jacobians, Torelli problem, Complete intersections, Riemann surfaces Le théorème de Torelli pour les intersections de trois quadriques. (Torelli theorem for intersections of three quadrics) | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians Let \(X\) be a smooth projective curve of genus \(g\geq 2\) defined over a number field \(K\). The \(p\)-adic method of Chabauty and Coleman gives an effective way of computing the finite set \(X(K)\), under some conditions, the main one being that the Mordell-Weil rank of the Jacobian be less than \(g\). Kim's non abelian approach replaces the Jacobian by the Selmer variety [\textit{M. Kim}, Invent. Math. 161, No. 3, 629--656 (2005; Zbl 1090.14006)]. In the paper under review, the authors introduce new techniques for studying Selmer varieties which enable them to produce new methods for determining the rational points of a variety over \({\mathbb{Q}}\) or over a quadratic field when the Mordell-Weil rank is equal to \(g\). The methods generalize those which were used for the study of integral points on hyperelliptic curves using \(p\)-adic heights in [\textit{J. S. Balakrishnan} et al., J. Reine Angew. Math. 720, 51--79 (2016; Zbl 1350.11067)]. They also use the results of the PhD Thesis of the second author [Topics in the theory of Selmer varieties. Oxford: Oxford University (2015)]. A crucial role in the proof is played by Nekovář's approach to the \(p\)-adic height pairing [\textit{J. Nekovář}, Prog. Math. 108, 127--202 (1993; Zbl 0859.11038)]. As an example, the authors show how to compute explicitly a finite set containing the rational points over \({\mathbb{Q}}\) or over a quadratic number field for a genus \(2\) bielliptic curve of Mordell-Weil rank \(2\). The example of \(X_0(37)\) over \({\mathbb{Q}}(i)\) is given; in an appendix to this paper, Stephen Müller carries out the Mordell-Weil sieve computations. Chabauty method; Chabauty-Coleman method; Chabauty-Kim method; Mordell's conjecture; Faltings's theorem; rational points; \(p\)-adic height; Selmer varieties; hyperelliptic curves; bielliptic curves; conjectures of Bloch and Kato; Nekovář's \(p\)-adic height pairing; mixed extensions; Mordell-Weil sieve J. S. Balakrishnan and N. Dogra, Quadratic Chabauty and rational points, I: \(p\)-adic heights, preprint, arXiv:1601.00388v2 [math.NT]. Rational points, Heights, Arithmetic varieties and schemes; Arakelov theory; heights Quadratic Chabauty and rational points. I: \(p\)-adic heights | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians In this doctoral dissertation, inspired by Professor Christopher Deninger (University of Münster, Germany) and Professor Annette Werner (University of Stuttgart, Germany), the author takes up the problem of describing vector bundles on certain algebraic curves over a non-Archimedean field by means of group representations. Whilst the classical case of vector bundles on a Riemann surface is well-understood, above all due to the fundamental work of \textit{A. Weil} (1938) and \textit{M. S. Narasimhan} and \textit{C. S. Seshadri} [Ann. Math. (2) 82, 540--567 (1965; Zbl 0171.04803)], the general case is still a subject of intensive contemporary research. During the past 20 years, some remarkable progress in the \(p\)-adic analysis for semistable vector bundles on a Mumford curve over a local number field has been achieved by \textit{G. Faltings} [Invent. Math. 74, 199--212 (1983; Zbl 0526.14018)] and, in the sequel, by \textit{M. van der Put} and \textit{M. Reversat} [Math. Ann. 273, 573--600 (1986; Zbl 0566.14008)]. Using the theory of formal schemes, Faltings constructed an equivalence of categories between semistable vector bundles of degree zero on a Mumford curve over a discrete non-Archimedean field and the so-called \(\Phi\)-bounded representations of its Schottky group, whereas M. van der Put and M. Reversat applied methods from rigid analytic geometry to generalize Faltings' result to arbitrary non-Archimedean fields. Their representations of the Schottky group (of the underlying Mumford curve) describing semistable vector bundles are very special, and will here be referred to as ``PW-representations''.
On the other hand, there is a very recent construction by \textit{Ch. Deninger} and \textit{A. Werner} [Ann. Sci. Éc. Norm. Supér., IV. Sér. 38, No.4, 553--597 (2005; Zbl 1087.14026)]. For any projective, smooth and geometrically connected algebraic curve over a finite field extension of \(\mathbb{C}_p\), where \(p\) denotes a prime number, these authors established a functor which, restricted to the algebraic fundamental group, associates a continuous \(\mathbb{C}_p\)-vector space representation to every semistable vector bundle of a certain, well-defined type. These representations, the so-called ``DW-representations'', have nice properties and, therefore, seem to be appropriate for the study of semistable vector bundles on a Mumford curve over a finite field extension of \(\mathbb{C}_p\) as well.
Now, with these developments in mind, the main topic of the thesis under review is a detailed comparison of the DW-representations attached to a certain class of semistable vector bundles of degree zero on a Mumford curve \(X_{\mathbb{C}_p}\) to the PR-representations defined for this class of vector bundles. The main result is that the DW-representation and the profinitely completed PR-representation attached to those vector bundles are isomorphic. The proof of this result is rather involved, as the functors of Deninger-Werner and van der Put-Reversat have different properties, and the author has to make extensive use of GAGA-type theorems in different geometries, including rigid analytic geometry, formal geometry, and algebraic geometry, in order to master all the technical difficulties.
These fundamental GAGA-type results (due to several authors) are reviewed in the first chapter of the present thesis, whereas the second chapter gives a detailed description of the functorial constructions of Faltings, Deninger-Werner and van der Put-Reversat in the author's special context.
The third chapter is dedicated of the above-mentioned main result, and the last chapter provides some concrete applications of it. The author studies various Galois groups of a Tate curve and investigates vector bundles on Tate curves and on Mumford curves of genus two as illustrating examples. algebraic curves; local ground fields; rigid analytic geometry; fundamental groups; formal methods Local ground fields in algebraic geometry, Rigid analytic geometry, Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20], Arithmetic ground fields for curves, Coverings of curves, fundamental group, Formal methods and deformations in algebraic geometry On representations attached to semistable vector bundles on Mumford curves | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians Suppose that \(S\) is a connected Dedekine scheme with \(\mathrm{dim}(S)=1\) and let \(\eta = \mathrm{Spec}(K)\) be its generic point. Let \(X\) be \(S\)-scheme that is faithfully flat of finite type. Let \(G\) be a finite \(K\)-group scheme and \(Y \to X_{\eta}\) a \(G\)-torsor. To extend the \(G\)-torsor \(Y \to X_{\eta}\) to \(X\) involves finding a finite flat \(S\)-group scheme \(G'\) with generic fibre \(G\), and a \(G'\)-torsor \(T \to X\) with generic fibre isomorphic to \(Y \to X_{\eta}\). \textit{A. Grothendieck} [Lecture Notes in Mathematics. 224. Berlin-Heidelberg-New York: Springer-Verlag. XXII, 447 p. (1971; Zbl 0234.14002)], \textit{M. Raynaud} [in: The Grothendieck Festschrift, Collect. Artic. in Honor of the 60th Birthday of A. Grothendieck. Vol. III, Prog. Math. 88, 179--197 (1990; Zbl 0722.14013)], \textit{M. Saïdi} [Manuscr. Math. 89, No. 2, 245--265 (1996; Zbl 0869.14010)] and the author have provided partial solutions before. In this paper, the author provides another possible solution to this problem as follows (Theorem 1.1):
Theorem. Let \(R\) be a complete discrete valuation ring with fraction field \(K\) and with algebraically closed residue field of characteristic \(p>0\). Let \(X\) be a smooth fibered surface over \(R\). Let \(G\) be a finite, étale and solvable \(K\)-group scheme. Then for every connected and pointed \(G\)-torsor \(Y\) over the generic fibre \(X_{\eta}\) of \(X\) there exists a regular fibered surface \(\tilde{X}\) over \(R\) and a model map \(\tilde{X} \to X\) such that \(Y\) can be extended to a torsor over \(\tilde{X}\) possibly after extending scalars in the following two cases: 1) \(|G| = p^n\); 2) \(G\) has a normal series of length \(2\).
To prove the theorem, first decompose \(Y \to X_{\eta}\) into a tower of quotient pointed \((\mathbb{Z}/p\mathbb{Z})_K\)-torsors after extending the scalars. The author (Theorem 3.12) has proved that when the Jacobian has abelian reduction, then the torsor can be extended to \(X\). If necessary, desingularize so that the torsor is regular (Theorem 3.8). Then proceed with induction. Antei, M., Extension of finite solvable torsors over a curve, Manuscripta Math., 140, 1, 179-194, (2013) Coverings of curves, fundamental group, Group schemes Extension of finite solvable torsors over a curve | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians In previous work [Ann. Inst. Fourier 55, No.~6, 1905--1941 (2005; Zbl 1095.14022)], the author extended cohomological obstruction calculus to group actions from smooth curves to stable curves. Here, he applies these results to the local structure of Hurwitz spaces, in particular to the stack of stable curves of given genus with faithful action of a given group in each fiber and semi-stable quotient of fixed genus. A local-global theorem for universal deformation rings is proven even in the case of a non-discrete set of fixed points, and some further interesting examples of quotients of versal deformation rings are calculated, in particular, for elements of order \(p\) acting in characteristic \(p\). group action, deformation theory, stable curve, Hurwitz space, stack Families, moduli of curves (algebraic), Coverings of curves, fundamental group, Arithmetic ground fields for curves Some results about equivariant deformations of stable curves | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians The study of the extrinsic geometry of the Jacobian locus \(\overline{J_g}\) in the moduli space \(A_g\) of principally polarized abelian varieties, namely of the closure of the locus of Jacobian varieties in \(A_g\), has been stimulated by different many problems throughout its history (two of the best known are the Torelli and Schottky problems). In particular, the theory of generalized Prym varieties as developed by [\textit{A. Beauville}, Invent. Math. 41, 149--196 (1977; Zbl 0333.14013)] highlighted a natural inclusion of the Jacobian locus inside boundary of the Prym locus \(\overline{P_{g+1}}\), which is the closure in \(A_g\) of the locus of Prym varieties associated to étale coverings of degree \(2\) onto smooth projective curves of genus \(g+1\). In terms of the study of the extrinsic geometry of \(\overline{J_g}\subset A_g\) such a theory suggested a refined study of \(\overline{J_g}\subset \overline{P_{g+1}}\) by using the parametrization of Jacobian varieties as generalized Prym varieties.
Let \(M_g\) be the moduli space of curves of genus \(g\) and let \(R_{g+1}\) be the moduli space of degree \(2\) étale coverings of curves of genus \(g+1\). For any \([C]\in M_g\), let \(JC\) be the Jacobian variety and let \(\mathrm{Cliff }C\) denote the Clifford index of \(C\) (see [\textit{E. Arbarello} et al., Geometry of algebraic curves. Volume II. With a contribution by Joseph Daniel Harris. Berlin: Springer (2011; Zbl 1235.14002)]). For any \(\zeta \in T_{[JC]}J_g\), let \(\xi \in H^1(C, T_C)\) such that \(\zeta \) can be identified with the cup product map \(\cup \xi: H^0(C, \omega_C)\to H^0(C, \omega_C)^\vee\), under the isomorphism \(T_{[JC]}A_g\simeq \mathrm{Sym}^2H^0(C, \omega_C)^\vee\). We define the rank of \(\zeta\) as the rank of \(\cup \xi\). The following is one of the main results of the paper.
Theorem 1. Let \(JC\) be a general Jacobian variety of dimension \(g\geq 7\). Then for any \(\zeta\in T_{[JC]}J_g\) of rank \(k=rk\zeta <\mathrm{Cliff }C-3\), the local geodesic in \(A_g\) at \([JC]\) with direction \(\zeta\) (defined with respect to the Siegel metric) is not contained in the Prym locus \(\overline{P_{g+1}}\) (in particular, also in \(\overline{J_g}\subset \overline{P_{g+1}}\)).
Assume \(g\geq 7\).
Let \(M_g\) be the moduli space of curves of genus \(g\) and let \(A_g\) be the moduli space of principally polarized abelian varieties.
The Torelli morphism
\[
j: M_g\to A_g , \quad [C]\mapsto [JC]
\]
sends a genus \(g\) smooth projective curve \(C\) to its Jacobian variety \(JC\) (up to isomorphism). Its image \(J_g=j(M_g)\subset A_g\) defines a proper locus for \(g>3\) and its closure \(\overline{J_g}\subset A_g\) is called the Jacobian locus. By using the isomorphisms
\[
T_{[C]}M_g\simeq H^1(C,T_C), \quad T_{[JC]}A_g\simeq \mathrm{Sym}^2H^0(C, \omega_C)^{\vee},
\]
the differential of the Torelli map
\[
dj:H^1(C,T_C)\to \mathrm{Sym}^2H^0(C, \omega_C)^{\vee},\quad \xi \mapsto \zeta
\]
is given by \(\zeta=\cup\xi:H^0(C, \omega_C)\to H^0(C,\omega_C)^\vee,\) the cup product map with \(\xi.\)
Let \(R_{g+1}\) be the moduli space parametrizing \(2\)-sheeted étale coverings \(\pi: \tilde{C}\to C'\) between smooth projective curves of genus \(\tilde{g}=g(\tilde{C})=2g(C')-1\) and \(g'=g(C')=g+1\), respectively (modulo isomorphism). A point in \(R_{g+1}\) is an isomorphism class assigned equivalently by
\begin{itemize}
\item[(i)] a pair \((\tilde{C}, i)\), where \(i:\tilde{C}\to \tilde{C}\) is an involution such that \(\tilde{C}/(i)=C'\),
\item[(ii)] a pair \((C', \eta)\), where \(\eta \in \mathrm{Pic}^0(C')\setminus \{\mathcal O_{C'}\}\) is a \(2\)-torsion point and \(\tilde{C}= \mathrm{spec} (\mathcal O_C\oplus \eta)\).
\end{itemize}
One can reconstruct the data \((i)\) from \((ii)\) and conversely (see [\textit{A. Beauville}, Invent. Math. 41, 149--196 (1977; Zbl 0333.14013)], [\textit{D. Mumford}, in: Contribut. to Analysis, Collect. of Papers dedicated to Lipman Bers, 325--350 (1974; Zbl 0299.14018)]) and so with a little abuse of notations we will describe a point of \(R_{g+1}\) in both ways depending on the setting.
The Prym morphism
\[
Pr: R_{g+1}\to A_g, \quad [(C', \eta)]\mapsto [P(C',\eta)]
\]
maps a pair \((C', \eta)\) to its Prym variety \(P(C',\eta)\)(up to isomorphisms).
By definition, the Prym variety is \(P(C',\eta)=\ker Nm^0,\) namely the connected component containing zero in the kernel of the norm map \(Nm: J\tilde{C}\to JC'\), and it is a principally polarized abelian variety of dimension \(g\), with the principal polarization given by one half of the restriction of that on \(J\tilde{C}\).
The image \(P_{g+1}=Pr(R_{g+1})\subset A_g\) of the Prym morphism defines a proper locus for \(g\geq 6\) and its closure \(\overline{P_{g+1}}\) is called the Prym locus.
By construction, the forgetful functor \(\pi_{R_{g+1}}: R_{g+1}\to M_{g+1},\) sending \([(C',\eta)]\mapsto [C']\), is an étale finite covering and so we can identify \(T_{[(C', \eta)]}R_{g+1}\simeq T_{[C']}M_g.\) Using the isomorphisms
\[
T_{[C', \eta]}R_{g+1}\simeq H^1(C', T_{C'}), \quad T_{[P(C, \eta)]}A_g\simeq \mathrm{Sym}^2H^1(C',\eta)\simeq \mathrm{Sym}^2 H^0(C', \omega_{C'}\otimes \eta)^{\vee},
\]
the differential of the Prym map
\[
d Pr:H^1(C', T_{C'})\to \mathrm{Sym}^2H^1(C',\eta), \quad \xi' \longmapsto \zeta',
\]
is given by \(\zeta'=\cup\xi':H^0(C', \omega_{C'}\otimes \eta)\to H^0(C', \omega_{C'}\otimes \eta)^{\vee}\), the cup product with \(\xi'.\)
Let \(C\) be a smooth projective curve. The gonality and the Clifford index of \(C\) are defined as
\begin{align*}
&\operatorname{gon} C= \min\{n\in\mathbb N \,| \, C \mbox{ has a } g^1_n\} ;\\
&\mathrm{Cliff }C = \min\{\mathrm{Cliff }D=\deg D- 2 (h^0(D)-1)\, | \, D\subset C \mbox{ divisor}, h^0(D)\geq 2, h^1(D)\geq 2 \}.
\end{align*}
By [\textit{M. Coppens} and \textit{G. Martens}, Compos. Math. 78, No. 2, 193--212 (1991; Zbl 0741.14035)], these are related by
\[
\operatorname{gon} C -3 \leq\mathrm{Cliff }C \leq \operatorname{gon} C- 2 ,
\]
where the second inequality is an equality on a general \(C\) in \(M_g\), while the first one is conjectured to be extremely rare [\textit{D. Eisenbud} et al., Compos. Math. 72, No. 2, 173--204 (1989; Zbl 0703.14020)]. The followings are well known.
\begin{itemize}
\item[1.] \(\operatorname{gon} C \leq \left\lfloor \frac{g+3}{2}\right \rfloor\) (and so \(\mathrm{Cliff} C \leq \left\lfloor \frac{g-1}{2}\right \rfloor\)) and the equality holds for a general curve \([C]\in M_g\) (see [\textit{E. Arbarello} et al., Geometry of algebraic curves. Volume II. With a contribution by Joseph Daniel Harris. Berlin: Springer (2011; Zbl 1235.14002)]);
\item[2.] let \(f:\mathcal C\to B\) be a fibration of projective curves over a complex curve \(B\) and such that the general fibre is smooth. The invariants \(\operatorname{gon} C_b \) and \(\mathrm{Cliff} C_b\) are maximal on a general fibre \(C_b\) of the set of smooth fibres ([\textit{K. Konno}, J. Algebr. Geom. 8, No. 2, 207--220 (1999; Zbl 0979.14004)]).
\end{itemize}
An \textit{admissible covering} of degree \(k\) with \(m\) ramification points is the data of
\begin{itemize}
\item[1.] a stable \(m\)-pointed reduced connected curve \((E, x_1, \dots ,x_m)\) of arithmetic genus \(0\) (i.e. a curve with ordinary double points where any rational component is smooth and stable and the dual graph is connected);
\item[2.] a reduced connected curve \(X\) with ordinary double points and a morphism \(\pi : X \to E\) of degree \(k\) (everywhere) such that over any marked point \(x_i\) of \(E\), \(X\) is smooth and \(f\) has a unique simple ramification point \(y_i\), on any smooth point of \(E\), \(f\) is étale and over double points \(p\) of \(E\), \(X\) has an ordinary double point \(q\) and \(f\) is locally described as
\[
X\,:\, xy=0; \quad E\,:\, uv=0;\quad \, f\,:\, u=x^{k'}\quad ,\, v= y^{k'},
\]
for some \(k'\leq k\).
\end{itemize}
We have the following
Lemma. Let \(C'\) be a stable nodal curve of genus \(g'\), let \(\nu': C^{\nu'}\to C'\) be its normalization and let \(C\subset C^{\nu'}\) be a smooth connected curve of genus \(g\). Consider \(f':\mathcal C'\to \Delta\), a family of smooth projective curves of genus \(g'\) over \(\Delta \setminus \{0\}\), the complex disk minus zero, such that \(C'=f^{-1}(0)\). Then any family of pencils \(g^1_k(t)\) over \( \Delta\setminus\{0\}\) compatible with \(f\) (i.e. \(g^1_k(t)\) is a pencil on \(C'_t=f^{-1}(t)\)) defines a pencil \(g^1_{k'}\) on \(C\) for some \(k'\leq k\). In particular, \(\operatorname{gon} C\leq \operatorname{gon} C'_t\) and \(\mathrm{Cliff }C\leq \mathrm{Cliff }C'_t+1\), for a general fibre \(C'_t=f^{-1}(t)\).
Let \(Y\) be a smooth complex variety and let
\((\mathbb H_{\mathbb Z}, \mathcal H^{1,0}, \mathcal Q) \) be a polarized variation of Hodge structures (pvhs, in short) of weight 1 on \(Y\). Namely, \(\mathbb H_Z\) is a local system of lattices, \(\mathcal H^{1,0}\) is a Hodge bundle of type \((1,0)\) (equivalently, the Hodge filtration in this case) and \(\mathcal Q\) is a polarization. Let \(\mathbb H_{\mathbb C}=\mathbb H_{\mathbb Z}\otimes_{\mathbb Z}\mathbb C\) and let \(\mathcal H=\mathbb H_{\mathbb C} \otimes \mathcal O_Y\) be the holomorphic flat bundle with the holomorphic flat connection \(\nabla\) defined by \(\ker \nabla \simeq \mathbb H_{\mathbb C}\), the Gauss-Manin connection. The holomorphic inclusion \(\mathcal H^{1,0}\subset\mathcal H\) of vector bundles induces the short exact sequence
\[
\begin{tikzcd}
0 \arrow{r} & \mathcal H^{1,0} \arrow{r} & \mathcal H \arrow{r}{\pi^{{0,1}}} &
\mathcal H/\mathcal H^{1,0} \arrow{r}& 0.
\end{tikzcd}
\]
Let \(\pi^{{0,1}'}: \mathcal H\otimes \Omega^1_Y\to\mathcal H/\mathcal H^{1,0}\otimes \Omega^1_Y\) be the map induced by \(\pi^{0,1}\) and \(\sigma =\pi^{{0,1}'}\circ \nabla: \mathcal H^{1,0}\to\mathcal H/\mathcal H^{1,0}\otimes \Omega_Y^1\) the second fundamental form of \(\mathcal H^{1,0}\subset\mathcal H\) with respect to \(\nabla\).
Following [\textit{A. Ghigi} et al., Commun. Contemp. Math. 23, No. 3, Article ID 2050020, 13 p. (2021; Zbl 1455.14013)], let \(\mathbb U=\ker \nabla_{|\mathcal H^{1,0}}\) and define
\[
\mathcal U:=\mathbb U\otimes
\mathcal O_Y,\quad \quad
\mathcal K:=\ker ( \sigma : \mathcal H^{1,0}\longrightarrow\mathcal H /\mathcal H^{1,0} \otimes
\Omega_Y^1).
\]
Then \(\mathcal U\) is a holomorphic vector bundle and \(\mathcal K\) is a coherent sheaf which is a vector bundle when \(\sigma\) is of constant rank.
Definition 1. We call \(\\mathcal U\) and \(\mathcal K\) as defined in (1), \textit{the unitary flat bundle} and \textit{the kernel sheaf} of the variation, respectively
Proposition 1. We have \( \mathcal U\subset \mathcal K\) and if \(\tau\equiv 0\), then \(\mathcal U=\mathcal K\).
For any \(B\subset A_g\) smooth complex curve, let \(f: \mathcal A\to B\) be the family of abelian varieties (defined up to finite base change) and let \(A_b\) be the fibre over \(b\in B\). Then the p.v.h.s. is defined by
\[
\mathbb H_{\mathbb Z}\simeq R^1f_\ast \mathbb Z, \quad \mathcal H^{1,0}= f_\ast \Omega^1_{\mathcal A/B}\subset \mathcal H= R^1f_\ast \mathbb C\otimes \mathcal O_{B};
\]
\[
\mbox{where }\quad (R^1f_\ast \mathbb Z)_b\simeq H^1(A_b,\mathbb Z), (f_\ast\Omega^1_{\mathcal A/B })_b\simeq H^0(A_b, \Omega^1_{A_b}), {(R^1f_\ast \mathbb C\otimes \mathcal O_{B})}_b\simeq H^1(A_b, \mathbb C).
\]
Let \(H_g\) denote the Siegel upper half space. As a symmetric space of non-compact type it is endowed by a symmetric metric \(h^s\), called the Siegel metric, defining a metric connection \(\nabla^{LC}\) on the tangent bundle \(TH_g\). As a parametrizing space of weight \(1\) p.v.h.s., it carries a universal p.v.h.s. \((\mathbb H_{\mathbb Z},\mathcal H^{1,0}, \mathcal Q)\) with its Hodge metric defined by \(Q\), inducing a metric \(h\) together with a metric connection \(\nabla^{hdg}\) on \(\mathrm{Hom} (\mathcal H^{1,0}, \mathcal H/\mathcal H^{1,0})\). There is a natural inclusion
\[
(TH_g, \nabla^{LC})\subset (\mathrm{Hom} (\mathcal H^{1,0}, \mathcal H/\mathcal H^{1,0}), \nabla^{hdg})
\]
compatible with the metric structure (see e.g. a classical reference [\textit{P. A. Griffiths}, in: Actes Congr. internat. Math. 1970, 1, 113--119 (1971; Zbl 0227.14008)] or some more recent references [\textit{A. Ghigi}, Boll. Unione Mat. Ital. 12, No. 1--2, 133--144 (2019; Zbl 1444.14028)], [\textit{A. Ghigi} et al., Commun. Contemp. Math. 23, No. 3, Article ID 2050020, 13 p. (2021; Zbl 1455.14013)]).
Consider the universal covering \(\psi: H_g\to A_g\) and the metric properties introduced before on \(H_g\). Let \([A]\in A_g\) and let \(\zeta\in T_{[A]}A_g\).
Take \(\tilde{A}\in \psi^{-1}([A])\) and consider \(\zeta\) as \(\zeta \in T_{\tilde{A}}H_g\simeq T_{[A]}A_g\). Then in \(H_g\) a (local) geodesic at \((\tilde{A}, \zeta)\) is simply a curve \(\gamma:(-\epsilon, \epsilon)\to H_g\) such that \(\gamma(0)=\tilde{A}\) and \(\gamma'(0)=\zeta\) satisfying \(\nabla^{LC}_{\gamma'}\gamma'=0\). Working locally, we can assume w.l.o.g that \(\gamma((-\epsilon, \epsilon))\) is contained in one sheet of \(\psi\).
Definition 2. Let \([A]\in A_g\) and let \(\zeta\in T_{[A]}A_g\)
A (local) geodesic associated to \(([A], \zeta)\) is a map \(\psi\circ \gamma: (-\epsilon, \epsilon)\to A_g\), where \(\gamma\) is a local geodesic in \(H_g\) defined as above.
We are interested in points of \(J_g\subset A_g\) and directions in \(T_{[A]}J_g\subset T_{[A]}A_g\). If \([A]=[JC]\), namely the Jacobian of some \([C]\in M_g\), and \(\zeta = \cup \xi\), for some \(\xi \in H^1(C, T_C)\), under the isomorphisms \(T_{[C]}M_g\simeq H^1(C, T_C)\) and \(T_{[J_g]}A_g\simeq \mathrm{Sym}^2H^1(C, \mathcal O_C)\simeq \mathrm{Sym}^2H^0(C, \omega_C)^\vee\), we will also refer to the geodesic at \((JC, \zeta)\) as the geodesic at \((C, \xi)\). This is admitted since the Torelli map is an immersion outside the Hyperelliptic locus and we are not considering hyperelliptic curves.
We have the following (see [\textit{A. Ghigi} et al., Commun. Contemp. Math. 23, No. 3, Article ID 2050020, 13 p. (2021; Zbl 1455.14013), Lemma 3.3 and Lemma 3.4] for the proof)
Lemma 2. Let \(\gamma:(-\epsilon, \epsilon)\to A_g\) be a local geodesic associated to \(([A], \zeta)\). Then there exists a complex curve \(B\subset H_g\) containing the geodesic and such that \(\mathcal K =\mathcal U\).
We can shrink \(B\) around \(\gamma((-\epsilon, \epsilon))\) in such a way that it is contained in one sheet on \(\psi\) and so we can then look at it as a curve in \(A_g\).
Let \(C\) be a smooth projective curve, let \(\mathcal F\) be a rank \(2\) vector bundle over \(C\) and let \(\alpha: \bigwedge^2H^0(C,\mathcal F)\to H^0(C,\det \mathcal F)\) be a linear map. A subspace \(W\subset H^0(C,\mathcal F) \) is called \textit{isoptropic} with respect to \(\alpha\) if \(\alpha_{|\bigwedge^2 W}\equiv 0\).
Let \(\mathcal L, \mathcal L'\) be two line bundles on \(C\), let \(0\to\mathcal L \to\mathcal F \to\mathcal L'\to 0\) be a s.e.s. associated to \(\xi\in \mathrm{Ext}^1_{\mathcal O_C}(\mathcal L', \mathcal L)\) and let \(\alpha: \bigwedge^2H^0(C,\mathcal F)\to H^0(C,\det \mathcal F)\simeq H^0(C,\mathcal L\otimes\mathcal L')\). A subspace \(V\subset H^0(C,\mathcal L')\) is called \textit{isotropic} with respect to \(\alpha\) if it lifts to a subspace \(W\subset H^0(C,\mathcal F)\) isotropic with respect to \(\alpha\).
Note that a subspace \(V \subset H^0(C,\mathcal L')\) lifts to \(W\subset H^0(C,\mathcal F)\) if and only if it lies in the kernel of the coboundary morphism \(\delta: H^0(C,\mathcal L')\to H^1(C,\mathcal L)\) on the long exact sequence in cohomology.
Theorem 2. Let \(f: \mathcal C\to B\) be a fibration of smooth projective curves over a smooth complex curve \(B\) and let \(\mathcal U\subset f_\ast \omega_{\mathcal C/B}\) be the associated unitary flat bundle (Definition 1). Assume that the fibres \(U_b\subset H^0(\omega_{C_b})\) are isotropic subspaces with respect to \(\alpha_b\) for any \(b\in B\). Then (up to a finite base change) there exists a smooth projective curve \(\Sigma\) of genus \(g'=rk \mathcal U\) and a non constant fibre-preserving map \(\varphi: \mathcal C \to \Sigma\) such that \(U_b\simeq \varphi^\ast H^0(\Sigma,\omega_{\Sigma})\). moduli space of curves and abelian varieties; geodesics; Prym locus; Jacobian locus; generalized Prym varieties; admissible coverings Jacobians, Prym varieties, Transcendental methods, Hodge theory (algebro-geometric aspects), Families, moduli of curves (algebraic), Families, moduli of curves (analytic), Period matrices, variation of Hodge structure; degenerations, Subvarieties of abelian varieties On the Jacobian locus in the Prym locus and geodesics | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians Let \(X/k\) be a smooth, geometrically connected, projective curve over a finite field \(k\), and let \(\mathcal{G}\) be a semisimple group scheme over \(X\) with generic fiber \(G\). This paper proves a formula relating the Tamagawa number \(\tau(G)\) (an arithmetic invariant) to the number of connected components of Bun\(_{\mathcal{G}}\), the moduli stack of \(\mathcal{G}\)-torsors on \(X\). In particular (Theorem 3.3), these two numbers are equal when \(k\) is large enough, the Tamagawa number of the universal cover of \(G\) is \(1\), and \(G\) satisfies the Hasse principle. Furthermore, the connected components of Bun\(_{\mathcal{G}}\) are geometrically connected. The necessary assumptions are satisfied when \(G\) is a Chevalley group.
The proof begins by exhibiting a bijection between isomorphism classes of \(\mathcal{G}\)-torsors over \(X\) with certain double cosets of the adelic points of \(G\) (the adeles being taken with respect to the function field of \(X\)). This allows one to relate \(\tau(G)\) to the number of \(k\)-points, appropriately counted, of Bun\(_{\mathcal{G}}\). This in turn can be computed, via a generalized Lefschetz trace formula, by computing the trace of Frobenius on the étale cohomology of Bun\(_{\mathcal{G}}\). Two important results of the paper are that these cohomology vector spaces are finite dimensional and that the absolute values of the eigenvalues of Frobenius on the \(i\)th cohomology group are bounded by \(|k|^{-i/2}\). Additionally, the paper proves, using a formula of Ono, that under the hypotheses of the main theorem, the Tamagawa numbers of \(G\) are eventually stable for larger and larger base changes of \(G\) by constant field extensions. Putting all these facts together, the paper relates \(\tau(G)\) directly to the trace of powers of Frobenius on \(H^0(\text{Bun}_{\mathcal{G}}, \mathbb{Q}_{\ell})\), which proves the theorem. Tamagawa number; Chevalley group; trace formula; G-bundles K. Behrend and A. Dhillon, ''Connected components of moduli stacks of torsors via Tamagawa numbers,'' Canad. J. Math., vol. 61, iss. 1, pp. 3-28, 2009. Finite ground fields in algebraic geometry, Galois cohomology of linear algebraic groups, Galois cohomology, Algebraic moduli problems, moduli of vector bundles, Vector bundles on curves and their moduli, Group schemes Connected components of moduli stacks of torsors via Tamagawa numbers | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians In an important paper [Invent. Math. 181, No. 3, 449--465 (2010; Zbl 1203.14029)], the first author and \textit{V. Mehta} proved a conjecture of Gieseker according to which for smooth projective varieties \(X\) over an algebraically closed field of positive characteristic triviality of the étale fundamental group implies triviality of the Tannakian fundamental group classifying stratified vector bundles on \(X\).
Here the authors prove a relative version: given a morphism \(Y\to X\) of smooth projective varieties over an algebraically closed field of positive characteristic inducing the trival map on étale fundamental groups, the induced map on the fundamental groups classifying stratified vector bundles is trivial as well.
Using Lefschetz theorems, the proof proceeds by reduction to the case where \(Y\) is a curve and \(X\) is a surface. As in the paper of Esnault and Mehta, a key tool then is an application, after specializing to \(\overline{\mathbf F}_p\), of a theorem of Hrushovski about the density of periodic points to the Frobenius map on certain subschemes of moduli spaces of vector bundles on \(X\). stratified bundle; fundamental group Vector bundles on surfaces and higher-dimensional varieties, and their moduli, Homotopy theory and fundamental groups in algebraic geometry A relative version of Gieseker's problem on stratifications in characteristic \(p>0\) | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians Form the author's introduction: ``Some of the most beautiful chapters of pure mathematics of the 20th century were motivated by considerations around Weil conjectures for zeta functions of algebraic varieties over finite fields. The purpose of these notes is to introduce the reader to enough algebraic geometry to be able to understand some elementary aspects of their proof for the case of algebraic curves and appreciate their number-theoretic consequences.''
``We focus on the problem of counting points on varieties over finite fields and the associated Weil conjectures in Section 4. We prove the rationality conjecture in an elementary way, as well as using ètale cohomology as a `black box', and we outline the approach to the functional equation.'' Weil conjectures History of algebraic geometry, History of number theory, History of mathematics in the 20th century, Langlands-Weil conjectures, nonabelian class field theory, Langlands-Weil conjectures, nonabelian class field theory, Zeta functions and related questions in algebraic geometry (e.g., Birch-Swinnerton-Dyer conjecture) Algebraic geometry | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians We establish a Lefschetz hyperplane theorem for the Berkovich analytifications of Jacobians of curves over an algebraically closed non-Archimedean field. Let \(J\) be the Jacobian of a curve \(X\), and let \(W_d \subset J\) be the locus of effective divisor classes of degree \(d\). We show that the pair \((J^{an},W_d^{an})\) is \(d\)-connected, and thus in particular the inclusion of the analytification of the theta divisor \(\Theta^{an}\) into \(J^{an}\) satisfies a Lefschetz hyperplane theorem for \(\mathbb{Z}\)-cohomology groups and homotopy groups. A key ingredient in our proof is a generalization, over arbitrary characteristics and allowing arbitrary singularities on the base, of a result of Brown and Foster for the homotopy type of analytic projective bundles. Rigid analytic geometry, Jacobians, Prym varieties A Lefschetz hyperplane theorem for non-Archimedean Jacobians | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians The author proposes a generalization of the Deligne-Grothendieck- MacPherson theory of Chern classes for possibly singular complex algebraic varieties. The article doesn't contain proofs and more details are announced for later. Let \({\mathcal V}\) be the category of compact complex algebraic varieties and \({\mathcal A}\) the category of abelian groups. Let F be the covariant functor from \({\mathcal V}\) to \({\mathcal A}\) whose value on a variety is the abelian group of constructible functions from that variety to \({\mathbb{Z}}\). Let \(H_*(V,{\mathbb{Z}})\) (resp. \(H_*(V,{\mathbb{Z}})[t])\) be the usual \({\mathbb{Z}}\)-homology (resp. \({\mathbb{Z}}[t]\)-homology) and \(H_*(-,{\mathbb{Z}})\) (resp. \(H_*(- ,{\mathbb{Z}}[t])\) the corresponding functor. One denotes by \(C_*\) the natural transformation from F to \(H_*(-,{\mathbb{Z}})\) which was defined by \textit{R. D. MacPherson} in Ann. Math., II. Ser. 100, 423-432 (1974; Zbl 0311.14001). The main result announced in this paper is the following theorem 1:
There exists a covariant functor \(F^ t\) from \({\mathcal V}\) to \({\mathcal A}\) such that:
1. \(F^ t(V)=F(V)\otimes_{{\mathbb{Z}}}{\mathbb{Z}}[t]\) and \(F^ 1=F;\)
2. There exists a unique natural transformation \(C_{t*}\) from \(F^ t\) to \(H_*(-,{\mathbb{Z}})[t]\) such that
(a) If V is smooth, then \(C_{t*}(V)(1_ V)=c_ t(TV)\cap [V]\) where \(1_ V\) is the characteristic function of V and \(c_ t(TV)\) is the Chern polynomial of the tangent bundle TV.
(b) \(C_{1*}\) coincides with the natural transformation \(C_*\). Chern classes for possibly singular complex algebraic varieties Yokura, S., On a generalization of MacPherson's Chern homology class, Proc. Japan Acad., 65, Ser. A (1989), 242-244. (Co)homology theory in algebraic geometry, Characteristic classes and numbers in differential topology, Natural morphisms, dinatural morphisms, Transcendental methods of algebraic geometry (complex-analytic aspects) On a generalization of MacPherson's Chern homology class | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians Let \(C\) be a complete nonsingular curve of genus \(g\) over an algebraically closed field \(k\), and \({\mathcal L}\) an invertible sheaf over \(C\) of positive degree. Generalising a construction of \textit{J. Wahl} [J. Differ. Geom. 32, No. 1, 77-98 (1990; Zbl 0724.14022)] the author defines a map
\[
\Phi^{(n)} : \bigwedge^n \Gamma (C, {\mathcal L}) \to \Gamma (C, \omega_C^{ \otimes 1/2n (n - 1)} \otimes {\mathcal L}^{\otimes n})
\]
whose surjectivity is related to linear normality of generalised dual curves of \(C\). His main theorem states that if \(\text{char} k > \deg {\mathcal L} > (g - 1) (2n^2 - 2n + 3) + 2(n^2 - 1)\) then \(\Phi^{(n)}\) is surjective.
The proof seems to be similar to arguments in Wahl's paper; the presentation of the paper is rather formal. Wahl map; dual curves Vector bundles on curves and their moduli, Embeddings in algebraic geometry On the higher Wahl maps | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians In this doctoral thesis the author proves sufficient criteria for the representability of the Weil-Restriction functor, on the one hand for the category of adic spaces after R. Huber, on the other hand for the category of analytic spaces after V. Berkovich. As an application he proves that adic Néron models with base change relative to a finite Galois extension of degree prime to the residue characteristic are compatible. Huber space; Berkovich space; Néron models; Weil-Restriction functor; base change Varieties over finite and local fields, Rigid analytic geometry Weyl restriction technique for Huber and Berkovich spaces and applications to adic Néron models | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians This comprehensive memoir provides a profound contribution to the currently developing topic of homotopical and higher categorical structures in algebraic geometry. Its main purpose is to generalize the concept of affine schemes to an adequate construction in homotopical algebraic geometry, namely to the concept of affine stacks, and to show how these new objects can be used to treat several questions in rational and \(p\)-adic homotopy theory from a novel point of view.
One of the motivating ideas of the present work can be traced back to A. Grothendieck's monumental, visionary manuscript ``Pursuing Stacks'' (unpublished) from the early 1980s. In this famous program, Grothendieck sketched a problem which he called the ``schematization problem for homotopy types'' stating that for any affine scheme \(\text{Spec\,}A\) there should be an appropriate notion of ``\(\infty\)-stack in groupoids over \(\text{Spec\,}A\)'' generalizing all sheaves and stacks in groupoids over the grand fpqc-site of \(\text{Spec\,}A\). Later on, it was shown by \textit{A. Joyal} (Letter to Grothendieck, 1984, unpublished) and by \textit{J. F. Jardine} [Homology Homotopy Appl. 3, No. 2, 361--384 (2001; Zbl 0995.18006)] that an adequate model for a theory of \(\infty\)-stacks in groupoids would be given by the theory of simplicial presheaves. According to this fundamental insight, the word ``stack'' in the present memoir, is used for an object in the homotopical category of simplicial sheaves (à la A. Joyal and J. F. Jardine).
After a thorough introduction to the contents of the present treatise, including a historical sketch of the developments leading to its subject, explanations of the basic conceptual framework, and an, outline of the links to related works by other researchers in the field. Section 1 recalls the fundamentals from the theory of simplicial presheaves on a Grothendieck site, that is from the theory of general stacks in the sense made precise above. Apart from an introduction to the basic definitions and results, homotopic limits, Postnikov decompositions, the cohomology of simplicial presheaves, and schemes in affine groups are the main topics of this preparatory section.
Section 2 introduces the first of the two novel fundamental notions of the memoir under review: affine stacks. These objects appear as a homotopic version of ordinary affine schemes, obtained from a model category with simplicial structure over the category of co-simplicial \(A\)-algebras. This model category is then used to define a derived functor of the Spec-functor, which in turn induces a certain functor from the homotopical category of co-simplicial \(A\)-algebras to the homotopical category of simplicial sheaves over the site \((\text{Aff}/A)_{\text{fpgc}}\). The category of affine stacks is then defined to be the essential image category of the latter functor. In the sequel, it is proved that the category of affine stacks is equivalent to the opposite category of the homotopical category of co-simplicial \(A\)-algebras, thereby generalizing the analoguous property of the category of ordinary affine schemes.
Finally, the author invents another important construction, namely that of the ``affinization of a simplicial presheaf'', which appears to be significant for the study of homotopy sheaves, and he gives a concrete criterion for the existence of affinizations. In the special case of a base scheme \(\text{Spec\,}k\), where \(k\) is a field, the affine stacks are completely characterized, and analogues of the standard theorems on rational and \(p\)-adic homotopy of algebraic varieties are deduced by using affine stacks.
Section 3 deals with the second crucial novelty provided by the author's work. More precisely, he introduces the notion of so-called ``affine \(\infty\)-gerbes'' and the concept of ``schematic homotopy type''. The underlying idea is to use affine stacks in order to define a homotopical version of affine gerbes in the Tannakian formalism (à la P. Deligne). This is done by glueing affine stacks to obtain so-called ``\(\infty\)-geometric stacks'', which may be seen as a generalization of ordinary algebraic stacks assed in algebraic moduli theory and non-abelian Hodge theory (à la C. Simpson). This framework is then applied to give two different solutions to A. Grothendieck's ``schematization problem for homotopy types of algebraic varieties'' mentioned above. In this context, homotopy types with respect to various cohomology theories (Betti, de Rham, crystalline, \(\ell\)-adic, etc.) are described in greater detail.
The concluding Section 4 is exclusively devoted to the study of ``\(\infty\)-geometric stacks'' over a ground field \(k\), with applications to the construction of analogues of some moduli stacks in this extended homotopical context.
In an appendix of the present memoir, the author recalls A. Grothendieck's ``schematization problem for homotopy types'', together with his interpretation and reflection of Grothendieck's vague sketches.
Without any doubt, this memoir is of utmost fundamental and propelling character with regard to further developments in homotopical algebraic geometry. A wealth of significant new concepts, methods, and applications is presented in a very detailed and lucid manner, and an important new point of view towards Grothendieck's program of pursuing stacks is strikingly exhibited. homotopical algebraic geometry; gerbes; Grothendieck topologies; cohomology theories; simplicial sheaves; model categories Toën, Bertrand, Champs affines, Selecta Math. (N.S.), 1022-1824, 12, 1, 39-135, (2006) Generalizations (algebraic spaces, stacks), Classical real and complex (co)homology in algebraic geometry, Homotopy theory and fundamental groups in algebraic geometry, Étale and other Grothendieck topologies and (co)homologies, Presheaves and sheaves, stacks, descent conditions (category-theoretic aspects), Grothendieck topologies and Grothendieck topoi, Topoi Affine stacks | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians This paper gives an equivalence between the derived category of sheaves on an elliptic threefold \(X \to S\) without a section (here called a ``generic elliptic threefold'') and a derived category of twisted sheaves on any analytic small resolution \(\overline{J}\) of its relative Jacobian \(J \to S\). The twist depends on an element of the Brauer group of \(\overline{J}\) which is the obstruction to gluing universal sheaves on patches of \(X\) arising from open subsets of \(\overline{J}\). The replacement of \(J\) by the analytic resolution and the twisting together give a substitute construction \(\mathcal{U}\) for a universal sheaf on \(X \times_{S} J\), which does not exist in the situation considered by the author. The equivalence of derived categories is achieved by extending \(\mathcal{U}\) by \(0\) along a certain inclusion. sheaves on an elliptic threefold; relative Jacobian; Brauer group; derived category of sheaves Căldăraru, A., Derived categories of twisted sheaves on elliptic threefolds, J. Reine Angew. Math., 544, 161-179, (2002) \(3\)-folds, Derived categories, triangulated categories, Brauer groups of schemes, Vector bundles on surfaces and higher-dimensional varieties, and their moduli Derived categories of twisted sheaves on elliptic threefolds | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians The present paper generalizes a result of de Jong and Oort concerning the extension of a morphism (toward a moduli stack of stable curves) from \(X\setminus D\) to \(X\) with \(D\subseteq X\) being a NC divisor.
The new result is formulated in terms of regular log schemes: If \((X,M)\) is such an object, then denote by \(U\) the open part with \(M\) being trivial. Theorem A treats morphisms \(h_U:U\to \overline{\mathcal M}_{g,r}\) into the Deligne-Mumford stack of stable pointed curves. If \(h_U(U)\subseteq\mathcal M_{g,r}\) and \(h_U\) extends over the generic points of \(X\setminus U\), then the author gives sufficient conditions for the extensibility of \(h_U\) at all. Moreover, without assuming these additional conditions, one has at least extensions after log étale base change or toward the coarse moduli space.
The proof is done by reducing the claim to the original result of de Jong and Oort. However, it requires a technical tool, the so-called log purity theorem which is presented as theorem B in the paper.
As an application, theorem C shows that hyperbolic polycurves (i.e.\ finite compositions of families of smooth genus \(g\) curves with \(r\) points removed) may be compactified to a familiy living on a regular log scheme. regular log schemes; extension of maps; stable curves; Deligne-Mumford stack; coarse moduli space; hyperbolic polycurves Arbarello, E., Cornalba, M., Griffiths, P.A.: Geometry of Algebraic Curves, II. Grundlehren der mathematischen Wissenschaften \textbf{268}. Springer, Berlin (2011) Families, moduli of curves (algebraic), Toric varieties, Newton polyhedra, Okounkov bodies Extending families of curves over log regular schemes | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians The author studies the Galois module structure of spaces of holomorphic polydifferentials of certain curves with function field \(F\) of the form
\[
\Omega_F(m)=\{ f dg^{\otimes m}: f,g\in F \text{ and } \mathrm{div}(f dg^{\otimes m}) \geq 0\},
\]
where \(g\) is a separating element. This is a classical object of study when the field is algebraically closed of characteristic zero, that goes back to Hurwitz.
The author focusses on the case of an algebraically closed field \(K\) of positive characteristic. The situation is much more difficult than the characteristic zero: there is wild ramification in the covering and also phenomena of modular representation theory occur. Describing the Galois module structure means describing the indecomposable summands that appear in the \(K[G]\)-module \(\Omega_F(m)\) and their multiplicities.
The method of study is based on previous work of Valentini, Madan, Calderon, Salvador and Madden [\textit{D. J. Madden}, J. Number Theory 10, 303--323 (1978; Zbl 0384.12008); \textit{M. Rzedowski-Calderón, G. Villa-Salvador} and \textit{M. L. Madan}, Arch. Math. 66, No.2, 150--156 (1996; Zbl 0854.11061); \textit{R. C. Valentini} and \textit{M. L. Madan}, J. Number Theory 13, 106--115 (1981Zbl 0468.14008)] who considered the \(m=1\) case. First a basis of holomorphic polydifferentials is described and then the \(K[G]\)-summands are studied. The multiplicities are described in terms of Boseck invariants. These are invariants first introduced by \textit{H. Boseck} [``Zur Theorie der Weierstrasspunkte'', Math. Nachr. 19, 29--63 (1958; Zbl 0085.02701)]. The author extends their definition to any ramified cover of the projective line. The cases that are studied are the two extreme cases of \(p\)-abelian covers of the projective line, namely elementary abelian groups, that are expressed in terms of Artin-Schreier extensions and cyclic groups that are described as a tower of normalized \(p\)-cyclic covers of the projective line. The method of proof in these two cases are different but the final results are similar.
The main application given in this article is the computation of the tangent space of the global deformation functor of curves with automorphisms. According to [\textit{A. Kontogeorgis}, J. Pure Appl. Algebra 210, No. 2, 551--558 (2007; Zbl 1120.14020)] this dimension is equal to the space of coinvariants
\[
\mathrm{dim}_K ( \Omega_F(2) \otimes_{K[G]} K).
\]
A closed formula for the above dimension is given using the Galois module structure previously computed. automorphisms; curves; differentials; Galois module structure; deformation theory of curves with automorphisms Karanikolopoulos, S., On holomorphic polydifferentials in positive characteristic, Math. Nachr., 285, 852-877, (2012) Automorphisms of curves, Curves over finite and local fields On holomorphic polydifferentials in positive characteristic | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians A classical result from Hurwitz asserts that the endomorphism ring of the Jacobian of a very general curve \(C\) of genus at least 2 is the smallest possible, i.e. it is \(\mathbb{Z}\). \textit{S. Lefschetz} [Am. J. Math. 50, 159--166 (1928; JFM 54.0410.02)] proved the same in the case of hyperelliptic curves. \textit{C. Ciliberto} et al. [J. Algebr. Geom. 1, No. 2, 215--229 (1992; Zbl 0806.14020)] studied curves whose Jacobian has endomorphism ring larger than \(\mathbb{Z}\). Finally, \textit{Y. G. Zarhin} [Math. Proc. Camb. Philos. Soc. 136, No. 2, 257--267 (2004; Zbl 1058.14064)] considered curves with automorphisms (of a specific type) and showed that the endomorphism ring of their Jacobians is as smallest as possible.
In the paper under review, the authors show that such a minimality property still holds for curves endowed with an action of a given (but arbitrary) finite group \(G\). Indeed, they show that \(\text{End}_{\mathbb{Q}}(JC)\cong \mathbb{Q}[G]\) for a very general \(G\)-curve, with quotient curve of genus at least 3.
As an application, they also obtain interesting consequences on the natural representation of the centralizer of \(G\) in \(\text{Mod}(C)\), \(\rho_G: \text{Mod}(C)^G\rightarrow \text{Sp}(H^1(C,\mathbb{Q}))^G\), where \(\text{Sp}(H^1(C,\mathbb{Q}))^G\) stands for the centralizer of \(G\) in \(\text{Sp}(H^1(C,\mathbb{Q}))\), regarded as a virtual
linear representation the mapping class group \(\text{Mod}(C/G)\). Indeed, let \(X(\mathbb{Q}[G])\) be the set of rational irreducible characters of \( G \), take the isomorphism \[
\text{Sp}(H^1(C,\mathbb{Q}))^G\cong \prod_{\chi \in X(\mathbb{Q}[G]) }\text{Sp}(H^1(C,\mathbb{Q})_{\chi})^G,
\]
which mirrors the isotypical decomposition of the Jacobian of a \(G\)-curve \(C\), and denote by \(Mon^0(C)\) the identity component of the Zariski closure of the
image of \( \rho_G \) in \(\text{Sp}(H^1(C,\mathbb{Q}))^G\) and by \(\text{Mon}^0(C)_\chi\) the projection of \(\text{Mon}^0(C)\) to the factor \( \text{Sp}(H^1(C,\mathbb{Q})_{\chi})^G \). Then, the authors deduce different interesting properties on \(\text{Mon}^0(C)\) and on its factors \(\text{Mon}^0(C)_\chi\). moduli of curves; automorphisms of curves and Jacobians Families, moduli of curves (algebraic), Automorphisms of curves, Group actions on manifolds and cell complexes in low dimensions, Jacobians, Prym varieties Curves with prescribed symmetry and associated representations of mapping class groups | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians This paper covers a particular topic in the interplay between the divisor theory on algebraic curves and the corresponding theory on finite graphs: the Brill--Noether Theory.
Recall that, if \(C\) is an algebraic curve of genus \(g\) over an algebraically closed field, then for any couple of positive integers \(r\) and \(d\), one defines the Brill-Noether variety as
\[
W^r_d(C) := \left\{L \in \mathrm{Pic}^dC : r(C,L) \geq r \right\}
\]
where \(\mathrm{Pic}^dC\) denotes the set of invertible sheaves of degree \(d\) on \(C\) and \(r(C,L):= {\dim}(H^0(C,L)) -1\); and the Brill--Noether number as
\[
\rho^r_d(g) := g-(r+1)(g-d+r).
\]
The link between them has been widely studied and the first remarkable results are the following (see [\textit{G. Kempf}, Schubert methods with an application to algebraic curves. Math. Centrum, Amsterdam, Afd. zuivere Wisk. ZW 6/71 (1971; Zbl 0223.14018)], \textit{S. L. Kleiman} and \textit{D. Laksov} [Am. J. Math. 94, 431--436 (1972; Zbl 0251.14005), Acta Math. 132, 163--176 (1974; Zbl 0286.14005)], \textit{P. Griffiths} and \textit{J. Harris} [Duke Math. J. 47, 233--272 (1980; Zbl 0446.14011)]):
Existence Theorem. If \(\rho^r_d(g) \geq 0\) and \(C\) is a smooth curve of genus \(g\), then \(W^r_d(C) \neq \emptyset\).
Brill-Noether Theorem. If \(\rho^r_d(g) < 0\) and \(C\) is a \textit{general} smooth curve of genus \(g\), then \(W^r_d(C) = \emptyset\).
In this paper a rather synthetic overview of the divisor theory on graphs is given. In analogy with the algebro--geometric setting, the notion of divisor (its genus, its degree, the intersection law, the Jacobian group, a ``combinatorial'' rank) on a finite connected graph \(\Gamma\) is reminded. So one can define \(W^r_d(\Gamma)\) as the set of the (classes of) divisors of degree \(d\) having combinatorial rank not exceeding \(r\). One should say that the notion of combinatorial rank is more subtle when \(\Gamma\) contains loops: it is necessary to introduce ``refinements'' of \(\Gamma\) as graphs with further vertices on its loops.
Since to a nodal curve \(X\) one can associate the dual graph, whose vertices (respectively, edges) represent the irreducible components of \(X\) (resp. the nodes of \(X\)), the study of the analogous for graphs of the two theorems above is very natural.
The main result of this paper is the Existence Theorem for graphs which claims that, if \(\rho^r_d(g) \geq 0\), then for every graph \(\Gamma\) of genus \(g\) it holds \(W^r_d(\Gamma) \neq \emptyset\). The proof skillfully uses the cited corresponding result for algebraic curves and a suitable version of the Baker specialization Lemma. But the problem of finding a purely combinatorial proof of such result still stands.
Finally, the author observes that a sort of Brill--Noether Theorem for graphs also holds: if \(\rho^r_d(g) < 0\) then there exists a graph \(\Gamma\) of genus \(g\) such that \(W^r_d(\Gamma) = \emptyset\) (this follows from a result of \textit{F. Cools} et al., [Adv. Math. 230, No. 2, 759--776 (2012; Zbl 1325.14080)]).
Hence the author conjectures that some assumption on a specific graph \(\Gamma\) could imply \(W^r_d(\Gamma) = \emptyset\), provided that \(\rho^r_d(g) < 0\).
The final interesting result deeply relates the subjects of the two Brill--Noether Theorems: if there exists a graph \(\Gamma\) of genus \(g\) such that \(W^r_d(\Gamma) = \emptyset\) then \(W^r_d(C) = \emptyset\) for a general projective curve \(C\).
In particular, if \(\rho^r_d(g) < 0\) then the Brill--Noether Theorem for curves holds. algebraic curves; divisors; Brill-Noether theory; graph Caporaso, L., Algebraic and combinatorial brill-Noether theory, (Alexeev, V.; Izadi, E.; Gibney, A.; Kollàr, J.; Loojenga, E., Compact Moduli Spaces and Vector Bundles, Contemporary Mathematics, vol. 564, (2012)), 69-85 Special divisors on curves (gonality, Brill-Noether theory), Graph theory, Fibrations, degenerations in algebraic geometry Algebraic and combinatorial Brill-Noether theory | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians \textit{A. Grothendieck} [Séminaire de géométrie algébrique du Bois Marie 1960/61 (SGA 1). New York: Springer-Verlag (1971; Zbl 0234.14002)] showed that for a proper separable morphism \(X\to S\) of connected schemes with geometrically connected fibres there is an exact sequence of étale fundamental groups
\[
\pi_1(X_{\bar s}, \bar x)\to \pi_1(X, \bar x)\to \pi_1(S, \bar s)\to 1
\]
where \(\bar x\) is a geometric point of \(X\) with image \(\bar s\) in \(S\), and \(X_{\bar s}\) is the geometric fibre. In a preprint, \textit{H. Esnault, P.-H. Hai} and \textit{E. Viehweg} [``On the homotopy exact sequence for Nori's fundamental group'', \url{arXiv:0908.0498}] gave a counterexample showing that such an exact sequence does not necessarily exist for \textit{M. V. Nori}'s fundamental group scheme [Proc. Indian Acad. Sci., Math. Sci. 91, 73--122 (1982; Zbl 0586.14006)], even for \(X\) and \(S\) projective and smooth over an algebraically closed field of positive characteristic.
In the present paper, the author gives necessarily and sufficient conditions (a bit complicated to check in practice) for the corresponding sequence of Nori fundamental group schemes to be exact. As applications, he gives partly new proofs of Grothendieck's sequence above and of a result of \textit{V. B. Mehta} and \textit{S. Subramanian} [Invent. Math. 148, No. 1, 143--150 (2002; Zbl 1020.14006)] according to which Nori's fundamental group scheme is compatible with products of proper connected schemes. The paper also contains a counterexample to product compatibility with one of the schemes non-proper: a product of the affine line with a supersingular elliptic curve in characteristic 2. Nori's fundamental group scheme; torsors under finite group schemes; homotopy exact sequence Zhang, L., \textit{the homotopy sequence of nori's fundamental group}, J. Algebra, 393, 79-91, (2013) Homotopy theory and fundamental groups in algebraic geometry, Vector bundles on surfaces and higher-dimensional varieties, and their moduli, Group schemes The homotopy sequence of Nori's fundamental group | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians In the paper under review, the authors prove a representability theorem in derived analytic geometry, analogous to
Lurie's generalization of Artin's representablility criteria to derived algebraic geometry.
This is an important, standard type result for the study of moduli problems and
a crucial step towards a solid theory of derived analytic geometry.
More specifically, the authors show that a derived stack for the étale site of derived analytic spaces is a derived analytic stack
if and only if
it is compatible with Postnikov towers, has a global analytic cotangent complex, and its truncation is an analytic stack in the classical
(underived) sense.
The result applies both to complex analytic geometry and non-archimedean analytic geometry.
Central to representability results as in the present paper is deformation theory, which the authors develop here for the derived analytic setup.
The authors define an analytic version of the cotangent complex which controls the deformation theory of the derived stack.
As in the algebraic setting, the cotangent complex represents a functor of derivations.
One key step in order to define the analytic cotangent complex is the elegant description of the \(\infty\)-category of modules over a
derived analytic space \(X\) as the \(\infty\)-category of spectrum objects of a certain \(\infty\)-category associated with \(X\).
Another important construction is the analytification functor which they establish in the derived setting.
To apply derived geometry to classical moduli problems, one may try to enrich classical moduli spaces with derived structures. The paper under review is an important tool in verifying when such enrichments are indeed the correct ones. representability; deformation theory; analytic cotangent complex; derived geometry; rigid analytic geometry; complex geometry; derived stacks Stacks and moduli problems, Rigid analytic geometry, Complex-analytic moduli problems, Fundamental constructions in algebraic geometry involving higher and derived categories (homotopical algebraic geometry, derived algebraic geometry, etc.) Representability theorem in derived analytic geometry | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians The paper considers a germ \(X\) of a complex (or real) analytic manifold \(M\) at a point \(x_0 \in M\) and a divisor \(D \subset X\). On it, an explicit construction and a description of the singularities of all finite normal coverings \(Y \rightarrow X\) ramified along a divisor is given, provided that there exists a closed subgerm \(N \subset D\) of codimension \(\geq 2\) in \(D\) such that \(D\) is a divisor with normal crossings outside \(N\). This result extends the classical Abhyankar's Lemma and it is applied to prove the existence of a diagonalizing morphism \(u_c: U_c \rightarrow\text{GL}(n)\), \(U_c\) regular, for some types of analytic morphisms \(g:X \rightarrow\text{GL}(n)\) over a suitable finite étale cover \(a_c:U_c \rightarrow U=X-D\). Finally, the stable triviality of the eigenbundles of \(g\) on \(U\) and vanishing of their Chern classes are proved. singularities; diagonalization morphism; eigenbundles; ramified eigenvalues; perturbation theory Selten, R; Warglien, M, The emergence of simple languages in an experimental coordination game, Proc. Natl. Acad. Sci. USA, 104, 7361-7366, (2007) Germs of analytic sets, local parametrization, Singularities of surfaces or higher-dimensional varieties The structure of a class of finite ramified coverings and canonical forms of analytic matrix-functions in a neighborhood of a ramified turning point | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians Let \(X\) be a smooth projective curve of genus 2 or 3 over an algebraically closed field of characteristic \(p> 0\) and denote by \(F: X\to X_1\) the relative Frobenius map. Let \(S_X\) and \(S_{X_1}\) denote the coarse moduli space of semistable rank-2 vector bundles on \(X\) and \(X_1\). Finally, let \(\Theta\) and \(\Theta_1\) denote the Theta divisors on the Jacobians of \(X\) and \(X_1\). Narasimhan and Ramanan showed that \(S_X\) is isomorphic to \(|2\Theta|\) in genus 2 and can be identified with the Coble quartic in \(|2\Theta|\) in the non-hyperelliptc genus-3 case. The map \(V: S){X_1}\to S_X\), defined by \(E\mapsto F^* E\), is called the (generalized) Verschiebung. \textit{Y. Laszlo} and \textit{C. Pauly} showed [Adv. Math. 185, No. 2, 246--269 (2004; Zbl 1055.14038)] that \(V\) is defined by polynomials in some cases.
The present paper investigates the map \(V\) further. The main result is the following theorem:
For \(p= 3,5\) or \(7\) and \(X\) general of genus 2 or 3 there is a rational map \(V:|2\Theta_1|\to |2\Theta|\) extending \(V\) which is completely determined by its restriction to the \(\tau_1\)-invariant locus of \(|2\Theta_1|\), where \(\tau_1\) ranges over the non zero 2-torsion points of \(J(X_1)\).
The proof uses representations of Heisenberg groups. This is used to compute explicit equations for the map \(V\) in these cases, which allows to draw some geometric consequences. about the base locus of the map \(V\). vector bundles; Frobenius; Prym varieties L. Ducrohet, Frobenius et fibrés sur les courbes, PhD thesis, Université de Paris 6. Vector bundles on curves and their moduli The Frobenius action on rank 2 vector bundles over curves in small genus and small characteristic | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians ''... the main purpose of this paper is to give a unification of the following two basic theories for coherent sheaves on analytic varieties: (1) a type of cohomology theory in which what we call polynomial growth \((=p.g.\)) conditions on cochains and coverings are involved; and (2) completion theory along subvarieties of a given analytic variety. Our theory is given to affine varieties and their analytic analogues... The main body of this paper is devoted to certain explicit uniform estimations... Our main application is to give an analogue of Theorems A and B of H. Cartan in our unified theory... We will apply such a result to generalize, to varieties with singularities, the theorems of A. Grothendieck on algebraic and analytic de Rham theory... Cohomology theories with p.g. conditions were studied by Deligne-Maltsiniotis [\textit{G. Maltsiniotis}, Astérisque 17, 141-160 (1974; Zbl 0297.14006)] and \textit{M. Cornalba} and \textit{P. Griffiths} [Invent. Math. 28, 11-106 (1975; Zbl 0293.32026)] for locally free sheaves over smooth affine varieties by \({\bar \partial}\)-estimations. The situation in our theory is more general than theirs. Our method, depending on Cousin integrals, differs from theirs...''
Partly because of the complexity of notation and the author's style of exposition, the reviewer is unable to quote even a small sample of the results in the paper. theorem A; theorem B; coherent sheaves; polynomial growth; p.g.; completion theory; affine varieties; de Rham theory; Cousin integrals N. Sasakura , Cohomology with polynomial growth and completion theory . Publ. Res. Inst. Math. Sci. Kyoto Univ. 17 ( 1981 ), 371 - 552 . Article | MR 642649 | Zbl 0561.32005 Analytic sheaves and cohomology groups, de Rham cohomology and algebraic geometry, Stein spaces, Sheaves and cohomology of sections of holomorphic vector bundles, general results Cohomology with polynomial growth and completion theory | 0 |
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