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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 The main result of the paper is a generalization of the theorem of \textit{M. Artin} and \textit{J. J. Zhang} [Adv. Math. 109, 228--287 (1994; Zbl 0833.14002)] characterizing certain classes of abelian categories that can be viewed as noncommutative analogues of categories of coherent sheaves on projective schemes. The main idea of this approach to noncommutative geometry is to associate to a noncommutative graded algebra the quotient category \(\text{QGRA}(A)\) of of the category of graded \(A\)-modules by the subcategory of torsion modules. This generalizes a theorem of Serre who proves that in the commutative case the category of quasicoherent sheaves on \(\text{Proj}(A)\) is equivalent to \(\text{QGRA} (A)\). Then Artin and Zhang [loc. cit.] gave a criterion (the AZ-theorem) for a locally Noetherian abelian category \(\mathcal C\) to be equivalent to \(\text{QGRA} (A)\) for some \(A\), namely that the category has the essential properties of coherent sheaves on a projective scheme \(X\) (i.e. has an ample sequence).
The main goal of the present paper is to prove that if one removes the assumption that the category is Noetherian in the AZ-theorem, then the corresponding graded algebra (constructed from the ample sequence) is still coherent and the abelian category in question is equivalent to the quotient of the category of coherent modules by the subcategory of finite-dimensional modules. The article proves partial results to develop techniques for checking whether a given graded algebra is coherent. This gives a connection between the coherency of a graded algebra and its Veronese subalgebras. Notice that the paper extends the class of graded algebras to the wider class consisting of \(\mathbb{Z}\)-algebras. This makes some connections to the non-obstructed cases of O.A. Laudals noncommutative geometry (which is not referred to). The paper is nicely written, and also explicit examples are given. noncommutative geometry; ample sequence; AZ-theorem A. Polishchuk, Noncommutative proj and coherent algebras. Math. Res. Lett. 12 (2005), 63-74. Noncommutative algebraic geometry Noncommutative proj and coherent algebras | 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 use in public key cryptography of Jacobian varieties of curves defined over finite fields stimulated the search of efficient algorithms to perform the addition in those Jacobian i.e. in the divisor class group of nonsingular curves (equivalently the ideal class group of its coordinate ring). In the case of hyperelliptic curves such an algorithm was proposed by \textit{D.G. Cantor} [Math. Comput. 48, 95--101(1987, Zbl 0613.14022)].
Instead of working with nonsingular models of curves the present paper deals with singular plane models, using what the authors call the generalized Jacobian variety of a curve \(C\)\, (the Picard group of invertible sheaves of degree zero on \(C\)).
Section 2 studies the relationship between the Jacobian of a nonsingular curve \(C\)\, and a certain singular plane model of \(C\), the so-called \(C_A\) model in the sense of \textit{S. Miura} [Shingakuron (A), J-181A, vol. 10, 1398--1421 (1998)]. This \(C_A\) curve, reviewed in section 3, has only one infinity point which is at most a cuspidal singularity.
Section 4 gives the addition algorithm. The authors show that an algorithm due to the first of them, the Arita's algorithm for nonsingular \(C_A\)\, curves, can also compute addition on the Jacobian variety of any curve using its singular \(C_A\) model. Since the generalized Jacobian of a singular \(C_A\)\, curve is a group isomorphic to the ideal class group of the coordinate ring, the addition reduces to multiplication in that ideal class group. Finally section 5 gives an explicit example. singular plane models; Arita's algorithm S. Arita, S. Miura, T. Sekiguchi, On the addition algorithm on the Jacobian varieties of curves, Preprint 2003. Jacobians, Prym varieties, Applications to coding theory and cryptography of arithmetic geometry, Cryptography An addition algorithm on the Jacobian varieties 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 The main aim of the book under review is to study a class of functors between derived categories of coherent sheaves of smooth varieties, known as integral (or, in some cases, Fourier-Mukai) functors. Recently, this subject is rapidly developing and the book under review contains a valuable survey of the known results.
The book contains 7 chapters and 4 appendices. The authors essentially assume that the reader is familiar with homological algebra at the level of the classical book by \textit{S. Gelfand} and \textit{Yu. I. Manin} [Methods of homological algebra. Berlin: Springer (2003; Zbl 1006.18001)]. But to make the book more self-contained, basic notions and results of homological algebra are quickly recalled in Appendix A written by F. Sancho.
Chapter 1 contains definition of integral functors and their first basic properties (also in the equivariant case).
Most of Chapter 2 is devoted to the proof of Orlov's theorem describing fully faithful exact functors between derived categories of coherent sheaves on smooth varieties as integral functors. It also contains characterization of Fourier-Mukai functors among all integral functors and Bondal and Orlov's reconstruction theorem for varieties with ample canonical or anticanonical divisor.
In Chapter 3 the authors study Fourier-Mukai transforms on abelian varieties (in characteristic zero). Many but not all results from this chapter can be found in \textit{A. Polishchuk}'s book [Abelian varieties, theta functions, and the Fourier transform. Cambridge: Cambridge University Press (2003; Zbl 1018.14016)]. In particular, the authors study in some cases behaviour of slope stability under integral transforms.
Chapter 4 is devoted to Fourier-Mukai transforms on complex \(K3\) surfaces. The main part of this chapter is devoted to construction of Fourier-Mukai transforms on reflexive surfaces.
In Chapter 5 the authors try to justify the words ``Nahm transforms'' in the title of the book but contrary to other chapters, this one uses analytic methods and it does not contain any proofs.
Chapters 6 studies relative Fourier-Mukai functors and their application to (usually relative) moduli spaces of semistable sheaves on (usually Weierstrass) elliptic fibrations.
Chapter 7 contains other applications of Fourier-Mukai transforms: classification of Fourier-Mukai partners of complex projective surfaces, Bridgeland's moduli space interpretation for smooth 3-dimensional flops and the derived version of McKay correspondence between the equivariant derived category and the derived category of a crepant resolution of a quotient.
Appendices A, B and C contain auxiliary results. The last appendix by E. Macri contains a survey on Bridgeland's stability conditions for derived categories with special emphasis on stability conditions on \(K3\) surfaces.
Each chapter finishes with useful comments containing history and indicating further development.
There is another book of \textit{D. Huybrechts} [Fourier-Mukai transforms in algebraic geometry. Oxford: Clarendon Press (2006; Zbl 1095.14002)] on a very similar subject. When compared to it, the book under review requires more prerequisities and it is more technical. But to reward the reader it contains some difficult proofs that are skipped in [loc. cit.] (e.g., Kawamata's proof of Orlov's representability theorem). Unlike Huybrecht's book this one contains many results concerning moduli spaces of semistable sheaves, especially related to elliptic fibrations.
When reading the book it is safer to assume that the base field has characteristic zero. Sometimes this assumption is hidden so well that it is difficult to spot (e.g., in Chapter 3 it appears at the beginning of 3.2 and lasts for the rest of the chapter) and sometimes it is necessary although the authors do not make this assumption (e.g., in positive characteristic Theorem C.6 is no longer true as stated).
As every long book the book under review still contains some misprints (e.g., on p. 66 the inequality between numerical and usual Kodaira dimensions is in the opposite direction) but there are few of them and usually they are not so disturbing as the one mentioned above.
In spite of these small omissions the book is very well written and it will certainly be very useful to researchers in algebraic geometry and mathematical physics. coherent sheaves; derived category; Fourier-Mukai functor; abelian varieties; \(K3\) surfaces; elliptic fibration; Nahm transform; stability conditions Bartocci, C., Bruzzo, U., Ruipérez, D.H: Fourier-Mukai and Nahm Transforms in Geometry and Mathematical Physics, volume 276 of Progress in Mathematics. Birkhäuser, Boston (2009). https://doi.org/10.1007/b11801 Research exposition (monographs, survey articles) pertaining to algebraic geometry, Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20], Fibrations, degenerations in algebraic geometry, Vector bundles on surfaces and higher-dimensional varieties, and their moduli, Derived categories, triangulated categories, Special connections and metrics on vector bundles (Hermite-Einstein, Yang-Mills) Fourier-Mukai and Nahm transforms in geometry and mathematical physics | 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 first main result of the paper is an equivalent formulation of the Jacobian conjecture in the two-dimensional case. It follows from the following general result on arbitrary polynomial mappings. Let \((f_{1},f_{2} ):\mathbb{C}^{2}\rightarrow\mathbb{C}^{2}\) be a polynomial mapping with a finite set of zeros. By simple modifications (i.e. by using compositions with polynomial automorphisms) the author can assume that \(f_{1}(x,y)\) and \(f_{2}(x,y)\) have the form \(x^{n}+(\)terms of lower degree) and that the intersection points of the curves \(f_{1}=0\) and \(f_{2}=0\) lie on the \(x\)-axis. Then
(1) there exists a unique polynomial solution \(g=(g_{1},g_{2})\) of the equation \(y\text{Jac}(f)=f_{1}g_{1}+f_{2}g_{2}\), \(\deg g_{i}=n-1\), and (2) the total intersection number (counting multiplicity) of the affine curves \(f_{1}=0\) and \(f_{2}=0\) equals to the coefficient of \(x^{n-1}\) of \(g_{2}.\) In particular if \(\text{Jac}(f)\equiv1\) then to prove the Jacobian conjecture it suffices to check that the coefficient in \(2.\) is equal to \(1.\) The second main result \ gives an explicite form for \((f_{1},f_{2})\) under the assumption that all intersection points of the curves \(f_{1}=0\) and \(f_{2}=0\) are normal crossings. In the proofs of the results the author uses the multidimensional residue theory. polynomial map; residue; intersection multiplicity Jacobian problem Some reductions on Jacobian problem in two variables. | 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 establish a version of the Becker-Gottlieb transfer in the context of etale homotopy. The transfer map we obtain generalizes the Becker-Gottlieb transfer to smooth quasi-projective varieties over any algebraically closed field of arbitrary characteristic.
Let F denote the fiber of the map \(E\to B\) in both cases. Then both these transfer maps have the following Lefschetz-number property:
\[
tr(f)^*p^*(x)=\wedge_ hx,
\]
for \(x\in H^*(B;Z)\) (for \(x\in H^*(EG\times_ GB;Z)\) in the equivariant case, respectively) and \(\wedge_ h\) is the Lefschetz-number of \(h=f| F\). In section 1 we construct a Becker-Gottlieb transfer for proper and smooth maps of quasi- projective varieties. This will be a stable map of their etale topological types. In section 2 we establish a similar transfer map in equivariant etale-cohomology for G-equivariant proper and smooth maps when G is an elementary abelian group of order prime to the characteristic. We obtain the main properties of the transfer in the next section. We first observe the naturality of the transfer with respect to base-change. The multiplicative property of the transfer is then obtained as a projection formula. We next obtain the Lefschetz-number property of the transfer map. This is done by making use of the above multiplicative property and by comparing the transfer with another transfer obtained using Spanier-Whitehead duality.
In section 4 we sheafify the transfer map and relate it to the trace map. [See \textit{J.-L. Verdier}, Astérisque 36-37, 101-151 (1976; Zbl 0346.14005), p. 128, and \textit{J. S. Milne}: Étale cohomology (1980; Zbl 0433.14012), p. 168] This sheafification enables us to obtain a transfer map in intersection homology for proper and smooth maps. The last section deals with various applications of the transfer. First we carry over many of the standard applications of the Becker-Gottlieb transfer to algebraic geometry. In addition to these we obtain other applications to K-theory and intersection homology. etale topological types; applications of the Becker-Gottlieb transfer to algebraic geometry; intersection homology Joshua, R.: Becker-gottlieb transfers in etale homotopy. Amer. J. Math. 109, 453-497 (1987) Transfer for fiber spaces and bundles in algebraic topology, Étale and other Grothendieck topologies and (co)homologies, Other algebro-geometric (co)homologies (e.g., intersection, equivariant, Lawson, Deligne (co)homologies) Becker-Gottlieb transfer in etale homotopy | 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 one-semester course given by the author at Concordia University in 1986. The aim of these lectures was to introduce graduate students and researchers in other fields to the basic theory of abelian varieties, starting from the classical theory of compact Riemann surfaces and their Jacobians and ending up with a glance at some of the most famous recent topics in the arithmetic realm. The notes of this course, which have been distributed informally over the past few years, have gained a remarkable popularity among students and teachers, mainly for their efficient arrangement, enlightening style, self-containedness and -- nevertheless -- manageable conciseness. The book at issue preserves the style of these lectures and, in this way, makes them available to a wider class of readers who wish an independent, brief introduction to the subject of abelian varieties.
The first six chapters provide the basics on compact Riemann surfaces and their Jacobians. This includes the Riemann-Hurwitz formula for finite coverings, the existence of Weierstraß points, Schwarz's theorem on the finiteness of the automorphism group of a compact Riemann surface and other consequences from the Riemann-Roch theorem, the Abel-Jacobi theorem and the Riemann period relations, the construction of the Jacobian of a complex curve, divisors on complex tori and theta functions, spaces of theta functions and, finally, projective embeddings of complex tori with positive theta divisors (Lefschetz's theorem). -- This first part follows the elegant approach of \textit{E. Artin} and \textit{A. Weil}, as it was elaborated and presented in \textit{S. Lang}'s classic book ``Introduction to algebraic and abelian functions'' (1972; Zbl 0255.14001); 2nd edition 1982].
Chapter 7 gives a nice illustration of Lefschetz's embedding theorem by explicitly describing the embedding of an elliptic curve into \(\mathbb{P}^ 3\) as an intersection of two quadrics. This is a more down-to-earth version of D. Mumford's general approach [cf. \textit{D. Mumford}, ``Tata lectures on theta. I'', Prog. Math. 28 (1983; Zbl 0509.14049)], particularly suited for beginners. -- Chapter 8 discusses the Jacobian of the Fermat curve \(x^ n+y^ n+z^ n=0\), while the next three chapters deal with the modular curves associated to congruence subgroups of \(SL_ 2(R)\) and their Jacobians. This already arithmetically flavoured topic is treated by following parts of [\textit{G. Shimura}'s comprehensive book ``Introduction to the arithmetic theory of automorphic functions'' (1971; Zbl 0221.10029)] and adapting them to the more elementary purpose of the present introductory notes. -- Chapter 12 provides a very brief introduction to arbitrary abelian varieties and their basic properties, giving proofs in the special case of the complex groundfield and compact connected complex Lie groups as an illustration.
The final part of the book, chapters 13 to 15, gives an outlook to the theory of abelian varieties over number fields. This includes the concept of Tate modules of an abelian variety and the statement of the Tate conjecture, the deduction of Mordell's conjecture from Tate's conjecture via Arakelov theory and the Kodaira-Parshin construction, and the relation to Shafarevich's finiteness conjecture on the number of isomorphism classes of algebraic curves of genus \(g\) with good reduction over an algebraic number field. -- Of course, this final part is more sketchy than the others, but all the same very enlightening and motivating for deeper studies guided by more advanced textbooks and the very recent research literature.
Altogether, this book really leads the non-specialist from the origins of the theory of abelian varieties to the frontiers of todays research in the field. Lefschetz theorem; Riemann surfaces; Jacobians; Riemann-Roch theorem; Abel-Jacobi theorem; Riemann period relations; theta functions; embeddings of complex tori; modular curves; Tate conjecture; Arakelov theory V. Kumar Murty, Introduction to abelian varieties, CRM Monograph Series, vol. 3, American Mathematical Society, Providence, RI, 1993. Analytic theory of abelian varieties; abelian integrals and differentials, Theta functions and abelian varieties, Introductory exposition (textbooks, tutorial papers, etc.) pertaining to algebraic geometry, Research exposition (monographs, survey articles) pertaining to algebraic geometry, Jacobians, Prym varieties, Abelian varieties of dimension \(> 1\), Arithmetic varieties and schemes; Arakelov theory; heights Introduction to 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 his paper [J. Differ. Geom. 61, No.1, 147--171 (2002; Zbl 1056.14021)], the author has studied aspects of the conjecture that birationally equivalent smooth projective varieties have equivalent derived categories iff they have equivalent canonical divisors. The article under review is dealing with singular instead of only smooth varieties and moreover with pairs of varieties and \(\mathbb Q\)-divisors, where at most log-terminal singularities are allowed:
Derived equivalence conjecture. Let \((X,B)\) and \((Y,C)\) be such pairs (and suppose some technical conditions). Denote \(\mathcal X\) and \(\mathcal Y\) the associated stacks, and assume there are proper birational morphisms \(\mu :W\to X\) and \(\nu :W\to Y\) such that \(\mu ^* (K_X+B)=\nu ^* (K_y+C)\). Then there exists an equivalence \(D^b({\mathcal X}) \to D^b({\mathcal Y})\) of triangulated categories .
After formulating the conjecture, first of all a converse statement is given. Main result of the article is a proof of the conjecture for toroidal varieties. The theorem generalizes an earlier result of the author [in: Algebraic geometry. A volume in memory of Paolo Francia. Berlin: de Gruyter. 197--215 (2002; Zbl 1092.14023)]. It implies the McKay correspondence for abelian quotient singularities as a special case. derived category of coherent sheaves; derived equivalence conjecture Kawamata Y., Log crepant birational maps and derived categories, J. Math. Sci. Univ. Tokyo, 2005, 12(2), 211--231 Minimal model program (Mori theory, extremal rays), Derived categories, triangulated categories Log crepant birational maps and derived categories | 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 result of Serre says that if an automorphism of finite order of a semi-abelian variety induces the identity on the scheme-theoretic kernel of multiplication by \(n\), then it is the identity if \(n\geq 3\), and its square is the identity if \(n=2\). This result is useful in the study of moduli spaces of abelian varieties with full level \(n\geq 3\) structure. Serre's lemma relies on the fact that if \(n\geq 3\) then every root of unity which is congruent to 1 modulo \(n\) is 1. This idea dates back to Minkowski, who proved that an integral matrix of finite order, which is congruent to the identity modulo \(n\), is the identity if \(n\geq 3\).
In this paper we give generalizations and variations of the Serre-Minkowski results. For example, if \(k\) and \(n\) are positive integers, \(\alpha\) is a root of unity in an integral domain \({\mathfrak O}\) of characteristic zero, \((\alpha-1)^k\) is divisible by \(n\), and no prime divisor of \(n\) is a unit in \({\mathfrak O}\), then \(\alpha=1\) if \(n\) is outside of a certain finite set of prime powers determined by \(k\). (The case \(k=1\) is the Serre-Minkowski case.) The proof relies on the arithmetic of cyclotomic integers. Although the ideas are simple, the result is useful and does not seem to have been noticed before. We give best-possible restrictions on \(\alpha\) when \(n\) is in the finite exceptional set. We have results for rings that are not necessarily integral domains (section 4), and we have applications to matrix rings, eigenvalues, projective modules, and quasi-unipotent elements (see sections 6 and 7). We provide additional information when \(n\) is in the exceptional finite set and give examples which show that our results are sharp. In section 8 we give applications to endomorphisms of semi-abelian varieties which generalize Serre's result. In section 9 we give applications to compatible systems of \(\ell\)-adic representations and the cohomology of projective varieties. automorphism of finite order; cyclotomic integers; endomorphisms of semi-abelian varieties A. Silverberg and Y. G. Zarhin, ''Variations on a theme of Minkowski and Serre,'' J. Pure Appl. Algebra, vol. 111, iss. 1-3, pp. 285-302, 1996. Automorphisms of curves, Abelian varieties and schemes, Abelian varieties of dimension \(> 1\), Cyclotomic extensions Variations on a theme of Minkowski and Serre | 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 book introduces the reader to modern algebraic geometry. It presents Grothendieck's technically demanding language of schemes that is the basis of the most important developments in the last fifty years within this area. A systematic treatment and motivation of the theory is emphasized, using concrete examples to illustrate its usefulness. Several examples from the realm of Hilbert modular surfaces and of determinantal varieties are used methodically to discuss the covered techniques. Thus the reader experiences that the further development of the theory yields an ever better understanding of these fascinating objects. The text is complemented by many exercises that serve to check the comprehension of the text, treat further examples, or give an outlook on further results. The volume at hand is an introduction to schemes. To get started, it requires only basic knowledge in abstract algebra and topology. Essential facts from commutative algebra are assembled in an appendix. It will be complemented by a second volume on the cohomology of schemes.
For the second edition, several mistakes and many smaller errors and misprints have been corrected.
See the review of the first edition in [Zbl 1213.14001]. textbook (algebraic geometry); schemes and morphisms; prevarieties; quasi-coherent sheaves; vector bundles; divisors; algebraic curves; determinantal varieties; singularities Introductory exposition (textbooks, tutorial papers, etc.) pertaining to algebraic geometry, Schemes and morphisms, Divisors, linear systems, invertible sheaves, Sheaves in algebraic geometry, Group schemes, Determinantal varieties, Group actions on varieties or schemes (quotients), Singularities in algebraic geometry Algebraic geometry I. Schemes. With examples and exercises | 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 article under review is based on two courses of the authors at the Centre Émile Borel of the Insrtitut Henri Poincaré during the ``Semestre \(p\)-adique'' of 1997. It contains a brief survey of results of Fontaine-Laffaille and Fontaine-Messing concerning the study of \(p\)-torsion étale cohomology of varieties with good reduction over the ring of Witt vectors with coefficients in a perfect field of characteristic \(p > 0\) [see \textit{J.-M. Fontaine} and \textit{G. Laffaille}, Ann. Sci. Éc. Norm. Supér., IV. Sér. 15, 547--608 (1982; Zbl 0579.14037); \textit{J.-M. Fontaine} and \textit{W. Messing}, Contemp. Math. 67, 179--207 (1987; Zbl 0632.14016)].
The authors then discuss an approach to the study of \(p\)-torsion étale cohomology and crystalline cohomology making use of log-syntomic methods developed in a series of publications by B. Mazur and his followers. In fact, they describe an extension of previous results of J.-M. Fontaine and W. Messing to the semi-stable case following ideas of \textit{Ch. Breuil} [Ann. Sci. Éc. Norm. Supér., IV. Sér. 31, No. 3, 281--327 (1998; Zbl 0907.14006) and Duke Math. J. 95, No. 3, 523--620 (1998; Zbl 0961.14010)]. In the conclusion, some applications are considered. Among them there are conjectures of \textit{J.-M. Fontaine} [in: Périodes \(p\)-adiques,Sémin. Bures-sur-Yvette 1988, Exposé III, Astérisque 223, 113--184 (1994; Zbl 0865.14009); (5.4.4)] and Fontaine-Jannsen [see \textit{K. Kato}, ibid., Exposé VI, Astérisque 223, 269--293 (1994; Zbl 0847.14009); (1.1)]. The authors also raise open questions concerning generalizations to the semi-stable case of known results by G. Faltings, V. A. Abrashkin, and others; they also present a detailed list of references containing 75 items. crystalline cohomology; étale cohomology; de Rham cohomology; Grothendieck topology; nearby cycles; log-schemes; log-syntomic topology; semi-stable reduction; torsion invariants; ramified extension; crystalline representations; Hodge-Tate weights; Galois theory for local fields; Bibliography Breuil, Christophe; Messing, William, Torsion étale and crystalline cohomologies, Astérisque, 279, 81-124, (2002) \(p\)-adic cohomology, crystalline cohomology, Varieties over finite and local fields, Galois theory, Étale and other Grothendieck topologies and (co)homologies, Abelian varieties of dimension \(> 1\), de Rham cohomology and algebraic geometry, Local ground fields in algebraic geometry Torsion étale and crystalline cohomologies. | 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 develop a power structure over the Grothendieck ring of varieties relative to an abelian monoid, which provides a systematic method to compute the class of the generalized Kummer scheme in the Grothendieck ring of Hodge structures. We obtain a generalized version of Cheah's formula for the Hilbert scheme of points, which specializes to Gulbrandsen's conjecture for Euler characteristics. Moreover, in the surface case we prove a conjecture of Göttsche for geometrically ruled surfaces. power structure; Hodge polynomial; Donaldson-Thomas invariant; generalized Kummer scheme Parametrization (Chow and Hilbert schemes), Algebraic theory of abelian varieties, Derived categories of sheaves, dg categories, and related constructions in algebraic geometry Hodge numbers of generalized Kummer schemes via relative power structures | 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 Given a Serre fibration \(f\:X\to S\) with connected base and fibers \(X_-:=f^{-1}(-)\) and a section \(p\:S\to X\), the fundamental groups \(\pi_1(X_{-},p({-}))\) constitute a (nonabelian) local system over~\(S\), and any other section \(q\:S\to X\) gives rise to an element of the classifying space \(H^1(S;\pi_1(X_{-},p({-})))\). \textit{M. Kim} [Publ. Res. Inst. Math. Sci. 45, No. 1, 89--133 (2009; Zbl 1165.14020)], using a certain ``de Rham fundamental group'', employs these ideas in his study of rational points of algebraic curves defined over an algebraic number field~\(K\). In the paper under review, the author makes an attempt to translate Kim's arguments to the case when \(K\) is a finite extension of the function field \(\mathbb F_p(t)\) of positive characteristic. He shows that, ``under very restrictive hypotheses'', the Frobenius invariant part of the classifying space has a structure of an algebraic variety. According to the author, there is still ``a very large amount of work to be done'' in order to make Kim's methods work in the case of positive characteristic. unipotent fundamental group; non-abelian Jacobian C.~Lazda, \emph{Relative fundamental groups and rational points}, Rend. Sem. Mat. Univ. Padova \textbf{134} (2015), 1--45. DOI 10.4171/RSMUP/134-1; zbl 1335.11050; MR3428414; arxiv 1303.6484 Varieties over global fields, Rational points Relative fundamental groups and rational points | 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 the entire collection see Zbl 0711.00011.]
If one considers curves of genus g defined over some algebraically closed field k, the generic such curve C is (loosely speaking) the curve corresponding to the generic point of the relevant moduli space. It has a field of definition K (depending on k and g), and with \(\bar K\) a separable closure of K, \(Gal(\bar K/K)\) acts on the torsion in \(Pic(C\otimes \bar K)\). The results of this (in the reviewer's opinion somewhat too sketchy) paper is that the image of this action is as large as it could possibly be. For the action on the \(\ell\)-primary part of this torsion this is well known, provided \(\ell \neq char(k)\). However, it should be emphasized that the known proofs of even this special case all contained a transcendental ingredient: somewhere in the proof one restricts oneself to the case \(k={\mathbb{C}}\) and uses methods there which do not translate easily (if at all) into more general algebraic geometric language. - The proofs given here are completely algebraic.
As an application, the author obtains (probably the first ever given) completely algebraic proof of irreducibility of moduli spaces of curves in arbitrary characteristics. - The proof mainly consists of first restating everything in terms of a monodromy action. By restricting to a base over which a family consisting of genus 2 curves with two `chains' of elliptic curves attached to it exists, the author verifies that it suffices to prove his main result in case \(g=2\). The latter case is then reduced to consider the action on the \(p=char(k)\)-torsion only, which can be settled by explicit local monodromy calculations. torsion points of Jacobians; algebraic fundamental group; \(\ell \)-adic representation; irreducibility of moduli spaces of curves; monodromy T. Ekedahl , The action of monodromy on torsion points of Jacobians. Arithmetic algebraic geometry (Texel, 1989) . Birkhäuser Boston ( 1991 ), 41 - 49 . MR 1085255 | Zbl 0728.14028 Families, moduli of curves (algebraic), Jacobians, Prym varieties, Coverings of curves, fundamental group The action of monodromy on torsion points of 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 book starts with a preface written by Yuri I. Manin, and so this review should start with a resume of that:
Grothendieck gave his definition of \textit{motives} in the 1960s. This was based on the various cohomology theories that existed for algebraic varieties over a field \(k\), and made use of A. Weil's counting of points over finite fields via the Lefschetz trace formula. Grothendieck gave the definition of a universal cohomology theory (with coefficients in \(R\)) as a functor \(h\) from the category \(\text{Var}(k)\) of smooth varieties to an abelian tensor category \(\text{Mot}(k)\) of \textit{pure motives} or \(\text{Mot}(k)_R\), satisfying a minimal list of expected properties.
The definition of motives is given in several steps based on enriching categories; the objects of \(\text{Var}(k)\) is kept, but the morphisms are replaced by classes of cycles on \(X\times Y\). Then the morphisms form an additive group or an \(R\)-module. Such a (coarsest) relation is the numerical equivalence on cycles. Here two equidimensional cycles are equivalent if their intersection indices with each cycle of complementary dimension coincide. A finest relation is the rational Chow equivalence, when equivalent cycles are fibres of a family parametrized by a chain of rational curves. The direct product of varieties induces the tensor product structure on the category.
The definition of the category of pure motives consists of a formal construction (that is a limit of) of new objects and morphisms that are pieces of varieties, i.e. kernels and images of projectors; correspondences \(p:X\rightarrow X\) with \(p^2=0\). This is what is called a \textit{pseudo-abelian} or \textit{Karoubian} completion of the category. The constructed category gives the projective line \(\mathbb P^1\) as the direct sum, i.e. \textit{motive of}, a point and the Lefschetz motive \(\mathbb L\).
For the final step in the definition, the class of objects is enhanced formally once more. They are now supposed to include all integer tensor powers \(\mathbb L^{\otimes n}\), \(n\in\mathbb Z\), and tensor products of these with other motives. In particular, \(\mathbb L^{-1}=\mathbb T\) is called the Tate motive.
The main challenge within the theory of motives, following Y. Manin, is the unification of the theory. The standard conjectures of Grothendieck still resisted all effords by the 1990's.
As is written by Y. Manin, the book by Gonçalo Tabuada is a dense combination of a survey paper and a research monograph dedicated to the development of the theory of motives during the last 25 years. The author contributed many important results and techniques in the theory in recent years, and in this book, he focuses on the \textit{noncommutative motives}. Y. Manin gives the following brief comments about this subject as treated in the book:
New age motivic constructions starts with triangulated categories and enhancement of such, i.e. differential graded (dg) categories. The classical categories of varieties are embedded there by their general enhanced derived categories, such as the categories of perfect complexes. In these derived categories, morphisms rather than objects are treated as complexes, modulo homotopy. Thus it is necessary to work with 2-categories and even categories of higher level.
Correspondences between such varieties (categories) are introduced by equivalences of module categories, that is Morita-like constructions. Here morphisms between (not necessarily commutative) rings \(A\rightarrow B\) are replaced with \((A,B)\)-bimodules, and the difference between commutative and noncommutative rings disappear because every commutative ring is Morita equivalent to the ring of matrices of any given order over it.
\textit{A. A. Beilinson} (see [Funct. Anal. Appl. 12, 214--216 (1979; Zbl 0424.14003)]) discovered that the derived category of coherent sheaves on a projective space can be described as a triangulated category of modules over a Grassmann algebra. Thus the projective space is affine in this noncommutative geometry. Beilinson's technique led to a general machinery describing triangulated categories in terms of exceptional systems and expanding the realm of candidates to the role of noncommutative motives.
The abstract properties of the categories constructed in this way gives the terminology for noncommutative geometry, and was one motivation of M. Kontsevich's project of Noncommutative Motives which is the central subject of Tabuada's book.
The shift of viewpoint required work to establish how much is lost by passing from the classical picture to the new one, and what is gained in understanding both the old and new universes of algebraic geometry. Some of these results are surveyed in Tabuada's monograph, and the book contains an ample reference list for those who want to follow specific directions of research. Y. Manin ends his preface by stating that ``This stimulating book will be a precious source of information for all researchers interested in algebraic geometry''.
This was a resume of Manin's review of this book, in which we totally agree. Some further comments can be given:
This is not a textbook for learning motives. The first chapter recall the notion of \textit{Differential Graded Categories}, and can be understood by readers with knowledge of Quillens derived categories and model categories. Then we learn about \textit{Drinfeld's DG quotient}, \textit{Bondal-Kapranov's pretriangulated envelope} and \textit{Kontsevich's smooth proper dg categories}. The second chapter treats \textit{Additive invariants}. This is the basis for motives, and a result of this chapter is the establishment of a general \textit{Lefshcetz's fixed point formula}.
Chapter two ends the ``text-book'' part of the book. Chapter three, which gives the background on pure motives, is based on André's and Manin's introductions to \textit{Chow motives}, and the construction is not recalled in this book. The chapter contains the definitions of \textit{Homological motives}, \textit{Numerical motives}, and \textit{Artin motives}, and ends the review of the commutative situation.
Chapter four is the engine in the book. It introduce several categories of noncommutative pure motives: \textit{Noncommutative Chow motives}, \textit{Noncommutative \(\otimes\)-nilpotent motives}, \textit{Noncommutative homological Motives}, and \textit{Noncommutative numerical Motives}. These are related to their commutative counterparts. In the case of noncommutative Chow notives, this gives applications to motivic measures, motivic zeta functions, motivic decompositions, and the simplification of Dubrovin's conjecture. Nice developments in the chapter leads to the definition of \textit{noncommutative motivic Galois groups} and the extension of the classical theory of intermediate Jacobians to noncommutative Chow motives.
Chapter five introduces the noncommutative analogues of the standard conjectures of type C and D, of Voevodsky's nilpotence conjecture, and of Kimura-finiteness. These noncommutative analogues are related with their commutative counterparts, chapter six introduce the noncommutative motivic Galois (super-)groups, relates them with their commutative counterparts, and establish a base-change short exact sequence. An unconditional version of the noncommutative motivic Galois groups is discussed.
Chapter seven contains the definition and recent results on the Jacobians of noncommutative Chow motives, and the references must be consulted to understand this chapter. The same is for chapter eight on \textit{Localizing invariants}, which use algebraic \(K\)-theory as a central element.
In the final chapter, nine, the author associate to each of the universal invariants given in chapter 8 a triangulated category of \textit{noncommutative mixed motives}. In the case of the \textit{universal additive invariant}, the associated category of noncommutative mixed motives comes equipped with a weight structure whose heart is the category of noncommutative Chow motives. This leads to \textit{weight spectral sequences} and to the computation of the Grothendieck ring of several triangulated categories in this study. The chapter includes applications to noncommutative mixed Artin motives and to Kimura-finiteness. differential graded categories; Drinfeld's DG quotient; Morita equivalence; noncommutaive Chow motives; noncommutative mixed motives; noncommutative motivic Galois group; noncommutative pure motives; witt vectors; Morel-Voevodsky's motivic homotopy theory; noncommutative mixed Artin motives; Kimura finiteness; Grothendieck derivators Tabuada, Gonçalo, Noncommutative motives, with a preface by Yuri I. Manin, University Lecture Series 63, x+114 pp., (2015), American Mathematical Society, Providence, RI Research exposition (monographs, survey articles) pertaining to algebraic geometry, Noncommutative algebraic geometry, (Equivariant) Chow groups and rings; motives, Enriched categories (over closed or monoidal categories), Derived categories, triangulated categories, Nonabelian homotopical algebra, \(K\)-theory and homology; cyclic homology and cohomology Noncommutative motives | 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 a quasi-quadratic time algorithm to compute the canonical lift of the Jacobian of an ordinary hyperelliptic curve in characteristic \(3\). This is done by an approximation process analogue to the characteristic \(2\) case developed in [Ramanujan J. 12, No. 3, 399--423 (2006; Zbl 1166.11021)]. The authors use the language of algebraic theta constants and work out equations for the higher dimensional analogue of the modular curve \(X_0(3)\). Actually these formulas can be seen as a \(3\)-adic analogue of Mestre's generalized AGM equations for hyperelliptic curves. One of the applications of the paper is then to derive a method for the construction of genus \(2\) curves over small degree number fields whose Jacobian has complex multiplication and good ordinary reduction at the prime \(3\). Tables can be found at \url{http://echidna.maths.usyd.edu.au/~kohel/dbs/complex_multiplication2.html}.
As the authors point out, over finite fields, their methods do not contain information about the Weil polynomial of the curve and cannot be used for point counting in characteristic greater than \(2\). However, this seems to be solved in a forthcoming paper. CM-methods; canonical lift; theta functions; modular equations Carls, R; Kohel, D; Lubicz, D, Higher-dimensional 3-adic CM construction, J. Algebra, 319, 971-1006, (2008) Theta functions and abelian varieties, Theta functions and curves; Schottky problem Higher-dimensional 3-adic CM construction | 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 a series of foregoing papers, the author has already very extensively pursued the general problem of developing an appropriate and efficient approach to representing, in the framework of abstract category theory, various scheme-theoretic geometries, among them being the absolute anabelian geometry of hyperbolic curves over \(p\)-adic local fields, the geometry of locally noetherian log schemes, the geometry of log schemes with archimedean structure, and others. Many of the author's constructions, especially those concerning Grothendieck's theory of the algebraic fundamental group of a Galois category as developed in SGA 1, turned out to be valid on a rather abstract category-theoretic level, at least so when working with the category of finite étale covers of a given base scheme.
In this context, the author invented his so-called ``anabelioids'', which are basically Cartesian products of categories of finite sets equipped with a continuous action by a fixed profinite group \(G\), and which serve to formalize the concept of a finite étale cover in a purely category-theoretic way the author [Publ. Res. Inst. Math. Sci. 40, No. 3, 819--881 (2004; Zbl 1113.14021)]. On the other hand, in his work on the anabelian geometry of hyperbolic curves over \(p\)-adic local fields, the author developed a certain geometry of ``semi-graphs of profinite groups'' [in: Galois theory and modular forms, Dev. Math. 11, 77--122 (2004; Zbl 1062.14031)]. Now, in the paper under review, these previously elaborated abstract theories are brought together in order to create an even more abstract, unifying framework, namely the formalism of ``semi-graphs of anabelioids''. As the author points out, the aim of the present paper is to deliver ``a piece of mathematical infrastructure'' in maximal possible generality, that is, to exhibit both the formal basic properties of semi-graphs of anabelioids and the ``general nonsense'' they transpire, just to quote his own keywords. However, the author's highly abstract approach leads to several interesting new results and deeper insights, including
(1) an analogue of Zariski's main theorem for certain types of morphisms of semi-graphs;
(2) certain properties of the profinite fundamental group associated to a graph of anabelioids;
(3) a generalization of \textit{Y. André's} concept of tempered, fundamental groups [Duke Math. J. 119, No. 1, 1--39 (2003; Zbl 1155.11356)] to so-called ``temperoids'';
(4) a localization theory for semi-graphs of anabelioids, with applications to the geometry of formal localizations of stable log curves;
(5) an analogue of the author's anabelian geometry \textit{S. Mochizuki} [Nagoya Math. J. 179, 17--45 (2005; Zbl 1129.14042)] with respect to tempered fundamental groups; and
(6) some new graph-theoretic applications to the study of free groups.
As for the presentation of all these abstract new concepts, methods, techniques, and results, the exhibition is fairly self-contained, lucid, detailed and well-structured. The author explains once again the basic notions from his foregoing papers as used in the present text, and he gives a plenty of motivations, supplementary remarks, hints for further reading, and concrete algebro-geometric examples. curves over arithmetic ground fields; fundamental groups; categories; localization of categories; graph theory; profinite groups; free nonabelian groups Mochizuki S., Semi-graphs of anabelioids, Publ. Res. Inst. Math. Sci. 42 (2006), no. 1, 221-322. Coverings of curves, fundamental group, Universal profinite groups (relationship to moduli spaces, projective and moduli towers, Galois theory), Homotopy theory and fundamental groups in algebraic geometry, Arithmetic ground fields for curves, Localization of categories, calculus of fractions, Free nonabelian groups Semi-graphs of anabelioids | 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' motivation for the work under review comes from a possible resolution of the so-called Zilber-Pink conjecture, for the coarse moduli space of principally polarised abelian surfaces. If we denote the moduli space of principally polarised abelian surfaces over \(\mathbb{C}\) by \(\mathcal{A}_2\), then this conjecture asserts that any irreducible algebraic curve \(V\) in \(\mathcal{A}_2\) that is not contained in any proper special subvariety, should have only finitely many intersections with the special curves of \(\mathcal{A}_2\).
A general method of resolving such problems that involve unlikely intersections -- known as the Pila-Zannier strategy -- has two components: one concerns the Galois orbits of points arising from such unlikely intersections, and the other arising from the parametrisation of special subvarieties using reduction theory of arithmetic groups. The special curves alluded to above are of three types, and in their earlier work, the authors have treated one type. Here, they extend that work to other cases -- a task that is by no means straightforward.
The authors start by proving quantitative bounds for the group elements used in Borel and Harish-Chandra's construction of the Siegel sets. More precisely, they obtain polynomial bounds on the lengths of integral vectors in a rational representation of a reductive group, and polynomial bounds in the construction of fundamental sets for arithmetic subgroups of reductive groups, when the latter vary in a real conjugacy class of subgroups of a fixed reductive group.
Using these quantitative results on reduction theory, they prove the first of the two main theorems which is a proof of the Zilber-Pink conjecture for \(\mathcal{A}_2\) assuming the truth of the so-called large Galois orbits hypothesis -- as mentioned above, one of the three cases had been treated by them in their earlier paper.
The validity of the Galois orbits hypothesis (which is a conjecture on certain Galois orbit being large in terms of a notion called its `complexity') for one of the three cases had been established in the earlier paper, and in the paper under review, the authors prove the analogous result for one of the other two cases left out -- namely, for points corresponding to abelian surfaces with quaternionic multiplication, under certain geometric conditions.
The authors remark that these tools would have wide-ranging applications in the future too. To quote them verbatim:
``We note that the results on quantitative reduction theory in this paper will be an important tool for proving the Zilber-Pink conjecture for other Shimura varieties, which will be the subject of future work by the authors. We expect these results to have further applications, for example a uniform version of the second-named author's bounds for polarisations and isogenies of abelian varieties and bounds for the heights of generators of arithmetic groups by combining them with some of the techniques of homogeneous dynamics.'' Borel; Harish-Chandra; reduction theory; principally polarized; unlikely intersections; Shimura variety; Zilber-Pink conjecture Structure of modular groups and generalizations; arithmetic groups, Arithmetic aspects of modular and Shimura varieties, Moduli, classification: analytic theory; relations with modular forms, Discrete subgroups of Lie groups Quantitative reduction theory and unlikely intersections | 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 rational map from \(\mathbb P^n\) to itself can be described by \(n+1\) homogeneous polynomials of degree \(d\) in \(n+1\) variables, so coefficients of the polynomials define a point on \(\mathbb P^N\) for \(N+1 = (n+1)\binom {n+d}d\). Within this \(\mathbb P^N\), the set of morphisms forms an open affine variety \(\mathrm{Hom}_d^n\). Choosing a different coordinate on \(\mathbb P^n\) amounts to a conjugation by an element of \(\mathrm{PGL}(n+1)\), so the natural moduli space for dynamical systems on \(\mathbb P^n\) is the quotient of \(\mathrm{Hom}_d^n\) by \(\mathrm{PGL}(n+1)\). \textit{C. Petsche, L. Szpiro}, and \textit{M. Tepper} [J. Algebra 322, No. 9, 3345--3365 (2009; Zbl 1190.14013)] have shown that this exists as a geometric quotient in the sense of Geometric Invariant Theory, generalizing the result of \textit{J. Silverman} [Duke Math. J. 94, No. 1, 41--77 (1998; Zbl 0966.14031)] for \(n=1\). There is also a previous work by the author [Acta Arith. 146, No. 1, 13--31 (2011; Zbl 1285.37020)], explicitly describing stable and semistable loci and giving a bound on on the stabilizer group of a point in the moduli space.
In this article, the author restates semistable reduction theorem for a closely-related moduli space given by the quotient of \(\mathrm{Hom}_d^n\) by \(\mathrm{SL}(n+1)\), and further proves some refinements. The semistable reduction theorem is a standard result in GIT, stating that the semistability on the generic fiber implies that on the special fiber, possibly after taking a finite cover. In the current context, this means that given a complete curve \(C\) and a semistable map \(\varphi\) on \(\mathbb P^n_{K(C)}\), there exist an abstract curve \(D\) mapping finite-to-one onto \(C\) and a self-map \(\Phi\) on a \(\mathbb P^n\)-bundle on \(D\) such that \(\Phi\) is equivalent to the pullback of \(\varphi\) to \(D\) upon some coordinate changes and the reduction of \(\Phi\) at each point of \(D\) is semistable. In particular, one can apply this theorem to a curve \(C\) contained inside the semistable part \(M_d^{n,ss}\) of the moduli space, since each point of \(C\) represents a semistable self-map of \(\mathbb P^n\). The author then answers several natural questions arising from this setup. First, he proves that for any \(n\) and \(d\), there exists a curve in \(M_d^{n,ss}\) such that no trivial bundle on a finite cover satisfies the semistable reduction. As a result, such a curve does not have a complete curve in \(\mathrm{Hom}_d^{n,ss}\) as a finite cover. Secondly, the author shows that an infinitely many non-isomorphic bundle classes can occur for some rational curves in \(M_d^{n,ss}\). Thirdly, the author proves that if the trivial bundle on \(D\) satisfies semistable reduction, the degree of the cover \(D\to C\) is bounded in terms of the sizes of the stabilizer groups for the points on \(D\). The proofs involve standard GIT arguments together with explicit analyses of the closures of \(\mathrm{PGL}(n+1)\)-orbits of some polynomial families. GIT; semistable reduction; moduli space; dynamical system [10]A. Levy, The semistable reduction problem for the space of morphisms on Pn, Algebra Number Theory 6 (2012), 1483--1501. Geometric invariant theory, Families and moduli spaces in arithmetic and non-Archimedean dynamical systems, Arithmetic dynamics on general algebraic varieties The semistable reduction problem for the space of morphisms on \(\mathbb P^n\) | 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 finite extension of \(\mathbb{Q}_p\), \(G\) be its absolute Galois group, \(A\) be an abelian variety defined over \(K\), \(\mathcal A\) be its Néron model (over the valuation ring of \(K\)) and \(\Gamma\) be the geometric points of the group of components of \({\mathcal A}_0\), this last one being the special fiber of \(\mathcal A\). One knows, following Grothendieck, that for all prime \(l\neq p\), \(\Gamma(l)\simeq H^1(I,T_l(A))_{\text{tors}}\), where \(I\) is the inertia subgroup of \(G\) and \(T_l(A)\) is the \(l\)-adic Tate module of \(A\). One knows also that this formula is wrong for \(l = p\). The aim of the authors is to obtain such formula when \(l = p\).
To do that, they define the functors \(\text{Chrys}_h\) on the category of \(\mathbb{Q}_p\)-representation of \(G\): if \(V\) is such a representation, \(\text{Chrys}_h(V)\) is the maximal crystalline subrepresentation of \(V\) with Hodge-Tate weights in \([0,h]\). The authors prove that if \(K/\mathbb{Q}_p\) is unramified, then \(\Gamma(p) \simeq R^1\text{Chrys}(T_p(A))_{\text{tors}}\). group of connected components; crystalline subrepresentation; abelian variety; Néron model; \(l\)-adic Tate module; Hodge-Tate weights Abelian varieties of dimension \(> 1\), Arithmetic ground fields for abelian varieties Crystalline subrepresentations and Neron 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 The aim of this paper is to give a new approach to the relative analogue of the theory of \((\varphi,\Gamma)\)-modules. In the classical setting, \((\varphi,\Gamma)\)-modules are modules over certain rings of power series, equipped with a semi-linear Frobenius map \(\varphi\) and a compatible action of a group \(\Gamma\). They were defined by \textit{J.-M. Fontaine} [Prog. Math. 87, 249--309 (1990; Zbl 0743.11066)] who used them to describe the \(p\)-adic representations of the Galois group of a \(p\)-adic field. A key input in Fontaine's work is the theory of the field of norms, which gives an isomorphism between a subgroup of the Galois group of a \(p\)-adic field and the Galois group of a local field of characteristic \(p\).
In the relative setting, the Galois group of a \(p\)-adic field is replaced by the étale fundamental group \(G\) of an affinoid space over a finite extension of \(\mathbb{Q}_p\). One is still interested in the representations of \(G\) over finite-dimensional \(\mathbb{Q}_p\)-vector spaces. Thanks to the work of many people (including Andreatta, Brinon, Faltings and Scholl), it progressively became clear that these representations of \(G\) could be studied using a suitable generalization of Fontaine's \((\varphi,\Gamma)\)-modules. This paper provides such a generalization, relying heavily on the theory of Witt vectors and the geometric spaces attached to them.
In addition to the introduction, the paper has nine chapters. The introduction is informative and provides a good overview of the contents of the paper as well as a discussion of the relations between the theory developed in this paper, Scholze's theory of perfectoïd spaces, and Fargues-Fontaine's curve in \(p\)-adic Hodge theory. The first chapter contains some algebro-geometric preliminaries. The second chapter discusses the spectra of nonarchimedean Banach rings. In the paper, both the Gelfan'd spectrum and the adic spectrum are used. The third chapter is about strict \(p\)-rings, and contains the generalization of the theory of the field of norms, namely the perfectoïd correspondence. The fourth chapter is about Robba rings and slope theory, and the fifth chapter is about relative Robba rings. The sixth chapter concerns \(\varphi\)-modules and includes a discussion of the relation with Fargues and Fontaine's constructions. The seventh chapter is about slope filtrations in families, and the eighth is about perfectoïd spaces. The ninth chapter contains the construction of the relative analogue of the theory of \((\varphi,\Gamma)\)-modules and the various equivalences of categories that one gets. The classical \((\varphi,\Gamma)\)-modules are replaced by sheaves of \(\varphi\)-modules over rings of period sheaves. The sheaf axiom (for the pro-étale topology) replaces the classical action of \(\Gamma\).
The last paragraph of the introduction contains a list of ``further goals'' that the authors plan to work on. \((\varphi,\Gamma)\)-module; \(p\)-adic Hodge theory; Robba ring; perfectoid space; pro-etale topology; Witt vectors; adic spectrum; Gelfan'd spectrum; affinoid; Banach algebra; slope filtration; etale local system; Fargues-Fontaine curve K. Kedlaya et R. Liu, Relative \(p\)-adic Hodge theory: Foundations, Astérisque 371, Soc. Math. France, Paris, 2015. Rigid analytic geometry, Ramification and extension theory, Analytical algebras and rings, Witt vectors and related rings, Transcendental methods, Hodge theory (algebro-geometric aspects), \(p\)-adic cohomology, crystalline cohomology Relative \(p\)-adic Hodge theory: foundations | 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 \(Y\) be an irreducible reduced curve with nodes and cusps as singularities and let \(E\) be a globally generated torsion-free sheaf. The authors prove two theorems concerning the (semi)stability of the kernels of the maps \(H^0(Y,E)\otimes\mathcal{O}_Y\to E\to 0\) and \(H^0(Y,K_Y)\otimes\mathcal{O}_Y\to K_Y\to 0\). Using these results they study the Brill-Noether loci \(B(n,d,k)\) of stable torsion-free sheaves over Y of rank n, degree d and with at least k independent sections. In particular they give conditions for the non-emptyness of \(B(n,d,k)\), find its Zariski tangent space, its singular set and provide an example. As further result the authors study the stability properties of the restriction of the Picard bundle to the image of a map \(Y \to \overline{J}^d(Y)\), where \(\overline{J}^d(Y)\) denotes the compactified Jacobian of \(Y\). Brill-Noether loci; semistability; Picard bundle; globally generated torsion-free sheaves; nodal; cuspidal; curve Bhosle, UN; Singh, SK, Brill-Noether loci and generated torsion free sheaves over nodal or cuspidal curves, Manuscr. Math., 141, 241-271, (2013) Vector bundles on curves and their moduli, Algebraic moduli problems, moduli of vector bundles Brill-Noether loci and generated torsionfree sheaves over nodal and cuspidal 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 an abelian variety over an algebraically closed field \(k\). The author considers the functor of germs of formal curves on \(X\); on \(S\)-points, this is \({\mathcal C}X(S) = \text{ Hom}(\text{Spf}k[\![t]\!] \times S, X)\). For any \(k\)-scheme \(Y\), let \(Y[n] = Y\times \text{ Spec}k[t]/t^n\). The main theorem states that \({\mathcal C}X\) is represented by \(\text{Pic}^0(\lim_{\rightarrow n}X[n])\). The proof relies on a characterization of \(\ker (\text{Pic}(Y[n]) \rightarrow \text{ Pic}(Y))\) for a proper scheme \(Y\) and on the identification of \(\text{Hom}(S[n], \text{Pic}(\hat X))\) with \(\text{Pic}(\hat X[n])(S)\). (In fact, the proof of Theorem 3.4 shows that \({\mathcal C}X\) is represented by \(\text{Pic}^0(\lim_{\rightarrow n}\hat X[n])\); there is a missing duality in the last paragraph on page 101.) abelian variety; Picard scheme; formal curve Picard schemes, higher Jacobians, Algebraic theory of abelian varieties The scheme of formal curves on an Abelian 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 Let \(C\) be a smooth irreducible curve of genus \(g\geq 2\) over the complex numbers, and let \(M(n,\xi)\) be the moduli space of stable vector bundles \(V\) of rank \(n\) on \(C\) such that \(\text{det }V\cong \xi\), where \(\xi\) is a line bundle of degree \(d\) such that \(n\) and \(d\) are coprime. In this paper, the author proves the Hodge conjecture Hodge\((p,p)\) for the cases \(n=3\), \(g=2\) and \(n=2\), \(g=3,4\).
In the case \(n=2\), results of \textit{M. Thaddeus} [Invent. Math. 117, No. 2, 317-353 (1994)] show that \(M(2,\xi)\) is dominated by a variety that is obtained from a projective space by blowing up and down along smooth centers of dimension at most 3, and the Hodge conjecture follows by an argument of \textit{J. P. Murre} [Nederl. Akad. Wet., Proc., Ser. A 80, 230-232 (1977; Zbl 0352.14006)].
The paper is mainly devoted to the case \(n=3\). The author uses the method of normal functions (due to H. Poincaré and S. Lefschetz, and further developed by P. Griffiths and S. Zucker). In this case the normal functions are not associated to a Lefschetz pencil, but to a fibration over a curve whose fibers are projective bundles over \(M(2,\xi)\). The proof is finished using Hecke correspondences. moduli space; Hodge conjecture; normal functions; Hecke correspondences Transcendental methods, Hodge theory (algebro-geometric aspects), Families, moduli of curves (algebraic), Variation of Hodge structures (algebro-geometric aspects) The Hodge conjecture for certain moduli 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 This paper is devoted to the purely category theoretic background of Grothendieck's six functor formalism with a view towards applications in algebraic geometry and algebraic topology. The basic theory is developed without assuming the categories to be triangulated, so the results hold both before and after passing to derived categories.
The general setup considered in the paper is two pairs of adjoint functors between closed symmetric monoidal categories, which are suggestively denoted by \((f^*,f_*)\) and \((f_!,f^!)\), respectively. The main aim of the authors is to clarify which relations among these adjoint pairs and the \(\otimes\)- and Hom-functors can be deduced on a purely formal level, and in which points additional input from the concrete situation is needed.
Apart from the general situation (referred to as the Verdier-Grothendieck-context) the authors also consider two special cases, namely \(f_*=f_!\) (the Grothendieck-context), and \(f^*=f^!\) (the Wirthmüller-context). Relations between the functors are then studied using dualizing objects.
In the last two sections, the authors pass to the setting of triangulated categories and prove, under additional assumptions on the categories in question, formal versions of the Grothendieck-, the Verdier-, and the Wirthmüller-isomorphism theorems. closed symmetric monoidal category; pair of adjoint functors; triangulated category; Grothendieck duality; Verdier duality; Wirthmüller isomorphism H. Fausk, P. Hu and J.\ P. May, Isomorphisms between left and right adjoints, Theory Appl. Categ. 11 (2003), no. 4, 107-131. Closed categories (closed monoidal and Cartesian closed categories, etc.), Varieties and morphisms, Monoidal categories (= multiplicative categories) [See also 19D23], Derived categories, triangulated categories, Applied homological algebra and category theory in algebraic topology, Categories in geometry and topology Isomorphisms between left and right adjoints | 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 [Lond. Math. Soc. Lect. Note Ser. 242, 85--95 (1997; Zbl 0902.14019)], \textit{T. Oda} showed that the the profinite completions of the étale homotopy type of the moduli stack of hyperbolic curves of genus \(g\) with \(n\) marked points over \(\overline{\mathbb Q}\) and of the Eilenberg-Maclane space \(K(\Gamma_{g,n},1)\) of the corresponding mapping class group \(\Gamma_{g,n}\) are weakly equivalent. Here the authors prove an analogous result for the moduli stack of principally polarized abelian varieties of dimension \(g\geq 1\) over \(\overline{\mathbb Q}\): the profinite completion of its étale homotopy type is weakly equivalent to the profinite completion of the Eilenberg-Maclane space \(K(\text{Sp}(2g,{\mathbb Z}),1)\). In fact, the result is stated for more general moduli stacks of polarized abelian varieties but the argument is the same.
The strategy of the proof is the same as that of Oda: combining the Artin-Mazur comparison theorem between étale and complex analytic homotopy types together with a homotopy descent theorem of Cox, the authors reduce the computation of the étale homotopy type of the moduli stack to that of the homotopy type of its complex analytification. They then compute the latter by means of the analytic hypercovering obtained as the Čech nerve of the uniformization map by the Siegel upper half space.
The paper is very clearly written and recalls a lot of background material for the benefit of the reader. étale homotopy theory; algebraic stacks; moduli of abelian schemes; principally polarised abelian varieties Homotopy theory and fundamental groups in algebraic geometry, Families, moduli of curves (algebraic), Families, moduli of curves (analytic) Étale homotopy types of moduli stacks of polarised 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 This treatise is the author's doctoral dissertation accomplished in 2006 at the University of Münster, North-Rine Westfalia, Germany, under the academic advice of Professor Helmut A. Hamm.
There are two main problems investigated in this work, both of which deal with algebraic morphisms \(f: X\to S\) from a non-singular affine complex surface \(X\) to a smooth complex curve \(S\).
The first problem is to establish sufficient conditions under which the morphism \(f: X\to S\) defines a differentiable fiber bundle, and the second, closely related problem is to determine conditions implying that \(f: X\to S\) is even a locally trivial holomorphic fiber bundle. Both in the differential-geometric and in the complex-analytic framework, there are well-known criteria for local triviality of such a morphism, due to the classical Ehresmann Fibration Theorem, on the one hand, and to a more recent theorem by \textit{H. V. Hà} and \textit{D. T. Lê} [Acta Math. Vietnam. 9, 21--32 (1984; Zbl 0597.32005)], on the other hand.
Playing a key role in the author's present work, these two specific and meanwhile classical theorems are modified and extended to the algebraic case mentioned above. However, as the analytic methods of proof are not directly transferable to morphisms of algebraic schemes, the author has to apply heavy algebro-geometric, techniques and theorems in order to achieve analoguous results.
More precisely, after summarizing the necessary prerequisites on normalizations, blowing-ups, and general fiber bundles in Chapter 1, the existence of minimal regular models for fibered surfaces over a Henselian curve à la \textit{T. Sekiguchi}, \textit{F. Oort} and \textit{N. Suwa} [Ann. Sci. Éc. Norm. Supér. (4) 22, No. 3, 345--375 (1989; Zbl 0714.14024)] is applied to deduce, in Chapter 2, the author's main theorem on differentiable local triviality. This new theorem (Theorem 2.70) states the following: Let \(S\) be a smooth complex curve, and let \(X\) he regular affine, curve over \(S\) given by an algebraic morphism \(f: X\to S\). Suppose that the fibers of \(f\) are of positive geometric genus and that the fibers over closed points of \(S\) are irreducible. Then, if furthermore all fibers are pairwise homeomorphic, the relative curve \(f: X\to S\) defines a \(C^\infty\)-fiber bundle.
Chapter 3 turns then to the problem of holomorphic local triviality, which is actually investigated in three parts, namely for families of hyperbolic Riemann surfaces, for families of elliptic curves, and for families of genus-zero curves (spheres), respectively.
In all these special cases, the author obtains a sufficient criterion for holomorphic local triviality, mainly by appropriately modifying the classical local triviality theorem of \textit{W. Fischer} and \textit{H. Grauert} [Nachr. Akad. Wiss. Göttingen, II. Math.-Phys. Kl. 1965, 89--94 (1965; Zbl 0135.12601)] and by exploiting the famous rigidity theorem of Arakelov-Parshin-Manin-Grauert for relative algebraic curves [cf. \textit{D. Mumford}, ``The red book of varieties and schemes.'' Lect. Notes Math. 1358 (1999; Zbl 0945.14001)].
As for families of elliptic curves, a crucial theorem on singular fibers due to \textit{A. Beauville} [Astérisque 86, 97--108 (1981; Zbl 0502.14009)] is deftly utilized. Finally, the last section of the present Ph.D. thesis addresses the problem of global triviality of holomorphic fiber bundles having Riemann surfaces as fibers, where the total space \(X\) is not necessarily affine. Using the framework of non-abelian cohomology and \textit{H. Grauert's} structure theorem for analytic fibrations [Math. Ann. 135, 263--273 (1958; Zbl 0081.07401)], a new cohomological global triviality criterion is derived. families of curves; holomorphic fiber bundles; structure of families; differentiable fiber bundles; minimal model program Families, moduli of curves (algebraic), Structure of families (Picard-Lefschetz, monodromy, etc.), Fibrations, degenerations in algebraic geometry, Minimal model program (Mori theory, extremal rays), Holomorphic bundles and generalizations, Fiber bundles in algebraic topology Locally trivial families of complex 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 According to the Grothendieck philosophy, the algebraic stacks behave like the schemes. The Diophantine approximations in the stack of \(G\)-torsors give results about the 1-cohomology of the algebraic group \(G\). We first establish results (of the type ``Hasse principle'' for example) about the 2-cohomology of an algebraic group \(G\) obtained by Diophantine approximations in the \(G\)-gerbs. Next, we show that the Diophantine approximations in the 2-gerbs and 2-stacks provide results about the 3-cohomology. Our general approach unifies several partial results of \textit{L. Moret-Bailly} [Compos. Math. 125, No. 1, 1-30 (2001; Zbl 1106.11022)], \textit{C. Scheiderer} [Invent. Math. 125, No. 2, 307-365 (1996; Zbl 0857.20024)], and others. non-Abelian cohomology of algebraic groups; Diophantine approximations; 2-gerbs; 2-stacks Cohomology theory for linear algebraic groups, Generalizations (algebraic spaces, stacks), Linear algebraic groups over global fields and their integers, Galois cohomology Diophantine approximations on algebraic stacks and non-Abelian 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 authors study a sequence of quotients \(\text{Coh}_{(c)}(X)\) of the category \(\text{Coh}(X)\) of coherent sheaves on a variety \(X\), obtained by modding out by sheaves supported in codimension~\(>c\). Assuming \(X\) irreducible smooth projective, it is shown that these categories have homological dimension \(c\), making them potentially more tractable than \(\text{Coh}(X)\) for many purposes. The authors give an application to Bridgeland stability conditions, after introducing similar quotients of the derived category \(\text{D}^b(X)\). They also give an interpretation of these categories in terms of birational geometry.
Some main results are as follows, where we write \(\text{D}^b_{(c)}(X)\) for the quotient of \(\text{D}^b(X)\) by objects supported in codimension~\(>c\). Mumford's \(\mu\)-stability naturally induces a Bridgeland stability condition on the category \(\text{D}^b_{(1)}(X)\), and the authors give a full description of the space of such stability conditions: the usual action of the universal cover of the orientation-preserving subgroup of \(\text{GL}(2,\mathbb{R})\) is found to be free, and its orbits are irreducible components, parametrized by certain classes in the cone of curves~\(N_1(X)_{\mathbb{R}}\).
The authors describe, for each \(c\), a natural class of rational maps which induce exact (pullback) functors between categories \(\text{Coh}_{(c)}\). They show that the category \(\text{D}^b_{(0)}(X)\), obtained by modding out by torsion objects, determines the birational equivalence class of \(X\). They further show that \(\text{D}^b_{(1)}(X)\) determines \(X\) up to birational equivalences which are isomorphisms in codimension~ \(1\), and that the exact autoequivalences of this category are generated by pullbacks along such birational maps, twists by line bundles, and homological shifts. stability conditions; abelian or triangulated quotient categories; birational maps Meinhardt, Sven; Partsch, Holger. \(Quotient categories, stability conditions, and birational geometry\). Geom. Dedicata 173 (2014), 365-392. Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20], Rational and birational maps, Derived categories, triangulated categories Quotient categories, stability conditions, and birational 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 Let \(G\) be an algebraic group, \(\rho\) a representation of \(G\), \(V\) a vector bundle on a smooth projective curve. The authors consider deformations of pairs \((P,\phi)\) consisting of a principal \(G\)-bundle \(P\) and a section \(\phi\) of \(\rho P\otimes V\). Section 2 contains the infinitesimal computation, while in section 3 the question is considered whether the moduli functor is formally smooth.
Then \(\rho\) is taken to be the coadjoint representation and \(V=K\) and one defines a symplectic structure on the moduli space. Confining to the pairs where \(P\) is stable, the moduli space can be identified with the cotangent bundle of the moduli space of \(G\)-bundles and the symplectic structure can be identified with the Hamiltonian structure. In the case when \(G=\text{SL}(n)\), Hitchin considered the global analogue of the map, which maps the Lie algebra of \(G\) into \(\mathbb{C}^{n+1}\), given by the coefficients of the characteristic polynomial. The authors look at the analogue of the Kostant map of the Lie algebra of \(G\) into \(\mathbb{C}^ \ell\) for all semisimple groups and show that the fibres are Lagrangian at the smooth points of the fibre and also that the symplectic form vanishes on any smooth variety in the fibre over 0. In section 6, they extend these results to pairs with parabolic structures. coadjoint representation; Lie algebra; semisimple groups I. Biswas and S. Ramanan, ''An infinitesimal study of the moduli of Hitchin pairs,'' J. London Math. Soc., vol. 49, iss. 2, pp. 219-231, 1994. Dynamical aspects of finite-dimensional Hamiltonian and Lagrangian systems, Algebraic moduli problems, moduli of vector bundles, Semisimple Lie groups and their representations An infinitesimal study of the moduli of Hitchin pairs | 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 of classical results of Grothendieck on vanishing cycles. Consider a regular, proper and flat curve \(X\) over a strictly local trait \(S=(S,s,\eta )\), whose generic fiber is smooth and whose reduced special fiber is a divisor with normal crossings. One can analyse the difference between the (étale) cohomology of the special fiber \(H^*(X_s)\) and that of the generic geometric fiber \(H^*(X_{\eta})\), the coefficients ring being \(\mathbb{Z}_{\ell}\), \(\ell\) a prime number invertible on \(S\). Assuming that the action of the inertia group \(I\) on \(H^*(X_{\bar{\eta}})\) is tame, one shows that the defect of the specialization map \(H^*(X_s)\rightarrow H^*(X_{\bar{\eta}})\) is an isomorphism controlled by the vanishing cycles groups. This construction leads to Grothendieck's local monodromy theorem and his monodromy pairing for abelian varieties over local fields. The author discusses related current developments and questions and includes the proof of an unpublished result of Gabber giving a refined bound for the exponent of unipotence of the local monodromy for torsion coefficients. étale cohomology; monodromy; Milnor fiber; nearby and vanishing cycles; alteration; hypercovering; semistable reduction; intersection complex; abelian scheme; Picard functor; Jacobian; Néron model; Picard-Lefschetz formula; \(\ell\)-adic sheaf Research exposition (monographs, survey articles) pertaining to algebraic geometry, History of algebraic geometry, Development of contemporary mathematics, Galois representations, Abelian varieties of dimension \(> 1\), Derived categories and commutative rings, Structure of families (Picard-Lefschetz, monodromy, etc.), Étale and other Grothendieck topologies and (co)homologies, Local ground fields in algebraic geometry, Picard schemes, higher Jacobians, Arithmetic ground fields for curves, Formal groups, \(p\)-divisible groups, Group schemes Grothendieck and vanishing cycles | 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 \(g\geq 2\) and \(J\) its Jacobian variety. For a fixed point \(x_0\in C\), let \(\iota: C\to J\) denote the natural map given by \(x\mapsto{\mathcal O}_C(x- x_0)\). The tautological ring \({\mathcal T}\) is defined to be the smallest subring of the Chow ring \(\mathrm{CH}(J)\) containing the class \(\iota_*[C]\) which is closed under the pull-backs \(k^*\) and push-forwards \(k_*\) for all \(k\in\mathbb{Z}\), and under the Fourier transform \({\mathcal F}\). Let \(\iota_*[C]= \sum_{0\leq i\leq g-1} C_{(i)}\), where \(C_{(i)}\in \mathrm{CH}^{g-1}_{(i)}(J)\), the subspace on which \(k^*\) acts through the multiplication by \(k^{2(g-1)-i}\). For a symmetric theta divisor \(\theta\) on \(J\), one defines \(p_i= {\mathcal F}(C_{(i-1)})\in \mathrm{CH}^i_{(i-1)}(J)\) \((i\geq 1)\), \(q_i={\mathcal F}(C_{(i)}\cdot\theta)\in \mathrm{CH}_{(i)}\) \((i\geq 0)\).
The main purpose of this paper is to investigate which elements of the set \(\{p_i,q_j\}\) are needed to generate the tautological ring \({\mathcal T}\). Among other things the authors prove that if \(C\) admits a pure ramification of degree \(d\), then \({\mathcal T}\) is generated by the classes \(p_i\), \(i\leq d-1\) and \(q_1\). algebraic cycle; Jacobian variety; Chow ring Algebraic cycles, Subvarieties of abelian varieties, (Equivariant) Chow groups and rings; motives On the tautological ring of a Jacobian modulo rational equivalence | 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, geometrically irreducible variety \(X\) defined over a number field \(k\). The authors are interested in computing the Brauer--Manin obstruction to the Hasse principle and weak approximation. They present an efficient general procedure for computing the obstruction related to the algebraic part of the Brauer group (i.e. to the kernel of the map (\({\text{{Br}}}(X)\to {\text{{Br}}}(X\times _k\bar k)\)) under the assumptions that the geometric Picard group \({\text{{Pic}}}(X\times _k\bar k)\) is finitely generated, torsion-free, and explicitly given by generators (a collection of codimension one geometric cycles) and relations with an explicit Galois action. The authors use one of the earliest counter-examples to the Hasse principle (the Cassels--Guy diagonal cubic surface) to illustrate each step of the proposed algorithm. Brauer--Manin obstruction; Hasse principle; rational points Kresch A., Tschinkel Yu., Effectivity of Brauer-Manin obstructions, Adv. Math., 2008, 218(1), 1--27 Global ground fields in algebraic geometry, Brauer groups of schemes Effectivity of Brauer-Manin obstructions | 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 tautological cycle classes on the Jacobian of a curve getting some general results about the structure of the ring of tautological classes on a curve . Further he lifts a result of \textit{F. Herbaut} [Compos. Math. 143, 883--899 (2007; Zbl 1187.14006)] and \textit{G. van der Geer} and \textit{A. Kouvidakis} [Compos. Math. 143, 900--908 (2007; Zbl 1125.14005)] to the Chow ring (as opposed to its quotient modulo algebraic equivalence) and he gives a method to obtain further explicit cycle relations. As an ingredient for this he proves a theorem about how Polishchuk's operator \(\mathcal{D}\) lifts to the tautological subalgebra of the Chow ring CH\((J)\) of the Jacobian \(J\) of a curve. Jacobian varieties; Chow ring; tautological cycles Moonen B.: Relations between tautological cycles on Jacobians. Comment. Math. Helv. 84(3), 471--502 (2009) Algebraic cycles, Jacobians, Prym varieties Relations between tautological cycles on 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 We show, under certain assumptions, a result towards the Serre conjecture for \(\mathrm{GSp}\) as formulated in another paper with \textit{F. Herzig} [J. Reine Angew. Math. 676, 1--32 (2013; Zbl 1312.11042)]: if the residual representation associated to a genus two cusp form of \(p\)-small weight, \(p\)-ordinary of prime-to-\(p\) level, leaves stable two distinct lines (instead of one) in a Lagrangian plane, then this form admits a companion form of prescribed weight. Our proof produces only a \(p\)-adic eigenform. It consists in translating, thanks to Faltings' mod \(p\) comparison theorem, the existence of the companion form into that of a solution of a differential equation provided by the dual BGG complex on the ordinary locus of the Siegel variety. The main limitation of the method is that of the Fontaine-Laffaille theory. On the other hand, it should apply to other groups admitting PEL Shimura varieties. Siegel modular forms; Galois representations; Siegel varieties; Hecke correspondences; modulo \(p\) representations of algebraic groups; modulo \(p\) modular forms Galois representations, Siegel modular groups; Siegel and Hilbert-Siegel modular and automorphic forms, Modular correspondences, etc., Congruences for modular and \(p\)-adic modular forms, Modular and Shimura varieties Companion forms and the dual BGG complex for \(\mathrm{GSp}_4\) | 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 article under review, Poirier generalizes the geometric Langlands correspondence over \(\mathbb C\) from smooth curves to curves with ordinary multiple point singularities.
In the first section, the author works within the category of complex analytic spaces. For an integral projective singular curve \(X\) of positive genus, she considers the image \(\underline{\Omega}\) of the sheaf of holomorphic differentials on \(X\) within the direct image of the sheaf of holomorphic differentials of the normalization of \(X\). She defines a point \(P\) on \(X\) to be a mild singularity of \(X\) if the natural morphism \(\underline{d}_P:\mathcal O_{X,P}\rightarrow\underline{\Omega}_P\) satisfies \(\underline{d}_P\mathfrak m_P^s=\mathfrak m_P^{s-1}\underline{\Omega}_P\) for all \(s\geq 1\), where \(\mathfrak m_P\) denotes the maximal ideal of \(\mathcal O_{X,P}\). If \(P\) is mildly singular, then \(\underline{d}_P\) is surjective. Examples for mild singularities include special singularities, which are singular points \(P\) of \(X\) with the property that the conductor of \(\mathcal O_{X,P}\) equals \(\mathfrak m_P\); ordinary multiple points are special. The author gives an example of a curve \(X_{\text{not mild}}\) with a cusp singularity \(P\) where \(\underline{d}_P\) fails to be surjective. Poirier then defines connections \((\mathcal{M},\nabla)\) on general curves \(X\) as above, using the sheaf \(\underline{\Omega}\) instead of the sheaf of holomorphic differentials \(\Omega^1_{X/\mathbb C}\). For the curve \(X_{\text{not mild}}\), she gives an example of a connection whose kernel is not a local system, thereby showing that the equivalence of categories between connections and local systems, valid for smooth curves, fails for \(X_{\text{not mild}}\). She then proves the first main theorem of her paper, stating that if \(X\) has at most mildly singular points, then the functor \((\mathcal{M},\nabla)\mapsto\ker(\mathcal{M},\nabla)\) establishes an equivalence between the category of connections on \(X\) and the category of local systems on \(X\). Her proof consists in finding, locally at a singular point \(P\) and after a choice of basis for \(\mathcal{M}_P\), a fundamental matrix \(Y\) for \(\nabla_P\). Poirier first constructs a formal solution \(\hat{Y}\) to the problem, with entries in the formal completion \(\hat{\mathcal O}_{X,P}\) of \(\mathcal O_{X,P}\), using the mildness assumption; she then uses general results on differential equations to show that the entries of \(\hat{Y}\) are convergent.
In the second section, Poirier continues to work in the above setting, albeit imposing the stronger assumption that the singularities of \(X\) be at most multiple ordinary points. A local system \(E\) on \(X\) yields, via pullback, a local system \(s^*E\) on the normalization \(s:\tilde{X}\rightarrow X\) of \(X\). To go the opposite way, level structures are required. Poirier defines notions of level structures for local systems and connections on \(\tilde{X}\): a level structure on a local system \(E\) on \(\tilde{X}\) is defined to consist, for each singular point \(P\) of \(X\), of a compatible collection of isomorphisms between the stalks \(E_{P'}\) in the points \(P'\) of the \(s\)-fiber of \(P\). Level structures for connections on \(\tilde{X}\) are defined similarly; here the definition can be extended to any curve with at most special singularities. Poirier observes that the categories of local systems with level structure on \(\tilde{X}\) and connections with level structure on \(\tilde{X}\) are equivalent, and she also establishes an equivalence with the categories of local systems respectively connections on \(X\).
In section 3, Poirier works in the category of algebraic varieties over \(\mathbb C\); as before, she considers integral projective curves of positive genus. She first recalls the fact that given a smooth such curve together with a finite set of positive divisors \(D_1,\dots,D_r\) with disjoint supports, one can construct a curve \(X=Y_{D_1,\dots,D_r}\) with at most special singularities having \(Y\) as a normalization such the support of \(\sum_i D_i\) coincides with the preimage of the singular locus of \(X\). She then defines an equivalence relation on the group of divisors of \(X_{\text{smooth}}\) by declaring that two such divisors be equivalent if and only if they differ by a rational function that is constant modulo the \(D_i\); she defines \(J_{D_1,\dots,D_r}\) to be the resulting group of equivalence classes. Given a divisor \(M\) on \(X_{\text{smooth}}\), the author computes the space of positive divisors on \(X_{\text{smooth}}\) that are equivalent to \(M\) in the above sense, and she gives a criterion for this space being nonempty, using the generalized Riemann-Roch theorem and regular differential forms. Poirier then defines a notion of \((D_1,\dots,D_r)\)-level structure for invertible sheaves on \(\tilde{X}\), and she introduces an equivalence relation on the set of invertible sheaves on \(\tilde{X}\) with level structure. The resulting set \(F(D_1,\dots,D_r)\) of equivalence classes carries a natural group structure, and there is a canonical group homomorphism onto \(\text{Pic}(\tilde{X})\) whose kernel is computed explicitly. Poirier establishes a natural isomorphism \(J_{D_1,\dots,D_r}\overset{\sim}{\rightarrow}F(D_1,\dots,D_r)\); its surjectivity is based on the aforementioned criterion for the existence of certain positive divisors. The author then defines a functor \(\mathcal F(D_1,\dots,D_r)\) from the category of \(\mathbb C\)-schemes to the category of abelian groups whose group of \(\mathbb C\)-valued points is \(F(D_1,\dots,D_r)\); she exhibits a surjection of presheaves from \(\mathcal F(D_1,\dots,D_r)\) onto the relative Picard functor \(\text{Pic}_X\) of \(X\), and she computes its kernel. Using rigidified line bundels with level structure, Poirier shows that \(\mathcal F(D_1,\dots,D_r)\) is representable by a commutative group scheme of finite type \(J_{D_1,\dots,D_r}\) over \(\mathbb C\). The author then shows that \(J_{D_1,\dots,D_r}\) is canonically isomorphic to the Jacobian variety of \(X\); as a corollary, she gives a description of the Jacobian of an arbitrary integral projective curve over \(\mathbb C\) of positive genus, using birational approximation by a special singular curve.
In the fourth and final section, Poirier works in the \(\mathbb C\)-analytic category; she considers integral projective curves over \(\mathbb C\) with positive genus and only ordinary multiple points as singularities, and she generalizes the geometric Hecke correspondence to curves \(X\) of this type. More specifically, she defines, using the explicit description of \(J_{D_1,\dots,D_r}\) given in section 3, a morphism \(H\) from \((\tilde{X}\setminus S)\times J_{D_1,\dots,D_r}\) to \(J_{D_1,\dots,D_r}\), where \(\tilde{X}\) is the normalization of \(X\) and where \(S\subset\tilde{X}\) is the union of the fibers above the singular points of \(X\). She then shows the final main result of the paper, that a rank one connection with level structure on \(\tilde{X}\) admits a `preimage' under \(H\). The proof again uses the explicit structure of \(J_{D_1,\dots,D_r}\) and its relation to the relative Picard functor.
The paper is very well written; Poirier explains the arguments in detail, and her exposition is clear. singular curves; generalised Jacobian; Hecke correspondence Geometric Langlands program (algebro-geometric aspects), Singularities of curves, local rings The geometric correspondence for singular 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 \(f : S^\prime \rightarrow S\) be a finite and faithfully flat morphism of locally noetherian schemes of constant rank \(n\) and let \(G\) be a smooth, commutative and quasi-projective \(S\)-group scheme with connected fibers. For every \(r \geq 1\), let \(\operatorname{Res}_G^{(r)} : H^r(S_{\text{ét}}, G) \rightarrow H^r(S_{\text{ét}}^\prime, G)\) and \(\operatorname{Cores}_G^{(r)} : H^r(S_{\text{ét}}^\prime, G) \rightarrow H^r(S_{\text{ét}}, G)\) be, respectively, the restriction and corestriction maps in étale cohomology induced by \(f\). For certain pairs \((f, G)\), we construct maps \(\alpha_r : \operatorname{Ker} \operatorname{Cores}_G^{(r)} \rightarrow \operatorname{Coker} \operatorname{Res}_G^{(r)}\) and \(\beta_r : \operatorname{Coker} \operatorname{Res}_G^{(r)} \rightarrow \operatorname{Ker} \operatorname{Cores}_G^{(r)}\) such that \(\alpha_r \circ \beta_r = \beta_r \circ \alpha_r = n\). In the simplest nontrivial case, namely when \(f\) is a quadratic Galois covering, we identify the kernel and cokernel of \(\beta_r\) with the kernel and cokernel of another map \(\operatorname{Coker} \operatorname{Cores}_G^{(r - 1)} \rightarrow \operatorname{Ker} \operatorname{Res}_G^{(r + 1)}\). We then discuss several applications, for example to the problem of comparing the (cohomological) Brauer group of a scheme \(S\) to that of a quadratic Galois cover \(S^\prime\) of \(S\). restriction map; Norm map; quadratic Galois cover; relative ideal class group; Tate-Shafarevich group; relative Brauer group Étale and other Grothendieck topologies and (co)homologies, Class numbers, class groups, discriminants, Arithmetic ground fields for abelian varieties, Brauer groups of schemes Cokernels of restriction maps and subgroups of norm one, with applications to quadratic Galois coverings | 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 \(X_0\) be a smooth geometrically connected curve over a finite field \(\mathbb F_q\) with \(q\) elements and characteristic \(p\), let \(\ell\) be a prime number distinct from \(p\), and let \(\mathcal F_0\) be a lisse irreducible Weil \(\overline{\mathbb Q}_\ell\)-sheaf of rank \(r\) on \(X_0\) such that \(\mathrm{det}(\mathcal F_0)\) has finite order. Lafforgue proves that \(\mathcal F_0\) is a pure sheaf of weight \(0\), and furthermore:
(1) There exists a finite extension \(E\) of \(\mathbb Q\) contained in \(\overline{\mathbb Q}_\ell\) such that for any closed point \(x_0\) of \(X_0\), the coefficients of the characteristic polynomial \(\mathrm{det}(1-F_{x_0}t, \mathcal F_{0,\bar x_0})\) of the geometric Frobenius \(F_{x_0}\) on the stalk of \(\mathcal F_0\) at \(\bar x_0\) lie in \(E\).
(2) For any prime number \(\ell'\) distinct from \(p\) and any embedding \(\sigma:E\hookrightarrow \overline{\mathbb Q}_{\ell'}\), there exists a lisse Weil \(\overline{\mathbb Q}_{\ell'}\)-sheaf \(\mathcal F'_0\) on \(X_0\) such that
\[
\sigma (\mathrm{det}(1-F_{x_0}t, \mathcal F_{0, \bar x_0}))=\mathrm{det}(1-F_{x_0}t, \mathcal F'_{0,\bar x_0})
\]
for all closed points \(x_0\) of \(X_0\).
The author generalizes the above result (1) to the case where \(X_0\) is a normal connected scheme of finite type over \(\mathbb F_q\). Then by a result of Drinfeld, (2) remains true if \(X_0\) is smooth.
The main step in the proof is to find an integer \(N\) such that for any \(n>N\) and any \(\mathbb F_{q^n}\)-points \(x\) of \(X_0\), \(\mathrm{Tr}(F_x, \mathcal F_{0,\bar x})\) is contained in the extension over \(\mathbb Q\) generated by \(\mathrm{Tr}(F_{x'}, \mathcal F_{0,\bar x'})\) with \(x'\) being \(\mathbb F_{q^{n'}}\)-points of \(X_0\) with \(n'\leq N\). This is done by using the result of Lafforgue. \(\ell l\)-adic sheaves; Frebenius traces P. Deligne, Finitude de l'extension de \(\mathbb{Q}\) engendrée par des traces de Frobenius, en caractéristique finie, Moscow Math. J. 12 (2012), 497--514. Étale and other Grothendieck topologies and (co)homologies, Finite ground fields in algebraic geometry On the finiteness of \(\mathbb Q\)-extensions generated by traces of the Frobenius in finite 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 There are many papers devoted to compactifying (generalized) Jacobians of curves and families of curves. The approaches vary widely: some constructions use moduli of semistable rank-1 sheaves, some use semistable projective curves, some use combinatorics of cell decompositions; yet others use degenerations of principally polarized abelian varieties and various notions of stable varieties.
The aim of this survey is to give a definitive account in the case of nodal curves and to show, that in this case all of the known approaches are equivalent and produce isomorphic varieties, with the degeneration of PPAVs approach being the special case of degree \(g - 1.\) The degree \(g - 1\) case deserves a special attention since in this case the compactified Jacobian is unique and comes with a canonical theta divisor. It is also intimately connected with the Torelli map.
The author also gives a detailed description of the ``canonical compactified Jacobian'' in degree \(g - 1.\) Finally, he explains how Kapranov's compactification of configuration spaces can be understood as a toric analog of the extended Torelli map. compactifying (generalized) Jacobians of curves and families of curves; moduli of semistable rank-1 sheaves; semistable projetive curves; principally polarized abelian varieties Alexeev V.: Compactified Jacobians and Torelli map. Publ. RIMS Kyoto Univ. 40, 1241--1265 (2004) Fine and coarse moduli spaces, Torelli problem, Jacobians, Prym varieties, Algebraic moduli of abelian varieties, classification Compactified Jacobians and Torelli map | 1 |
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 is concerned with the existence of a geometric quotient \(X/R\) of a scheme \(X\) by a finite equivalence relation \(R\rightrightarrows X\). A typical example is the relation given by the normalisation morphism \(f:X\to Y\) of a non-normal scheme \(Y\).
In general, even if \(X\) is Noetherian and a categorical quotient \((X/R)^{\mathrm{cat}}\) exists, it may be non-Noetherian and the geometric quotient may either not exist or be different from \((X/R)^{cat}\). The author gives many examples in section 2 of this type of phenomenon. As one example, let \(f:X\to Y\) be a finite surjective morphism and \(R:=(X\times_YX)^{\mathrm{red}}\subset X\times X\) the corresponding set-theoretic equivalence relation (note that \(X\times_Y X\) may fail to be reduced). Then \(X/R\) exists and \(X/R\to(X/R)^{\mathrm{cat}}\) is a finite and universal homeomorphism, but need not be an isomorphism. In particular, if \(X\) is the normalisation of \(Y\), then \(X/R\) is the weak normalisation of \(Y\).
Section 3 contains some elementary results on the existence of geometric quotients. Section 4 consists of an inductive plan for constructing \(X/R\), which works well in finite characteristic; in characteristic \(0\), on the other hand, a certain inductive assumption proves quite restrictive. In section 5, the author proves that, if \(X\) is an algebraic space which is essentially of finite type over a Noetherian \({\mathbb F}_p\)-scheme \(S\) and \(R\rightrightarrows X\) is a finite set-theoretic equivalence relation, then \(X/R\) exists (Theorem 6). This result remains valid for algebraic spaces for which the Frobenius map \(F^q:X\to X^{(q)}\) is finite. Section 6 contains a discussion of gluing.
In the appendix, Claudiu Raicu constructs some interesting examples. The following question remains open in characteristic \(0\): let \(R\subset X\times X\) be a scheme-theoretic equivalence relation such that the projections \(R\rightrightarrows X\) are finite; does \(X/R\) exist? geometric quotient; categorical quotient; equivalence relation J. Kollár, Quotients by finite equivalence relations, preprint (arXiv: 0812.3608). Schemes and morphisms Quotients by finite 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 author considers proper smooth connected rigid analytic varieties with a strict semi-stable model such that the irreducible components of the special fiber are rational varieties; this is a higher-dimensional analogue of Mumford curves. His main result says that on the category of smooth and connected rigid analytic spaces, the rigid-analytic Picard functor of such a variety [see \textit{U.\ Hartl} and \textit{W.\ Lütkebohmert}, J.\ Reine Angew.\ Math.\ 528, 101--148 (2000; Zbl 1044.14007)] is represented by the quotient of a number of copies of the multiplicative group by a (not necessarily maximal-rank) lattice; this is a higher-dimensional analogue of the analytic uniformization of the Jacobian of a Mumford curve. This allows the author to describe line bundles on the variety in terms of rigid-analytic automorphic functions.
The results hold in arbitrary characteristic and the varieties do not have to be algebraic. In the algebraic \(p\)-adic setting, a less restrictive definition of total degeneracy was given by Raskind and Xarles. The higher-dimensional \(p\)-adically uniformized varieties of \textit{G.~A.\ Mustafin} [Mat.\ Sb., N.\ Ser.\ 105(147), 207--237 (1978; Zbl 0407.14006)] satisfy this latter condition, and the author expects that they also satisfy his condition. With his condition abelian varieties given as analytic tori are totally degenerate.
The paper ends with some instructive examples, e.g.\ of surfaces. rigid analytic variety; totally degenerate; Picard variety; torus Rigid analytic geometry, Picard schemes, higher Jacobians Line bundles on totally degenerated formal 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 Let \(X\) be a uniruled projective manifold, and let \(\mathcal K\) be a family of minimal rational curves, i.e. a covering family of rational curves that has minimal degree with respect to a fixed polarisation. For a general point \(x \in X\) the space \(\mathcal K_x\) parametrising elements of \(\mathcal K\) passing through \(x\) is projective and we can define a rational map \(\mathcal K_x \dashrightarrow \mathbb P T_x X\) by sending a point to the tangent line of the corresponding curve at the point \(x\). The strict transform \(\mathcal C_x \subset \mathbb P T_x X\) of \(\mathcal K_x\) is called the variety of minimal rational tangents VMRT at \(x\). \textit{J.-M. Hwang} and \textit{N. Mok} have shown in a series of papers (e.g. [J. Math. Pures Appl., IX. Sér. 80, No. 6, 563--575 (2001; Zbl 1033.32013)], [Asian J. Math. 8, No. 1, 51--64 (2004; Zbl 1072.14015)]) that the VMRTs contain a lot of interesting information about the geometry of \(X\). In particular they established an (equidimensional) Cartan-Fubini extension theorem saying that (under some mild hypothesis) an isomorphism between open analytic subsets of Fano manifolds with Picard number one that preserves the VMRTs extends to a global biholomorphism. In the paper under review the authors generalise this fundamental result to the case where the starting point is not an isomorphism but merely an immersion. More precisely they prove the following statement:
Let \(Z\) be a Fano manifold with Picard number one and \(\mathcal H\) a family of minimal rational curves such that \(\mathcal C_z=:\mathcal C_z Z\) is positive-dimensional at a general point \(z \in Z\). Let \(X\) be a uniruled projective manifold and \(\mathcal K\) a family of minimal rational curves on \(X\). Let furthermore \(f: U \rightarrow X\) be an immersion defined on a connected open subset \(U \subset Z\). If \(f\) respects the VMRTs (i.e. the image of the VMRT is the intersection of the VMRT with \(df(\mathbb P(T_z Z))\)) and is non-degenerate with respect to \((\mathcal K, \mathcal H)\), then \(f\) extends to a rational map \(Z \dashrightarrow X\). The non-degeneracy condition means that \(f(U)\) meets the locus of general points for \(\mathcal K\) and that for general points the restriction of the second fundamental form to the tangent space of \(df(\tilde C_z Z)\) has trivial kernel. In the equidimensional case this hypothesis corresponds to the generic finiteness of the Gauss map for \(C_z Z\), cf. [J. Math. Pures Appl., IX. Sér. 80, No. 6, 563--575 (2001; Zbl 1033.32013)]. We refer to the interesting introduction of the paper for a discussion of the differential-geometric background and the strategy of proof. \newline The authors give a geometric application of their main result. Suppose that \(X=G/P\) is a rational homogeneous manifold associated to a long simple root, and let \(Z=G_0/P_0\) be a non-linear, rational homogeneous manifold associated to a subdiagram of the marked Dynkin diagram of \(G/P\). If \(f: U \rightarrow X\) is a holomorphic embedding from a connected open subset \(U \subset Z\) which respects the VMRTs in a general point, then \(f\) extends to a standard embedding of \(Z\) into \(X\). minimal rational curves; analytic continuation; variety of minimal rational tangents; rational homogeneous manifold Hong, J; Mok, N, Analytic continuation of holomorphic maps respecting varieties of minimal rational tangents and applications to rational homogenous manifolds, J Diff Geom, 86, 539-567, (2010) \(n\)-folds (\(n>4\)), Fano varieties, Holomorphic mappings, (holomorphic) embeddings and related questions in several complex variables, Homogeneous complex manifolds Analytic continuation of holomorphic maps respecting varieties of minimal rational tangents and applications to rational homogeneous manifolds | 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 is a survey on recent achievements on arithmetic algebraic geometry due especially to Vojta and Faltings, as well as their links to classical Diophantine approximation results of Roth and Schmidt.
The first chapter deals with Roth's Theorem. In 1954 Roth proved the following statement concerning rational approximation to algebraic numbers: let \(\alpha\) be an irrational real algebraic number, \(\varepsilon\) a positive real number. Then for all but finitely many pairs of integers \((p,q)\), \(|\alpha-p/q|>q^{-2-\varepsilon}\). The author gives the idea of the proof, stressing the main technical difficulty, i.e. a ``nonvanishing lemma'' for polynomials in several variables. Here two main approaches are possible: the original one (Roth's lemma) and the more modern ``Dyson's lemma'' in several variables, proved by \textit{H. Esnault} and \textit{E. Viehweg} [Invent. Math. 78, 445-490 (1984; Zbl 0545.10021)], revisited by \textit{M. Nakamaye} himself [Invent. Math. 121, 355-377 (1995; Zbl 0855.11036)]. The author gives a clear account of the latter, explaining its relation with Faltings' product theorem.
The second part deals with Mordell conjecture (now Faltings' theorem) asserting the finiteness of rational points on an algebraic curve of genus greater than one. The link with Roth's theorem is provided by \textit{P. Vojta}'s proof of Faltings' theorem [Ann. Math. (2) 133, 509-548 (1991; Zbl 0774.14019)], which follows the main steps of classical proofs in Diophantine approximation. The author emphasises this link through the presentation of Vojta's generalization of Dyson's lemma to products of curves of arbitrary genus. The author also gives an overview of the subsequent proof by Faltings of the Lang conjecture on rational points on algebraic subvarieties of abelian varieties.
Finally, a chapter is devoted to the new proof of the Subspace Theorem by Faltings and Wüstholz, leading to interesting generalizations. The Subspace Theorem, which is the natural generalization of Roth's theorem to higher dimension, provides a lower bound for the rational approximation to a hyperplane (or a family of hyperplanes) defined over the algebraic numbers. The new ideas of the Faltings-Wüstholz proof are clearly presented; they led to new results concerning approximations by rational points on an algebraic subvarieties as well as approximation to nonlinear subspaces. Since any clear account of these interesting new applications is lacking both in the Faltings-Wüstholz paper and in the article under review, the interested reader is referred to articles by \textit{J.-H. Evertse} and \textit{R. G. Ferretti}, especially [Int. Math. Res. Notes 25, 1295-1330 (2002)] and [\textit{R. G. Ferretti}, Compos. Math. 121, 247-262 (2000; Zbl 0989.11034)]. Diophantine approximation; rational points on algebraic varieties; arithmetic algebraic geometry; Roth's theorem; nonvanishing lemma for polynomials in several variables; Roth's lemma; Dyson's lemma; Mordell conjecture; Faltings' theorem; finiteness of rational points; algebraic curve of genus greater than one; Vojta's generalization of Dyson's lemma; products of curves of arbitrary genus; Lang conjecture; Subspace Theorem; lower bound for the rational approximation to a hyperplane Results involving abelian varieties, Varieties over global fields, Abelian varieties of dimension \(> 1\), Rational points Diophantine approximation on algebraic 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 This book, written in Portuguese, provides an introduction to moduli spaces in algebraic geometry. It is designed for those readers who have a profound knowledge in basic algebraic geometry, including algebraic sheaves and their cohomology. In view of these assumed prerequisites, the text is highly non-elementary and, quite naturally, just as advanced as the topic of moduli itself. -- Out of the vast amount of aspects of moduli theory, the author has concentrated on two general strategic goals, namely
(i) explaining and illustrating the philosophy that underlies the concept of moduli in algebraic geometry, and
(ii) describing some of the most powerful general construction techniques for moduli spaces by means of two important examples.
The text is subdivided into five chapters. Chapter 1 introduces the concept of fine moduli spaces and universal families of algebro-geometric objects. This is then illustrated by showing that Grassmannians can be interpreted as fine moduli spaces for certain vector bundles over affine spaces (Plücker embedding). -- Chapter 2 discusses, in a pleasantly simplified manner, the construction of Hilbert schemes and Grothendieck's quotient schemes. -- This is used, in chapter 3, to construct the Jacobi scheme as a fine moduli space for degree-\(d\) line bundles over a smooth algebraic curve. At the end of this chapter, local moduli spaces and coarse global moduli spaces are introduced, in order to prepare the reader for the next steps into moduli theory. -- Chapter 4 provides some basic material from geometric invariant theory, including group actions on varieties, good quotients, geometric quotients, the concepts of stability and semi-stability, and the Mumford stability criterion. -- The concluding chapter 5 is devoted to another concrete moduli problem, namely to the moduli problem for vector bundles over a smooth projective curve. The construction of the coarse moduli space for rank-\(r\) vector bundles over a given genus-\(g\) curve, which is presented here, follows the more recent approach by \textit{C. T. Simpson} [Publ. Math., Inst. Hautes Étud. Sci. 79, 47-129 (1994; Zbl 0891.14005)].
The material presented here is arranged in a very thorough, detailed and lucid manner. The author has tried to simplify the matter wherever possible, and thereby achieved some remarkable methodological improvements. The book, though only available in Portuguese, may serve as a good introduction to the more advanced texts on moduli spaces, including those which also cover the moduli spaces for algebraic curves, algebraic surfaces, and abelian varieties, that is to topics which are not touched upon in the book under review.
Those readers who are not familiar with the Portuguese language can find a lot of the same material in the English standard texts by \textit{P. E. Newstead} [``Lectrues on introduction to moduli problems and orbit spaces'', Tata Lect. Math. Phys., Math. 51 (1978; Zbl 0411.14003)] and \textit{J. Le Potier} [``Fibres vectoriels sur les courbes algébriques'', Publ. Math. Univ. Paris VII, Vol. 35 (Paris 1995; Zbl 0842.14025) and: ``Lectures on vector bundles'', Camb. Stud. Adv. Math., Vol. 54 (Cambridge 1997; Zbl 0872.14003)]. Plücker embedding; moduli spaces; fine moduli spaces; gemetric invariant theory; coarse moduli space Algebraic moduli problems, moduli of vector bundles, Research exposition (monographs, survey articles) pertaining to algebraic geometry, Geometric invariant theory, Families, moduli of curves (algebraic), Families, moduli, classification: algebraic theory Construction of moduli spaces. | 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 context of so-called \` piecewise algebraic geometry\'\ in characteristic zero, there is a leading question of Larsen and Lunts: if two varieties have the same class in the Grothendieck ring of varieties, are they piecewise isomorphic (the converse being obvious)? In this paper the author studies a similar question, assuming now that the two varieties are \(K\)-equivalent.
We fix an algebraically closed base field of characteristic zero. Recall that two varieties \(X\) and \(Y\) are \textsl{piecewise isomorphic} if there exist finite partitions \((X_i)_{i\in I}\) and \((Y_i)_{i\in I}\) of \(X\) and \(Y\), respectively, in locally closed subvarieties such that \(X_i\) is isomorphic to \(Y_i\) for all \(i\in I\). Two smooth and irreducible varieties \(X\) and \(Y\), that are birationally equivalent, are called \textsl{\(K\)-equivalent} if there exists a smooth variety \(Z\) and proper and birational morphisms \(f:Z\to X\) and \(g:Z\to Y\), such that the relative canonical divisors \(K_{Z|X}\) and \(K_{Z|Y}\) on \(Z\) are equal. Denote here by \(C_X\) and \(C_Y\) closed subvarieties of \(X\) and \(Y\), respectively, such that their open complements are isomorphic via \(f\) and \(g\). The following is the main result of the paper. \newline Let \(X\) and \(Y\) be \(K\)-equivalent complete varieties. If \(\dim(C_X)=\dim(C_Y)\leq 2\), then \(X\) and \(Y\) are piecewise isomorphic. \newline The proof uses the geometry of jet spaces and certain \` birational simplification techniques\'\ from [Math. Z. 265, No. 2, 321--342 (2010; Zbl 1195.14003)] by \textit{Q. Liu} and the author. As a corollary, birationally equivalent complex (connected, smooth and complete) Calabi-Yau varieties of dimension at most \(4\) are always piecewise isomorphic. piecewise geometry; \(K\)-equivalence; Grothendieck ring; arc space Rational and birational maps, Algebraic cycles, Applications of methods of algebraic \(K\)-theory in algebraic geometry, Calabi-Yau manifolds (algebro-geometric aspects) \(K\)-equivalent varieties and piecewise 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 In this paper the authors identify two moduli spaces of sheaves, namely the moduli space of torsion free sheaves on \(\mathbb{P}^m\) with Chern character \((r,0, \dots, 0, -n)\) that are trivial along a hyperplane \(D\subset \mathbb{P}^m\) and the Grothendieck's Quot scheme \(Quot_{\mathbb{A}^m}(\mathcal{O}^{\oplus r},n)\) of 0-dimensional length \(n\) quotients of the free sheaf \( \mathcal{O}^{\oplus r}\) on \(\mathbb{A}^m\).
More precisely, denote by \(Fr_{r,n}(\mathbb{P}^m)\) the moduli space of torsion free sheaves \(E\) with framing at \(D \subset \mathbb{P}^m\) and Chern character \((r,0, \dots, 0, -n)\), that is we fix an isomorphism \(\phi: E|_D \to \mathcal{O}_D^{\oplus r}\). The authors prove the following result.
Fix \(m\geq 2, r\geq 1, n\geq 0\) and an hyperplane \(D\subset \mathbb{P}^m\). There is an injective morphism
\[
\eta: Quot_{\mathbb{A}^m}(\mathcal{O}^{\oplus r},n)\to Fr_{r,n}(\mathbb{P}^m),
\]
which is an isomorphism if and only if \(m\geq 3\) or \((m,r)=(2,1)\).
The proof goes by an infinitesimal argument, by comparing the two well-known tangent-obstruction theories on these moduli spaces.
It is relevant to notice that the interest in these moduli spaces comes from Enumerative Geometry (Quot schemes) and String Theory (moduli of framed sheaves), as they lead to higher rank versions of Donaldson-Thomas invariants, which had been intensively studied in the Math and Physics literature. Quot schemes; framed sheaves; deformation theory; tangent-obstruction theories; moduli of sheaves Parametrization (Chow and Hilbert schemes), Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects) Framed sheaves on projective space and Quot 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 In this well-written article, the authors continue the program of removing hypotheses from the Riemann-Roch theorems. The starting point is the Grothendieck-Riemann-Roch theorem [\textit{A. Borel} and \textit{J.-P. Serre}, Bull. Soc. Math. Fr. 86(1958), 97-136 (1959; Zbl 0091.330)] proved for proper morphisms between smooth, quasiprojective varieties. Then \textit{P. Baum}, \textit{W. Fulton} and \textit{R. D. MacPherson} [Publ. Math., Inst. Hautes Études Sci. 45, 101-145 (1975; Zbl 0332.14003)] eliminated the hypothesis that the schemes be smooth varieties, thus generalizing the Grothendieck-Riemann-Roch theorem to the category of quasiprojective schemes (over an arbitrary base field) and proper morphisms; the main point there was to consider homology, \(A_*X_ Q\), and to construct the natural (Riemann-Roch) transformation \(\tau_ X: K_ 0X\to A_*X_ Q\) such that \(\tau_ X({\mathcal O}_ X)=Td(X)\), the Todd class of X. In this article, the authors eliminate the assumption of quasiprojectivity, proving a Riemann-Roch theorem in the category of algebraic schemes (over an arbitrary base field) and proper morphisms. Their method also works to generalize the topological K-theory Riemann- Roch theorem of \textit{P. Baum}, \textit{W. Fulton} and \textit{R. D. MacPherson} [Acta Math. 143, 155-192 (1979; Zbl 0474.14004)] to algebraic \({\mathbb{C}}\)- schemes. The basic ingredients of the proof are the Riemann-Roch theorem for quasiprojective schemes, Chow's lemma allowing one to approximate any algebraic scheme X by a quasiprojective ''envelope'' X'\(\to X\), and an exact sequence from algebraic K-theory (involving only \(K_ 0\) and \(K_ 1)\), which relates \(\tau_{X'}\) to \(\tau_ X\). They also include a proof of an implicit assumption made by \textit{H. Gillet} [Algebraic K- theory, Proc. Conf. Evanston 1980, Lect. Notes Math. 854, 141-167 (1981; Zbl 0478.14011)], concerning the equality of the Riemann-Roch transformations \(\tau^ S_ X\) and \(\tau^ T_ X\) from \(K_ 0X\) to \(A^*X_ Q\) for a given scheme X, regarded as a quasiprojective S- scheme or as a quasiprojective T-scheme. Grothendieck-Riemann-Roch theorem; algebraic schemes; algebraic \({\mathbb{C}}\)-schemes [F-G]W. Fulton, H. Gillet: ``Riemann Roch for general Algebraic Varieties{'', Bull. Soc. Math. France 111 (1983) pp. 287--300.} Riemann-Roch theorems, Applications of methods of algebraic \(K\)-theory in algebraic geometry Riemann-Roch for general algebraic 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 ``functional method'', due to Cayley, supposes that every \(k\)-secant formula involving a smooth curve \(C\) in projective space should depend only on the degree of the curve and the number \(h\) of ``apparent'' double points of \(C\). (If \(C\) is irreducible of genus \(g\) and degree \(n\), then \(h= (n-1) (n-2)/ 2-g\).) Previously [C. R. Acad. Sci., Paris, Sér. I 303, 299-302 (1986; Zbl 0603.14040)], the author announced a theorem giving a modern justification of this classical hypothesis for space curves. The present paper presents the proof of this theorem. The key result is the expression of the rational equivalence class of the Hilbert scheme of \(k\)-uplets of \(C\) in terms of certain cycles in the Chow group of \(\text{Hilb}^k \mathbb{P}^3\). The methods used in the proof are similar to those used by \textit{P. Le Barz} in his paper ``Formules multisecantes pour les courbes gauches quelconques'' [in: Enumerative geometry and classical algebraic geometry, Prog. Math. 24, 165-197 (1982; Zbl 0514.14023)], which dealt with \(k\)-secant lines. As an application, the author computes the number of conics that meet \(C\) in \(k\) points and satisfy some other conditions as well. \(k\)-secant formula; space curves; Hilbert scheme; Chow group Vassallo, V.: Justification de la méthode fonctionnelle pour LES courbes gauches. Acta math. 172, 257-297 (1994) Enumerative problems (combinatorial problems) in algebraic geometry, Plane and space curves, Parametrization (Chow and Hilbert schemes) Justification for the functional method for space 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 In this paper, the author studies a condition known as ``strong geometricity'' that ensures that geometric quotients of algebraic spaces are categorical.
Recall that a groupoid (in the category of algebraic spaces) is defined by two algebraic spaces, \(R\) and \(X\); and two morphisms \(s,t:R \to X\), subject to a number of axioms. (The definition is given in full detail in Definition 1.1.) An equivariant morphism is defined as a morphism \(q:X \to Y\) such that \(q \circ s = q \circ t\). \(q\) is called a strongly geometric quotient if it is geometric, universally Zariski, universally constructible and the canonical morphism \(R \to X \times_Y X\) is universally submersive. In Theorem 3.16, the author proves that, a strongly geometric quotient \(q\) is categorical under any one of a list of conditions on \(q\). This generalizes Corollary 2.15 in [\textit{J. Kollár}, Ann. Math. (2) 145, No. 1, 33--79 (1997; Zbl 0881.14017)]. The author also gives results on the quotients of finite locally free groupoids of affine schemes in Section 4, and uses these to deduce existence of quotients of algebraic spaces by finite groups in Section 5. In Section 6, it is proved that algebraic stacks with finite inertia have coarse moduli spaces (Theorem 6.12). This generalizes a result of \textit{S. Keel} and \textit{S. Mori} [Ann. Math. (2) 145, No. 1, 193--213 (1997; Zbl 0881.14018)]. The Stacks Project Authors. \textit{Stacks Project}. http://stacks.math.columbia.edu (2015) Generalizations (algebraic spaces, stacks), Geometric invariant theory Existence and properties of geometric quotients | 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 under review is devoted to the restauration (in the frame of modern algebraic geometry) of some beautiful classical work mainly due to \textit{A. B. Coble} [cf. Coble's book: ``Algebraic geometry and theta functions'' (New York 1929)].
The main objects of this work are the moduli spaces \({\mathbb{P}}^ m_ n\) for the projective equivalence classes of ordered m-uples of points in an n-dimensional projective space \({\mathbb{P}}_ n\). One of the most exciting aspects concerning these moduli spaces is their relation with root systems (relation discovered by Coble, Kantor and Du Val), specifically the existence of a representation of a certain Weyl group \(W_{n,m}\) in the group of birational automorphisms of \(P^ m_ n\); this topic is discussed in the chapters V-VII of the book.
The last two chapters are devoted to the link between the spaces \(P^ m_ n\) and abelian varieties. The main (classical) observation is that an ordered set of \(2g+2\) points in \({\mathbb{P}}_ 1\) defines a hyperelliptic curve of genus g together with a level 2 structure on its Jacobian and similarly an ordered set of 7 points in \({\mathbb{P}}_ 2\) defines a curve of genus 3 together with a level 2 structure. It is shown for instance that there is a birational morphism from the moduli space \(M_ 3(2)\) of curves of genus 3 with level 2 structure to the space \({\mathbb{P}}^ 7_ 2\) defined (roughly speaking) by associating to each curve C in \(M_ 3(2)\) a ``geometrically marked'' Del Pezzo surface S of degree 1 which is a double cover of \({\mathbb{P}}_ 2\) branched along C (identified in an obvious way with a point in \({\mathbb{P}}^ 7_ 2)\). moduli spaces of ordered m-uples; root systems; Weyl group; group of birational automorphisms; abelian varieties I. Dolgachev, D. Ortland,'' Points Sets in Projective Spaces and Theta Functions.'' Ast'erisque 165, 1988. Projective techniques in algebraic geometry, Algebraic moduli problems, moduli of vector bundles, Research exposition (monographs, survey articles) pertaining to algebraic geometry, Theta functions and abelian varieties, Birational geometry Point sets in projective spaces and theta functions | 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 \(u : A \rightarrow A'\) be a regular morphism of Noetherian rings and \(B\) an \(A\)-algebra of finite type. Then any \(A\)-morphism \(v : B \rightarrow A'\) factors through a smooth \(A\)-algebra \(C\), that is \(v\) is a composite \(A\)-morphism \(B \rightarrow C \rightarrow A'\). This theorem called General Néron Desingularization was first proved by the second author [Nagoya Math. J. 100, 97--126 (1985; Zbl 0561.14008)]. Later different proofs were given by \textit{M. André} [Cinq exposés sur la désingularisation. Ecole Polytechnique Fédérale de Lausanne (Handwritten manuscript) (1991)], \textit{R. G. Swan} [in: Lectures in algebra and geometry. Proceedings of the international conference on algebra and geometry, National Taiwan University, Taipei, Taiwan, December 26--30, 1995. Cambridge, MA: International Press. 135--192 (1998; Zbl 0954.13003)] and \textit{M. Spivakovsky} [J. Am. Math. Soc. 12, No. 2, 381--444 (1999; Zbl 0919.13009)]. All the proofs are not constructive. In [J. Symb. Comput. 80, Part 3, 570--580 (2017; Zbl 1406.13006)], the authors gave a constructive proof together with an algorithm to compute the Néron Desingularization for 1-dimensional local rings. In this paper we go one step further. We give an algorithmic proof of the General Néron Desingularization theorem for 2-dimensional local rings and morphisms with small singular locus. The main idea of the proof is to reduce the problem to the one-dimensional case. Based on this proof we give an algorithm to compute the desingularization. smooth morphisms; regular morphisms; Néron desingularization Pfister, G.; Popescu, D., \textit{Construction of Neron Desingularization for Two Dimensional Rings}. arXiv:AC/1612.01827 Étale and flat extensions; Henselization; Artin approximation, Local structure of morphisms in algebraic geometry: étale, flat, etc., Regular local rings, Global theory and resolution of singularities (algebro-geometric aspects) Construction of Néron desingularization for two dimensional 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 This second edition (see [Zbl 1308.14001] for a review of the first edition) is extended by a chapter on recent developments.
After a short motivating first chapter that introduces the questions addressed in this book, follow two chapters dealing with background, one on sheaves, algebraic varieties and analytic spaces, and one on homological algebra and duality. The treatment includes spectral sequences. In these chapters the theorems are formulated in their natural generality. The definition of a singularity is initially given both for analytic spaces and for algebraic varieties over an arbitrary algebraically closed field. After stating Artin's Algebraization Theorem, that an isolated singularity of an analytic space is isomorphic to the germ of an algebraic variety over \(\mathbb{C}\), only the algebraic case is considered. After a chapter defining the canonical divisor for varieties over an arbitrary algebraically closed field the further discussion is restricted to the field of complex numbers.
The book defines log canonical, canonical, log terminal, terminal and rational singularities and provides a characterization of isolated such ones in terms of plurigenera. The classification is refined in the two-dimensional case, and rational surface singularities are described in some detail. Also the results of the Author on two-dimensional Du Bois singularities are introduced. The next chapter considers the analogous
questions for higher dimensional singularities, and in particular for the case of dimension three. It concludes with the list of the famous
95 families of simple \(K3\)-singularities.
The final chapter presents some developments after the publication of the first Japanese version [Zbl 1308.14002] of this book. These concern log discrepancies for pairs and the use of the space of arcs in their description. This opens up the possibility of proving results in positive characteristic. The book closes with a brief introduction to \(F\)-singularities, in positive characteristic. rational singularities; minimal model program; Du Bois singularities; arc spaces; F-singularities Research exposition (monographs, survey articles) pertaining to algebraic geometry, Singularities in algebraic geometry, Deformations of singularities, Global theory and resolution of singularities (algebro-geometric aspects), Local complex singularities Introduction to singularities | 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 Here it is proved that the moduli scheme \(\bar M_ g\) of genus-g stable curves is of general type if \(g>23\) and has Kodaira dimension \(>0\) if \(g=23\), refining the results and simplifying the proofs of the fundamental paper by \textit{J. Harris} and \textit{D. Mumford} in Invent. Math. 67, 23-86 (1982; Zbl 0506.14016)]. The proofs here use the theory of limit linear series on stable curves with \(Pic^ 0\) compact introduced by the authors in Invent. Math. 85, 337-371 (1986; Zbl 0598.14003) and applied by the authors (and others) to obtain several interesting results. The proof uses the computation in \(Pic(M_ g)\otimes {\mathbb{Q}}\) of the class of the following divisors: \(D^ r_ s\) (when \(g+1\) is not prime, \(g+1=(r+1)(s-1))\), \(E_ s\) (when g is even, \(g=2(s-1))\); \(E_ s\) is the closure of the set of genus \(g\) curves with a linear series of degree \(s\) and dimension 1 violating Petri's condition; \(D^ r_ s\) is the closure of the locus of curves with a \(g^ r_ d\) with \(d=rs-1\). Furthermore, when \(g=(r+1)(s-1)-1\), on a general curve C, \(W^ r_ d(C)\) is a smooth, irreducible curve; the authors need and compute its genus. Weierstrass point; moduli scheme; Kodaira dimension; limit linear series on stable curves; Pic David Eisenbud and Joe Harris. The {K}odaira dimension of the moduli space of curves of genus {\(\geq 23\)}. {Invent. Math.}, 90(2):359--387, 1987. DOI 10.1007/BF01388710; zbl 0631.14023; MR0910206 Families, moduli of curves (algebraic), Divisors, linear systems, invertible sheaves, Algebraic moduli problems, moduli of vector bundles The Kodaira dimension of the moduli space of curves of genus \(\geq 23\) | 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, geometrically irreducible, curve of genus at \(least\quad 2,\) defined over a number field K. For K large enough, the regular minimal model \({\mathfrak X}\) of C over Spec(\({\mathfrak O}_ K)\) has semistable reduction and the ratio \(e(C)=(\omega \cdot \omega)/[K:{\mathbb{Q}}]\) is then independent of the choice of K, where \(\omega\) is the relative dualizing sheaf of \({\mathfrak X}\) over Spec(\({\mathfrak O}_ K)\) and (\(\omega\cdot \omega)\) is an Arakelov intersection product. By generalizing a method of Arakelov in the function field situation, we obtain a proof of the strict positivity of e(C) if the stable model has at least one reducible fiber or if the set of places of completely supersingular reduction is infinite. The method uses Weierstrass points and leads also to a proof of the boundedness of the average height of Weierstrass points of powers of a given line bundle on C. regular minimal model of curve; Arakelov intersection; boundedness of the average height of Weierstrass points Jean-François Burnol, Weierstrass points on arithmetic surfaces, Invent. Math. 107 (1992), no. 2, 421 -- 432. Arithmetic varieties and schemes; Arakelov theory; heights, Riemann surfaces; Weierstrass points; gap sequences Weierstrass points on arithmetic 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 \(V\) be an irreducible algebraic variety over an algebraically closed field \(K\) of char 0. Let \(L_{V,q}\) the \(K\)-scheme of arcs traced on \(V\) passing by the point \(q\in V\), and \(C\) be an irreducible component of \(L_{V,q}\). The authors prove that \(C\) defines a valuation on \(K(V)\) and it is a divisor. The Arf characteristic with respect to the set of arc space valuations is called primary Puiseux exponents. They define the secondary Puiseux exponents of \(V_{q} \) in terms of the Fitting ideals of the differential modules \(\Omega^j_{V,K}\), \(j=1,\ldots,d\).
The authors prove that, if the local ring \({ \mathcal O }_{V_{q} }\) is Cohen-Macaulay then roughly speaking, the information on the primary Puiseux exponents of \(V_{q}\) is equivalent to:
(1) The residue fields of the centers of the arc space valuations at \(q\) by successive blow-ups of such centers.
(2) The adapted multiplicity tree of curve singularities \(\Gamma\) wich consists of one branch for each component \(C\) of the arc space \(L_{V,q}\) which is a generic arc in \(V\).
The information on the secondary Puiseux exponents of \(V_{q}\) relative to the ideal \(I\) is equivalent to:
(1) The residue fields of the centers of the Rees valuations of \(I\) by successive blow-ups of such centers.
(2) The multiplicity tree of the curve singularities given as complete intersections of \(r-1\) hypersurfaces defined by a general element of \(I\).
The subject is very interesting, it would be nice to have a more detailed exposition with complete proofs and examples. Arf closure; Rees valuation; arc space valuations; multiplicity tree Campillo, A., Castellanos, J.: On Puiseux exponents for higher dimensional singularities. In: Topics in Algebraic and Noncommutative Geometry. Volume 324 of Contemporary Mathematics, pp. 91-102. American Mathematical Society, Providence (2003) Singularities in algebraic geometry, Local complex singularities On Puiseux exponents for higher dimensional singularities. | 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 \(C\) denote a smooth projective algebraic curve given by the zeros of the polynomials \(P_1,\dots ,P_s \in {\mathbb L}[x_0,\dots ,x_n]\) where \(\mathbb{L}\) is an algebraic closure of a perfect field \(\mathbb{K}\) of characteristic \(q\neq 2\). If the polynomials \(P_1,\dots ,P_s \) are defined over a subfield \(\mathbb{F}\) of \(\mathbb{L}\), then we call \(\mathbb{F}\) a field of definition of \(C\). For \(\sigma \in \Gamma =\text{Gal} (\mathbb{L} /\mathbb{K})\), the polynomials \(P_1^{\sigma} ,\dots ,P_s^{\sigma}\) define a new smooth projective curve \(C^{\sigma}\) which is, in general, not isomorphic to \(C\). We define the moduli field \(\mathbb{M}\) of \(C\) relative to the extension \(\mathbb{L} /\mathbb{K}\) to be the fixed field of the group
\[
U=\left\{ \sigma \in \Gamma =\text{Gal} (\mathbb{L} /\mathbb{K}) \bigg| C \text{ is isomorphic to } C^{\sigma} \text{ over } \mathbb{L} \right\}.
\]
Clearly the moduli field is a subfield of any field of definition of \(C\). The moduli field however need not be a field of definition.
Necessary and sufficient conditions for when the field of moduli is a field of definition were given in [\textit{A. Weil}, Am. J. Math. 78, 509--524 (1956; Zbl 0072.16001)]. A consequence of Weil's conditions is that if the automorphism group \(\text{Aut} (C)\) of \(C\) is trivial, then the moduli field is a field of definition. Unfortunately, when \(\text{Aut} (C)\) is non-trivial, Weil's conditions are often difficult to test. This has motivated an interest in developing alternate conditions for when the moduli field is a field of definition, and this is the primary goal of the authors in the current article.
In order to present these conditions, we need the following definitions. The signature of a subgroup \(H\) of \(\text{Aut} (C)\) is defined to be the tuple \((g_0;m_1,\dots ,m_r)\) where the curve \(C/H\) has genus \(g_0\) and the quotient map \(C\rightarrow C/H\) is branched over \(r\) points with branching orders \(m_1,\dots ,m_r\). The subgroup \(H\) is said to be unique up to conjugation if for any other subgroup \(K\) of \(\text{Aut} (C)\) with the same signature as \(H\), \(K\) is conjugate to \(H\). In the article under review, the authors prove that if \(\text{Aut} (C)\) contains a subgroup \(H\) that is unique up to conjugation, the quotient surface \(C/H\) has genus \(0\), and the group \(N_{\text{Aut} (C)} (H)/H\) is neither trivial nor cyclic when the characteristic \(q=0\), nor cyclic of order relatively prime to \(q\) for \(q\neq 0\), then \(C\) can be defined over its field of moduli relative to the extension \(\mathbb{L} /\mathbb{K}\). The method of proof is direct -- the authors show that given these specific conditions, Weil's conditions hold and hence the field of moduli is a field of definition.
This result is a generalization of a result developed for hyperelliptic curves [\textit{B. Huggins}, Math. Res. Lett. 14, No. 2, 249--262 (2007; Zbl 1126.14036)] and cyclic \(p\)-gonal curves [\textit{A. Kontogeorgis}, J. Théor. Nombres Bordx. 21, No. 3, 679--692 (2009; Zbl 1201.14020)]. It should be noted that the conditions presented are not necessary -- that is, there exist explicit examples of curves which fail the conditions yet are still defined over their field of moduli. field of moduli; field of definition; automorphism Hidalgo, Rubén A.; Quispe, Saúl, Fields of moduli of some special curves, J. Pure Appl. Algebra, 220, 1, 55-60, (2016) Special algebraic curves and curves of low genus, Automorphisms of curves Fields of moduli of some special 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 [For the entire collection see Zbl 0542.00005.]
This paper is an exposition of the result, due to Harris and Mumford, that the moduli space of genus \(g\) algebraic curves over \(\mathbb C\) is of general type for \(g\) large enough [cf. \textit{J. Harris} and \textit{D. Mumford}, Invent. Math. 67, 23--86 (1982; Zbl 0506.14016) and \textit{J. Harris}, Invent. Math. 75, 437-466 (1984; Zbl 0542.14014)]. The first part of the paper describes the calculus of line bundles on the moduli functor and explains the strategy of the proof. In the second part of the paper sketches of proofs are given for almost all (or sample cases of almost all) the steps of the argument in the odd genus case. A large part of the paper is devoted to admissible coverings and their applications, and to examples of the technique of calculating relations among divisor classes on moduli space by ''restriction'' to suitable families of (generally singular) curves. Picard groups; moduli space of genus g algebraic curves; line bundles on the moduli functor; admissible coverings; divisor classes on moduli space M. Cornalba, Systèmes pluricanoniques sur l'espace des modules des courbes et diviseurs de courbes \(k\)-gonales (d'après Harris et Mumford) , Astérisque (1985), no. 121-122, 7-24, Séminiare Bourbaki, 1983/84, Soc. Math. France, Paris. Families, moduli of curves (algebraic), Algebraic moduli problems, moduli of vector bundles, Divisors, linear systems, invertible sheaves Pluricanonical systems on the moduli space of curves and divisors of \(k\)-gonal curves (following Harris and Mumford) | 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 concerns special subvarieties of moduli spaces. Let \(\mathcal A_g\) denote the moduli space of principally polarized abelian varieties of dimension \(g\). A classical problem asks whether \(\mathcal A_g\) contains any Shimura curves which are generically contained in the Torelli locus \(\mathcal J_g\), i.e.\ the locus of Jacobians of smooth curves of genus \(g\). Previously a series of examples have been constructed by de Jong, Moonen, Mumford, Noot, Oort and others, mostly arising as Galois covers of \(\mathbb P^1\) with varying branch points. Recently there have been several new developments, starting from the non-abelian Galois covers studied in [\textit{P. Frediani} et al., Int. Math. Res. Not. 2015, No. 20, 10595--10623 (2015; Zbl 1333.14023)] and the purely analytic constructions in genus 3 by the authors in [\textit{S. Grushevsky} and \textit{M. Möller}, Int. Math. Res. Not. 2016, No. 6, 1603--1639 (2016; Zbl 1338.14046)] (which produce, in fact, infinitely many Shimura curves contained in the locus of hyperelliptic Jacobians).
The present paper introduces a geometric construction for infinitely many Shimura curves generically contained in \(\mathcal J_4\), using \(\mathbb Z/3\mathbb Z\) Galois covers of elliptic curves following \textit{G. P. Pirola} [J. Reine Angew. Math. 431, 75--89 (1992; Zbl 0753.14040)]. The proof builds on a method going back to \textit{G. Shimura} [Ann. Math. (2) 78, 149--192 (1963; Zbl 0142.05402)] to work out moduli spaces of abelian varieties of given endomorphism ring and polarization (the so-called PEL-Shimura varieties). In particular, the authors compute explicitly the period matrices involved.
Independently, the same curves (and other covers of elliptic curves) were discovered by \textit{P. Frediani} et al. [Geom. Dedicata 181, 177--192 (2016; Zbl 1349.14104)].
Note that for sufficiently large \(g\), it has recently been proved that the Torelli locus does not contain special subvarieties of certain types, for instance Shimura curves of Mumford type, of maximal variation or of hyperelliptic Jacobians [\textit{X. Lu} and \textit{K. Zuo}, J. Math. Pures Appl. (9) 108, No. 4, 532--552 (2017; Zbl 1429.14016)], [\textit{P. Frediani} et al., Geom. Dedicata 181, 177--192 (2016; Zbl 1349.14104)]. Meanwhile the Shimura curves constructed presently contain some curves which are of neither type. abelian variety; Jacobian; Shimura curve; Torelli locus; Galois cover; period matrix Modular and Shimura varieties, Jacobians, Prym varieties Explicit formulas for infinitely many Shimura curves in genus \(4\) | 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 an interesting paper with many connections to other work on algebraic curves and group theory. The reviewer feels that the author's fine introduction cannot be improved upon.
The theory of algebraic curves associated with subgroups of finite index in the modular group \(\Gamma\) is highly developed for such subgroups of \(\Gamma\) as may be defined by means of congruences in the ring \({\mathbb{Z}}\) of rational integers. The situation in the case of non-congruence subgroups of \(\Gamma\), on the other hand, is drastically different. It reduces to a few isolated examples, two of which may be found in the author's paper in J. Reine Angew. Math. 268/269, 348-359 (1974; Zbl 0292.10021). Related research by \textit{A. O. L. Atkin} and \textit{H. P. F. Swinnerton-Dyer} began in Proc. Symp. Pure Math. 19, 1-25 (1971; Zbl 0235.10015).
Observing that Macbeath's curve [cf. \textit{A. M. Macbeath}, Proc. Lond. Math. Soc., III. Ser. 15, 527-542 (1965; Zbl 0146.427)] affords another pertinent example made us look at it more closely. Its automorphism group is isomorphic with the simple group PSL(2,8) of order 504. Accordingly, the associated subgroup \(\Delta\) of \(\Gamma\) is a maximal normal subgroup of index 504. We shall prove, in passing, that there are exactly two such subgroups in \(\Gamma\), neither of them a congruence group.
The view above renders Macbeath's curve as a covering of the projective line with Galois group G isomorphic to PSL(2,8). Corresponding to any of its Sylow 7-subgroups and its normalizer in G we find two intermediate curves \textbf{B} and \textbf{A}, respectively elliptic and rational, of which the former covers the latter 2-fold and with 4 branch points. In this classical situation the 4 points of \textbf{A} under the branch points may, moreover, be made explicit through Macbeath's model. Their cross ratio, or Legendre's modulus \(\lambda\), and in turn the absolute invariant J of \textbf{B} could then be calculated.
There is, however, more to gain with less effort. We find the said 4 points on \textbf{A}, after a Möbius transformation, to satisfy an algebraic 4-th-degree equation \(P(X)=0\) with integral rational coefficients. Thus \(Y^{2}=P(X)\) describes \textbf{B} as a curve over \({\mathbb{Q}}\). Its Weierstraß normal form yields the invariants \(g_ 2=196\), \(g_ 3=-196\), and \(J=(14/13)^ 2\). The Mordell-Weil rank of \textbf{B}(\({\mathbb{Q}})\) may then be seen to be greater than 1, by reduction modulo 29.
The genus \(g=7\) of Macbeath's curve and the order \(h=504\) of its automorphism group are related by \(h=84(g-1)\). G is then called a Hurwitz group. Many instances of such groups were recently constructed by \textit{J. M. Cohen} [Math. Proc. Camb. Philos. Soc. 86, 395-400 (1979; Zbl 0419.20034) and in ''The geometric Vein'', Coxeter Festschrift, 511-518 (1982; Zbl 0496.20033)], as abelian extensions of PSL(2,7). We should like to point out that all such extensions occur in the author's paper in Math. Ann. 174, 79-99 (1967; Zbl 0157.036); they correspond to the ideal I of algebraic integers in \({\mathbb{Q}}(\sqrt{-7})\), and their orders are \(h=168(N(I))^ 3\). This also accounts for the orders of some groups in a paper by \textit{A. Sinkov} [Ann. Math., II. Ser. 38, 577-584 (1937; Zbl 0017.05702)]. automorphisms group of Macbeath curve; algebraic curves associated with subgroups of finite index in the; modular group; non-congruence subgroups; covering of the projective line; Hurwitz group; algebraic curves associated with subgroups of finite index in the modular group Klaus Wohlfahrt, Macbeath's Curve and the Modular Group, Glasg. Math. J.27 (1985), p. 239-247 - ISSN : 2118-8572 (online) 1246-7405 (print) - Société Arithmétique de Bordeaux Coverings of curves, fundamental group, Arithmetic ground fields for curves, Structure of modular groups and generalizations; arithmetic groups, Special algebraic curves and curves of low genus, Singularities of curves, local rings, Unimodular groups, congruence subgroups (group-theoretic aspects), Subgroup theorems; subgroup growth, Finite automorphism groups of algebraic, geometric, or combinatorial structures Macbeath's curve and the modular 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 The first example of a non-reduced component of the Hilbert scheme corresponding to nonsingular space curves was constructed by \textit{D. Mumford} [Am. J. Math. 84, 642-648 (1962; Zbl 0114.131)]. The topic was later studied by \textit{J. Kleppe} (Thesis, Oslo 1982), who constructed a series of such components, the general curves of which are contained in a cubic surface. The present paper aimes at a conjecture by Kleppe on necessary and sufficient conditions for a family of curves on cubic surfaces to be such a non-reduced component of the Hilbert scheme. While pointing out that Kleppe's conjecture must be modified a little (adding the hypothesis of linear normality in certain cases), the author is able to prove the conjecture in a wider range of degree and genus than was known before, although not the full range.
To show that a given curve lies on a non-reduced component, one finds first the dimension of the tangent space to the Hilbert scheme by deformation theory. If the curve moves in a family of strictly smaller dimension, we are basically done. The hard part, therefore, is to verify that a given curve on a cubic is not a specialization of a curve not on a cubic. In the cases treated in this paper, the relative size of degree and genus guarantees that the general curve is at least on a quartic, and the proof is by a study of curves on quartic surfaces. space curves; non-reduced component of the Hilbert scheme; family of curves on cubic surfaces Ellia, P., D'autres composantes non réduites de \(\text{Hilb} \mathbf{P}^3\), Math. Ann., 277, 3, 433-446, (1987) Parametrization (Chow and Hilbert schemes), Families, moduli of curves (algebraic), Projective techniques in algebraic geometry, Fine and coarse moduli spaces D'autres composantes non réduites de Hilb\({\mathbb{P}}^ 3\). (Other non reduced components of Hilb\({\mathbb{P}}^ 3\)) | 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 show that the cohomology groups of a general stable sheaf \(\mathcal{V}\) on a Hirzebruch surface \(\mathbb{F}_e\) are determined by its Euler characteristic, under some necessary and sufficient conditions on the intersection number of the total slope of \(\mathcal{V}\) with the section \(E\) with self-intersection \(E^2=-e\) (see Theorem 1.1). This result can be compared with the Brill-Noether theorem by \textit{L. Göttsche} and \textit{A. Hirschowitz} [Lect. Notes Pure Appl. Math. 200, 63--74 (1998; Zbl 0937.14010)] stating that a general stable bundle on \(\mathbb{P}^2\) has at most one nonzero cohomology group.
As one of the main ingredients in the proof of the aforementioned result, the authors give for a general stable sheaf on \(\mathbb{F}_e\) a particular resolution in terms of direct sums of line bundles, in the spirit of \textit{F. Gaeta}'s resolution [C. R. Acad. Sci., Paris 233, 912--913 (1951; Zbl 0043.36104)] in the case of \(\mathbb{P}^2\) (see Theorem 1.4).
As an outcome, the paper provides a classification of the Chern characters such that the general stable bundle is globally generated. Hirzebruch surfaces; Brill-Noether theorems; stable and globally generated sheaves; II Vector bundles on surfaces and higher-dimensional varieties, and their moduli, Rational and ruled surfaces, Algebraic moduli problems, moduli of vector bundles Brill-Noether theorems and globally generated vector bundles on Hirzebruch 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 book under consideration appeared first 1952 in Japanese. A revised edition came 1973, of which the present one is a so far unaltered English translation. As the author notes, most of the material was written during the years 1947-1948.
As the title indicates, it deals with the theory of algebraic functions. It splits into an algebraic part of roughly one third of the volume (chapters 1 and 2) and a complex analytic part (chapter 3 to 5).
Chapter 1 provides the elementary prerequisites from valuation theory: valuations, prime divisors, extensions. In chapter 2, the algebraic theory of algebraic function fields (of one variable) over an arbitrary base field is developed: divisors, the adele ring (here called idele ring), Weil and Hasse differentials, theorem of Riemann-Roch. Chapter 3 deals with general (i.e., not necessarily closed) Riemann surfaces: analytic mappings, coverings, differentials, integrals. Following H. Weyl's potential theoretic approach, the existence of nontrivial global functions and integrals is shown, and a proof of the Riemann mapping theorem is given. In chapter 4, the Riemann surface associated with a complex function field is constructed, and it is shown that the concepts of compact Riemann surfaces and of complex function fields essentially agree.
Finally, in chapter 5, abelian integrals, the Jacobian variety, the Abel- Jacobi map are treated. In an appendix, added in 1973, the author introduces Tate's notion of differentials and residues over an algebraic curve, and discusses the simplification of the proof of chapter 2 thereby obtained.
The book reflects the point of view and the fashion of its time of writing, which creates some difficulties for the reader of the nineteen- nineties. For example, the notation \(\mathbb{N}\), \(\mathbb{Z}\), \(\mathbb{R} \dots\) is not used (instead, the field of complex numbers if mostly labelled by \(K)\), and the rational first homology of a Riemann surface \({\mathcal R}\) is \(B_ 1({\mathcal R})\). More important is the lack of the notion of sheaf [sheaves have been introduced into algebraic geometry in \textit{J.-P. Serre}'s epoch-making ``Faisceaux algébriques cohérents'', Ann. Math., II. Ser. 61, 197-278 (1955; Zbl 0067.162)], whose use would greatly simplify both the statements and the proofs of most of the results presented in the book. The reviewer (1994) doesn't agree with the assertion in the foreword to the revised edition (1973) that ``neither noteworthy developments nor crucial reforms have occurred during the last twenty years''. -- However, the reader not troubled with questions of yesterday's or today's taste will find a well-composed book that, requiring not more than basic algebra and function theory, yields access to substantial results. compact Riemann surfaces; complex function fields; abelian integrals; Jacobian variety; Abel-Jacobi map \textit{Iwasawa K.}, Algebraic Functions, Amer. Math. Soc., New York and Providence (1993) (Trans. Math. Monogr.; 188). Algebraic functions and function fields in algebraic geometry, Compact Riemann surfaces and uniformization Algebraic functions. Translated from the Japanese by Goro Kato | 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 These two volumes represent the second edition of the author's well-known and beautiful introductory book on algebraic geometry (1972; Zbl 0258.14001) which has been translated into several other languages (e.g. into English 1974, second edition 1977). This new edition differs from the first one because it was reworked and completed with great care. Here we only want to emphasize the differences between the two editions.
Chapter I begins with some new sections such as: an elementary discussion about plane curves, their singularities and the projective plane. Then some considerations about the zeta function, the theorem of Abhyankar- Moh, the Jacobian conjecture, the Grassmann variety and its Plücker embedding, associative algebras, determinantal varieties and the Tsen theorem and its application to the rationality of surfaces, are added.
Chapter II contains the following new things: more examples of smooth varieties, the varieties associated to associative algebras, Puiseux expansions, singularities of maps, the generic irreducibility or smoothness of morphisms, etc.
In chapter III the author added considerations about pencils of conics, a more detailed discussion about the cubic curves with emphasis to some arithmetical questions, and in chapter IV, the inequality of Riemann-Roch for surfaces, the geometry of the smooth cubic surface in \({\mathbb{P}}^ 3\), the singularities of a curve on a surface and their resolutions, and Du Val singularities of surfaces. The first volume ends with a (new) appendix of algebraic prerequisites.
The second volume contains the following new paragraphs: (1) The classification of the geometric objects, universal schemes, and the Hilbert scheme (in chapter VI; for this reason the paragraph in chapter I about Chow coordinates has been removed); (2) Connectivity of the fibers of a morphism of algebraic varieties; (3) The topology of the singularities of curves (both in chapter VII), and (4) Kähler manifolds and the Hodge theorem (in chapter VIII). A few exercises from the old edition disappeared, but many others have been included.
All in all these changes and completions made this remarkable textbook more updated and even more interesting. This second edition is highly recommended to every mathematician. plane curves; zeta function; Jacobian conjecture; Puiseux expansions; singularities; pencils of conics; universal schemes; Hilbert scheme I. R. Shafarevich, \textit{Basic Algebraic Geometry} (Nauka, Moscow, 1988; Springer, Berlin, 2013), Vol. 1. Research exposition (monographs, survey articles) pertaining to algebraic geometry, Foundations of algebraic geometry, Local theory in algebraic geometry, Cycles and subschemes Basic algebraic geometry. (Osnovy algebraicheskoj geometrii.) Vol. 1: Algebraic manifolds in projective space. (Tom 1: Algebraicheskie mnogoobraziya v proektivnom prostranstve). Vol. 2: Schemes. Complex manifolds. (Tom 2: Skhemy. Kompleksnye mnogoobraziya). 2nd ed., rev. and compl | 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 number field with algebraic closure \(F^{\text{al}}\). Given a nice curve \(X\) over \(F\), denote by \(J\) and \(J^{\text{al}}\) the Jacobian of \(X\) and its base change to \(F^{\text{al}}\). The practical computations of the geometric endomorphism ring \(\text{End}(J^{\text{al}})\) for the curve \(X\) of gens \(\geq 2\) is the main concern of the paper under review, which is done by several improvements and generalization of the existing algorithms and methods.
In Section 2, after setting up some notation and background, the authors discussed the representations of endomorphisms in bits. Sections 3-5 are devoted to describing the algorithms and methods of numerically computing the group law of the Jacobian by developing the methods of [\textit{K. Khuri-Makdisi}, Math. Comput. 73, No. 245, 333--357 (2004; Zbl 1095.14057)]. The key point of their methods is to use the Newton and Puiseux lifts to numerical inversion of the Abel-Jacobi map by working infinitesimally. Then, in Section 6, they prove the correctness of their methods and algorithm.
In Section 7, the upper bounds on the dimension of endomorphism algebra as a \(\mathbb Q\)-vector space has been considered. They showed how determining Frobenius action on \(X\) for a large set of primes often quickly leads to sharp upper bounds. The last section of the paper includes several worked examples of curves with genus \(\geq 2\), where the algorithms and methods are examined practically. nice curve; Jacobian; endomorphism ring Abelian varieties of dimension \(> 1\), Computational number theory, Jacobians, Prym varieties, Arithmetic ground fields for abelian varieties, Computational aspects of algebraic curves Rigorous computation of the endomorphism ring of a Jacobian | 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 Recently, the theory of logarithmic structures on algebraic varieties has been developed by \textit{I. Illusie} [``Logarithmic spaces (according to K. Kato)'' in Barsotti Sympos. Algebraic Geometry, Abano Terme 1991, Perspect. Math. 15, 183-203 (1994)] and \textit{K. Kato} [in Algebraic Analysis, Geometry and Number Theory, Proc. JAMI Inaugur. Conf., Baltimore 1988, 191-224 (1989; Zbl 0776.14004)]. Each embedding of a variety \(D\) as a divisor in a smooth ambient space \(X\) gives rise to logarithmic structures on \(D\) and \(X\). In this paper, the author considers logarithmic embeddings of varieties over \(\mathbb{C}\) with normal crossings; these are logarithmic structures which are locally isomorphic to the standard one obtained from an embedding of the variety as a divisor in a smooth space. To each logarithmic embedding of \(D\) is associated a filtered logarithmic de Rham complex and a counterpart of this with integral coefficients, which is a constructible complex. If \(D\) is compact, the hypercohomology groups of these form mixed Hodge structures, which in the case of genuine embeddings reduces to the mixed Hodge structure of a deleted neighborhood of \(D\) [cf. \textit{A. Durfee}, Duke Math. J. 50, 1017-1040 (1983; Zbl 0545.14005) and \textit{F. Elzein}, Trans. Am. Math. Soc. 275, 71-106 (1983; Zbl 0511.14003)].
A logarithmic embedding provided with a suitable global section of its logarithmic sheaf is called a logarithmic deformation. Such a structure exists exactly for \(d\)-semistable varieties in the sense of \textit{R. Friedman} [Ann. Math., II. Ser. 118, 75-114 (1983; Zbl 0569.14002)]. The previous result is used for the construction of a ``limit mixed Hodge structure'' associated to a logarithmic deformation, which reduces to the construction by the reviewer [Invent. Math. 31, 229-257 (1976; Zbl 0305.14002)] in the case of a one-parameter smoothing with smooth total space. logarithmic embeddings; logarithmic de Rham complex; hypercohomology groups; mixed Hodge structures; logarithmic deformation J. H. M. STEENBRINK, Logarithmic embeddings of varieties with normal crossings and mixed Hodge structures, Math. Ann., 301 (1995), pp. 105-118. Zbl0814.14010 MR1312571 Variation of Hodge structures (algebro-geometric aspects), Embeddings in algebraic geometry, (Co)homology theory in algebraic geometry, Transcendental methods, Hodge theory (algebro-geometric aspects) Logarithmic embeddings of varieties with normal crossings and mixed Hodge structures | 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 the entire collection see Zbl 0745.00052.]
For a positive integer \(N\), \(J_ 0(N)\) denotes the Néron model of the Jacobian of the modular curve \(X_ 0(N)_ \mathbb{Q}\), and for a prime \(p\), \(\Phi_{N,p}\) denotes the group of connected components of the reduction of \(J_ 0(N)\) modulo \(p\).
The result of this article is the following: if \(p>3\), and if \(l\) is a prime not dividing \(N\), then the Hecke operator \(T_ l\) acts on \(\Phi_{N,p}\) by multiplication by \(l+1\). This generalizes a theorem of Ribet (who had to assume \(p^ 2\not| N)\) which plays an important role in his proof that the Taniyama-Weil conjecture implies Fermat's Last Theorem [\textit{K. A. Ribet}, Invent. Math. 100, No. 2, 431-476 (1990; Zbl 0773.11039)]. -- The proof uses the description of \(\Phi_{N,p}\) given in \textit{M. Raynaud}'s article in the same volume [cf. Courbes modulaires et courbes de Shimura, C. R. Sémin., Orsay/Fr. 1987-88 Astérisque 196- 197, 9-25 (1991; see the following review)], and the description of the reduction of \(X_ 0(N)\) as given by \textit{P. Deligne} and \textit{M. Rapoport} [cf. Modular Functions one variable II, Proc. internat. Summer School, Univ. Antwerp 1972, Lect. Notes Math. 349, 143-316 (1973; Zbl 0281.14010)], both of which were extended to the present case by the author [Ann. Inst. Fourier 40, No. 1, 31-67 (1990; Zbl 0679.14009)]. reduction of modular curves; Néron model of the Jacobian of the modular curve; group of connected components; Hecke operator ; Edixhoven, Courbes modulaires et courbes de Shimura. Astérisque, 196-197, 159, (1991) Jacobians, Prym varieties, Modular and Shimura varieties, Hecke-Petersson operators, differential operators (one variable) The action of the Hecke algebra on the group of components of Jacobians of modular curve is ``Eisenstein'' | 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 \(n\in A:= {\mathbb F}_q[T]\) be a prime element. Let \(X_1(n)\) be the smooth projective curve over \({\mathbb F}_q(T)\) associated to the moduli problem of classifying pairs \((\phi,P)\), where \(\phi\) is a rank 2 Drinfeld \(A\)-module over an \({\mathbb F}_q(T)\)-scheme and \(P\) is a nowhere vanishing point of its \(n\)-torsion.
Let \(J_1(n)\) be the Jacobian of \(X_1(n)\). The main result of this paper is that the closed fibre of the Néron model of \(J_1(n)\) over \(A_{(n)}\) has trivial geometric component group. It follows that the Néron model of \(J_1(n)\) over \({\mathbb P}^1_{{\mathbb F}_q} - \infty\) has connected fibres, hence the title. (The component group of the fibre above \(\infty\) is a much more complicated beast, but is not part of the moduli problem). This work is a function field analogue of results of \textit{B. Conrad}, \textit{B. Edixhoven} and \textit{W. Stein} [Doc. Math., J. DMV 8, 331--408 (2003; Zbl 1101.14311)], and the proof strategy is similar, although the translation is far from trivial.
To prove the result, it suffices to show that \(X_1(n)\) has a regular proper model over \(A_{(n)}\) with geometrically integral special fibre. The author does this by first constructing a model with a unique non-regular point (corresponding to \((\phi,P)\), where \(j(\phi)=0\) and \(P\) is the kernel of Frobenius). This is a cyclic quotient singularity, which is resolved using the Jung-Hirzebruch resolution developed in [loc. cit.]. Contraction of the special fibre of this resolution then yields the desired integral model of \(X_1(n)\).
A result of independent interest obtained along the way is a function field analogue of the theory of Igusa curves. The paper is rather technical, but very well written. component groups; Drinfeld modular curves; Igusa curves Modular forms associated to Drinfel'd modules, Formal groups, \(p\)-divisible groups, Drinfel'd modules; higher-dimensional motives, etc. The Drinfeld modular Jacobian \(J_1 (N)\) has connected fibers | 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 the entire collection see Zbl 0623.00011.]
This paper represents a continuation and extension of the author's previous work on foliations of complex analytic spaces by curves [cf. the author, Aportaciones Mat., Notas Invest. 1, 36-64 (1985; Zbl 0627.14011)]. In the first section, he recapitulates the concept of a geometric, possibly singular foliation of a complex manifold by holomorphic curves, as well as its analytic properties [cf. reference above]. From this he derives a more general, purely analytic definition: A holomorphic foliation (with singularities) of a complex manifold M is a holomorphic map \(X: L\to TM\) from a line bundle \(L\in Pic(M)\) to the tangent bundle TM of M. The equivalence of two such holomorphic foliations, (L,X) and (L',X'), is just given by biholomorphic equivalence.
In Section 2, the author proves that there is a one-to-one correspondence between the holomorphic foliations of M by curves and the invertible subsheaves of the sheaf of holomorphic vector fields on M. This allows to define analytically parametrized families of such foliations and, by applying A. Douady's theorem on the existence of universal families of quotient sheaves, to prove the existence of a universal family of holomorphic foliations of M by curves.
Finally, Section 3 provides some insight into the geometry of the parameter spaces of the constructed universal families. It is shown that the Chern class of the line bundle tangent to a foliation remains constant on each connected component of the universal parameter space, furthermore that for a compact Kähler manifold M with vanishing first Betti number the universal parameter space is a disjoint union of projective spaces, and that for a projective manifold M the universal space of foliations with given Chern class is a projective variety. As for the latter case, it is shown, in addition, that for some fixed Chern classes the universal space of foliations admits the structure of a projective bundle over a complex torus, and its dimension is computed. The basic ingredients of proof are the Kodaira-Nakano vanishing theorem, the Hirzebruch-Riemann-Roch theorem, and the existence of the Poincaré bundle on Picard varieties. holomorphic curve; vector bundle; universal deformation; local moduli space; sheaf cohomology; foliations of complex analytic spaces by curves; complex manifold X. GOMEZ-MONT , Universal families of foliations by curves , Astérisque 150-151, 1987 , p. 109-129. MR 89d:32067 | Zbl 0641.32014 Deformations of submanifolds and subspaces, Dynamics induced by group actions other than \(\mathbb{Z}\) and \(\mathbb{R}\), and \(\mathbb{C}\), Parametrization (Chow and Hilbert schemes), Foliations in differential topology; geometric theory Universal families of foliations by 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 \(M_{\mathbb{P}^2}(r,\chi)\) denote the moduli space of semistable sheaves on \(\mathbb{P}^2\) with linear Hilbert polynomial \(P(m)=rm+\chi\). The generic stable sheaves in this moduli space are line bundles on smooth plane curves of degree \(r\). It is known that the spaces \(M_{\mathbb{P}^2}(r,\chi)\) are projective and irreducible of dimension \(r^2+1\).
The authors of this paper investigate the geometry of the spaces \(M_{\mathbb{P}^2}(4,\chi)\) for \(1\leq \chi\leq4\). They decompose each moduli space into locally closed subvarieties (or \textit{strata}) and describe each stratum as a good or geometric quotient of a set of morphisms of locally free sheaves.
The different strata are characterized by cohomological conditions and the resolutions of the sheaves contained in any stratum (except for one case) have been found by the second author in [``On two notions of semistability'', Pac. J. Math. 234, No. 1, 69--135 (2008; Zbl 1160.14007)] In the paper under review, one of the main difficulties is proving that the quotients are good or geometric, in particular when the group acting on the space of morphisms is not reductive. They are able to prove this result by means of a method already used by the first author in [``Varietes de modules extremales de faisceaux semi-stables sur \(\mathbb{P}^2\)'', Math. Ann. 290, No.4, 727--770 (1991; Zbl 0755.14005)].
The cohomology estimates proved in the paper show that Clifford's theorem is true not only for the generic sheaves in the moduli spaces, but for all sheaves in \(M_{\mathbb{P}^2}(r,\chi)\), when \(0 \leq \chi < r\leq 4\). Hence, the authors conjecture that a suitable generalization of Clifford's theorem holds for any \(r\geq1\). Moduli spaces of sheaves; semistable sheaves; plane quartics Drézet J.-M., Maican M., On the geometry of the moduli spaces of semi-stable sheaves supported on plane quartics, Geom. Dedicata, 2011, 152, 17-49 Algebraic moduli problems, moduli of vector bundles, Fine and coarse moduli spaces On the geometry of the moduli spaces of semi-stable sheaves supported on plane quartics | 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 curve, that is a connected, reduced, projective scheme of dimension 1 over an algebraically closed field, admits two types of compactifications of its generalized Jacobian: the moduli schemes of P-quasistable torsion-free, rank-1 sheaves and Seshadri's moduli schemes of S-equivalence classes of semistable torsion-free, rank-1 sheaves. Both are constructed with respect to a choice of polarization. The former are fine moduli spaces which were shown to be complete; here we show that they are actually projective. The latter are just coarse moduli spaces. Theorem 4.4, the main result of the paper, gives a sufficient condition under which these two types of moduli spaces are equal. curves; Jacobians; moduli spaces; compactifications Esteves, E, Compactified Jacobians of curves with spine decomposition, Geometriae Dedicata, 139, 167-181, (2009) Jacobians, Prym varieties, Theta functions and curves; Schottky problem, Vector bundles on curves and their moduli Compactified Jacobians of curves with spine decompositions | 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 \(\mathscr{X}/S\) be a flat algebraic stack of finite presentation. We define a new étale fundamental pro-groupoid \(\Pi_1(\mathscr{X}/S)\), generalizing Grothendieck's enlarged étale fundamental group from SGA\,3 to the relative situation. When \(S\) is of equal positive characteristic \(p\), we prove that \(\Pi_1(\mathscr{X}/S)\) naturally arises as colimit of the system of relative Frobenius morphisms \(\mathscr{X}\to \mathscr{X}^{p/S}\to \mathscr{X}^{p^2/S}\to\cdots\) in the pro-category of Deligne Mumford stacks. We give an interpretation of this result as an adjunction between \(\Pi_1\) and the stack \(\operatorname{Fdiv}\) of \(F\)-divided objects. In order to obtain these results, we study the existence and properties of relative perfection for algebras in characteristic \(p\). relative Frobenius; \(F\)-divided object; perfection; coperfection; étale fundamental group; étale affine hull Characteristic \(p\) methods (Frobenius endomorphism) and reduction to characteristic \(p\); tight closure, Stacks and moduli problems, Homotopy theory and fundamental groups in algebraic geometry, Positive characteristic ground fields in algebraic geometry Unramified \(F\)-divided objects and the étale fundamental pro-groupoid 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 Let \(g: Z \to \mathbb C\) be a polynomial mapping from a smooth affine variety \(Z\), with compactification \(\overline{g} : \overline{Z}\to \mathbb C\); let \(j: Z \to \overline Z\) be the inclusion. Suppose a bounded constructible complex of sheaves \(\mathbb F\) of modules over a PID \(R\) is given. The author studies the question under which conditions the sheaf complex of vanishing cycles \(\Phi_{\overline g}j_*\mathbb F\) is acyclic, under successively less restrictive conditions, but in more special situations. If \({\mathbb F}= \mathcal L\), a locally free \(R_Z\)-module of finite rank, then acyclicity of \(\Phi_{\overline g}j_*\mathcal L\) implies that \(g\) defines a smooth fibre bundle. Finally a condition is given which avoids a compactification.
As a corollary the following generalisation of the theorem of \textit{Hà Huy Vui} and \textit{Lê Dũng Tráng} [Acta Math. Vietnam. 9, No. 1, 21--32 (1984; Zbl 0597.32005)] is given. Suppose \(\dim Z=2\), that \(\overline{g^{-1}(0)}={\overline g}^{-1}(0)\) and that \(g^{-1}(0)\) is generically reduced. If the Euler characteristic \(\chi(\{g=t\})\) is independent of \(t\), then \(g\) defines a smooth fibre bundle. Examples show that none of the hypotheses can be dropped. atypical value; sheaf of vanishing cycles; topological triviality Structure of families (Picard-Lefschetz, monodromy, etc.), Global theory of complex singularities; cohomological properties, Equisingularity (topological and analytic), Topological properties of mappings on manifolds Euler characteristics and atypical values | 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 \(C/S\) be a relative smooth proper curve of genus \(g\geq 1\) over an irreducible base \(S\). Let \(\ell\) be a prime number which is invertible on \(S\), and let \(s\) be a geometric point of \(S\). The fundamental group \(\pi_1(S,s)\) acts linearly on the fiber \(\text{ Pic}^0(C)[\ell]_s\) of the \(\ell\)-torsion of the relative Jacobian of \(C\). If we identify the latter with \(\mathbb Z/\ell)^{2g}\), we obtain the mod-\(\ell\) monodromy representation \(\rho_{C/S, \ell}:\pi_1(S,s)\rightarrow \text{GL}_{2g}(\mathbb Z/\ell)\). By considering powers of \(\ell\) and taking the limit, we obtain the \(\ell\)-adic monodromy representation where the target is \(\text{GL}_{2g}(\mathbb Z_\ell)\).
If there is a primitive \(\ell\)-th root of unity globally on \(S\), then \(\text{Pic}^0(C)[\ell]_s\) carries an alternating form, and the image of \(\rho_{C/S,\ell}\) is contained in the corresponding symplectic group. Deligne and Mumford proved that for \(C/S\) sufficiently general, the image of \(\rho_{C/S,\ell}\) is equal to this symplectic group.
In the paper under review, the authors compute the image of \(\rho_{C/S,\ell}\) in the cases when \(S\) is an irreducible component of the moduli space of hyperelliptic or trielliptic curves, and \(C/S\) is the tautological curve. In the hyperelliptic case, the image of \(\rho_{C/S,\ell}\) is the full symplectic group. This result was previously obtained by J.-k. Yu (unpublished). In the trielliptic case, there is a \(\mathbb Z/3\) action which constrains the image of the monodromy representation to lie in a certain unitary group. The theorem in this case says that the image in fact is this unitary group. There are a number of related results in the literature; see the introduction of the paper under review for further details.
The proof proceeds by induction on the genus. Since every curve of genus \(g=1,2\) is hyperelliptic, the claim in this case follows from the analogous statement for the moduli space \(\mathcal M_g\) of all curves of genus \(g\). The base case \(g=3\) for the trielliptic case is done using a comparison with a Picard modular variety. monodromy; hyperelliptic curves; trielliptic curves J. D. Achter and R. Pries, ''The integral monodromy of hyperelliptic and trielliptic curves,'' Math. Ann., vol. 338, iss. 1, pp. 187-206, 2007. Arithmetic aspects of modular and Shimura varieties, Structure of families (Picard-Lefschetz, monodromy, etc.), Jacobians, Prym varieties The integral monodromy of hyperelliptic and trielliptic 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 \(\phi : D \to C\) be an unramified morphism of projective smooth curves defined over \(\mathbb{Q}\). By the Chevalley-Weil theorem, there is a number field \(K\) such that \(\phi^{-1}( C(\mathbb{Q}) ) \subset D(K)\). In the paper under review, the author considers a morphism \(\psi\) between two special families of affine curves \(V\) and \(W\), and explicitly computes a number field \(K\) such that
\[
\psi^{-1}(W(\mathbb{Q})) \subset V(K),
\]
and uses this setup to completely determine the rational solutions of three examples of curves and two examples of families of curves. This explicit computation is generalized in the recent works \textit{K. Draziotis} and \textit{D. Poulakis} [``Explicit Chevalley-Weil theorem for affine plane curves,'' Rocky Mt. J. Math. 39, No. 1, 49--70 (2009; Zbl 1222.14063) and An effective version of Chevalley-Weil theorem for projective plane curves, arxiv:0904.3845v1].
Let \(F(X,Y):= f(X,Y)^q - h(X)\, g(X,Y)\) be a polynomial in \(\mathbb{Z}[X,Y]\) where \(f\), \(h\), and \(g\) are fairly general polynomials. Let \(h(X)=h_1(X)\, h_2(X)\) be a certain factorization in \(\mathbb{Z}[X]\), which always is possible, such that \(q \mid \deg(h_1)\). See the paper for details. Consider the varieties
\[
\begin{aligned} V &: F(X,Y)= 0,\;T^q = h_1(X),\\ W &: F(X,Y)=0. \end{aligned}
\]
For \(q=2\) or 3, he explicitly computes a finite set \(S\) such that the number field \(K\) described above is given by \(\mathbb{Q}(\root q \of b \in S)\). More specifically, he shows that for each \((x,y) \in W(\mathbb{Q})\), there is a \(b \in S\) such that the twist \(b T^q = h_1(X)\) has a solution \((x,t)\) where \(t \in \mathbb{Q}\). Thus, it reduces the problem to that of solving finitely many twists of a superelliptic curve. It seems to the reviewer that the restriction on \(q\) in the theorem is only for practical computability purpose and that the theorem can be somewhat explicitly stated for \(q\) being a prime number.
He gives a geometric interpretation of this reduction of the original problem in which it is established that the morphism between the projective desingularizations \(D\) and \(C\) of \(V\) and \(W\), respectively, is unramified, and hence, being put into the context of the Chevalley-Weil Theorem. He uses the crucial condition \(q \mid \deg(h_1)\) to pull the unramified property out of the towers of the function fields \(\overline{\mathbb{Q}}(\mathbb{A})\), \(\overline{\mathbb{Q}}(W)\), and \(\overline{\mathbb{Q}}(Z)\) where \(Z\) is the affine variety \(T^q = h_1(X)\), and he uses the condition in the proof of the main theorem for the affine varieties as well. rational points on curves Curves of arbitrary genus or genus \(\ne 1\) over global fields, Rational points, Global ground fields in algebraic geometry On the rational points of the curve \(f(X,Y)^{q} = h(X)g(X,Y)\) | 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 From the foreword: Let \(K|k\) be a finite Galois extension of fields, \(X'\) a quasi-projective \(K\)-variety. Then there exists a quasi-projective \(k\)-variety \(W\) which has in particular the properties that \(X' (K)\simeq W(k)\) and \(W_K=W\otimes_kK\) is a product of Galois-conjugates of \(X'\). \(W\) is called the Weil restriction of \(X'\) with respect to \(K|k)\). After Weil-restrictions where successfully studied to solve problems of ``pure mathematics'' for decades, a new direction of research was shown by Frey in a talk in 1998. He suggested to use Weil-restrictions of elliptic curves both as a tool to construct as well as to break discrete-logarithm problems (D-L problems).
In this work we study Weil-restrictions of varieties both from a pure as well as from an applied point of view. In particular, we show how questions on Weil-restrictions of abelian varieties motivated by cryptoanalytical applications can often be proven directly from the defining functorial properties.
In chapter one, we first give basic definitions related to Weil-restrictions of varieties and schemes. We study the Weil-restriction of a projective variety \(X'/K\) with a rational point with respect to a Galois field extension \(K|k\), we analyze the Picard functor of the Weil-restriction \(W\) and we derive the structure of the endomorphism ring of Weil-restrictions of an abelian variety over finite fields.
For the second chapter, if \(A\) is an elliptic curve \(E\), then \(W\) is isogenous to the product of \(E\) and an abelian variety \(N\) called its trace-zero-hypersurface. We study the Néron-Severi group of \(N\) and in particular the polarizations of \(N\).
The third chapter is entirely devoted to cryptoanalytical applications. finite Galois extension; Weil restriction; cryptoanalytical applications; abelian variety; elliptic curve C. Diem, A study on theoretical and practical aspects of Weil-restrictions of varieties, Dissertation, 2001. http://www.math.uni-leipzig.de/~diem/dissertation\_diem.dvi Arithmetic varieties and schemes; Arakelov theory; heights, Arithmetic ground fields for abelian varieties, Abelian varieties of dimension \(> 1\), Applications to coding theory and cryptography of arithmetic geometry, Elliptic curves A study on theoretical and practical aspects of Weil-restrictions of 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 main purpose of the article is to construct, and study, Wronski systems on a family \(X/S\) of curves. When \(X/S\) is a family consisting of complete intersection reduced curves, the author constructs a Wronski system consisting of locally free \(\mathcal O_X\)-modules \(Q^i\) that have natural functorial properties and that fit into exact sequences
\[
0\to \omega^{\otimes i} \to Q^i \to Q^{i-1} \to 0,
\]
where \(\omega\) is the dualizing sheaf of the family. The \(Q^i\) are unique with the functorial properties. To show that Wronski systems exist is important because they give rise to Wronski determinants whose zero loci give the Weierstrass points of the family. For a single integral Gorenstein curve Weierstrass points have been defined and studied by Widland in his thesis [see also \textit{R. F. Lax} and \textit{C. Widland}, Pac. J. Math. 50, No. 1, 111-122 (1991; Zbl 0686.14033) and \textit{A. Garcia} and \textit{R. F. Lax}, Commun. Algebra 22, No. 12, 4841-4854 (1994; Zbl 0824.14033)], and Wronski systems have been constructed by \textit{D. Laksov} and \textit{A. Thorup} [Ark. Mat. 32, No. 2, 393-422 (1994; Zbl 0839.14020)]. The latter authors also got results for certain families as a consequence of their work on Wronskian systems on schemes of arbitrary dimension. \textit{E. Esteves} has later used residues to compare his approach with other approaches for families of curves [see Bol. Soc. Bras. Mat., Nova Sér. 26, No. 2, 229-243 (1995; Zbl 0855.14003)]. The present article shows that the previous results, in a natural way, can be extended to families of curves, and that reduced curves can be allowed in the family, as long as the family consists of complete intersections. The construction holds in any characteristic. Wronski systems; Weierstrass points; families of curves; complete intersections Esteves, E.: Wronski algebra systems on families of singular curves. Ann. sci. Éc. norm. Super. (4) 29, No. 1, 107-134 (1996) Riemann surfaces; Weierstrass points; gap sequences, Singularities of curves, local rings, Families, moduli of curves (algebraic), Complete intersections Wronski algebra systems on families of singular 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 Given a relative faithfully flat pointed scheme over the spectrum of a discrete valuation ring \(X\rightarrow S\), this paper is motivated by the study of the natural morphism from the fundamental group scheme of the generic fiber \(X_\eta\) to the generic fiber of the fundamental group scheme of \(X\). Given a torsor \(T\rightarrow X_\eta\) under an affine group scheme \(G\) over the generic fiber of \(X\), we address the question of finding a model of this torsor over \(X\), focusing in particular on the case where \(G\) is finite. We provide several answers to this question, showing for instance that, when \(X\) is integral and regular of relative dimension 1, such a model exists on some model \(X'\) of \(X_\eta\) obtained by performing a finite number of Néron blowups along a closed subset of the special fiber of \(X\). Furthermore, we show that when \(G\) is étale, then we can find a model of \(T\rightarrow X_\eta\) under the action of some smooth group scheme. In the first part of the paper, we show that the relative fundamental group scheme of \(X\) has an interpretation as the Tannaka Galois group of a Tannakian category constructed starting from the universal torsor. Group actions on varieties or schemes (quotients), Group schemes, Elliptic curves over global fields Models of torsors and the 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 \(X\) be a smooth commutative group scheme over a complex variety \(S\). Identify \(X\) with the induced representable functor from \(\ll S\)-schemes \(\gg ^ \circ\) to \(\ll\)groups\(\gg\). Let \(f:S \to \text{Spec}\mathbb{C}\) be the canonical map. Let \({\mathbf G} = f_*X\), which is the functor from \(\ll\mathbb{C}\)-schemes\(\gg^ \circ\) to \(\ll\)groups\(\gg\) given by \({\mathbf G}(T)=\{\)sections of \(X \times T\) over \(S \times T\}\). Our main theorem characterizes the restriction of \({\mathbf G}\) to the category of reduced \(\mathbb{C}\)-schemes:
We find that either \({\mathbf G}\) is representable by a complex Lie group \(H\), or else it is of the form \(H \times \bigl( \bigoplus^ \infty_{i=1}\mathbb{G}_ a\bigr)\). For example, if \(S=\mathbb{A}^ 1\), and \(X= \mathbb{G}_ a \times \mathbb{A}^ 1\), then \(G=\bigoplus^ \infty_{i=1}\mathbb{G}_ a\).
As a consequence we are able to describe the topology which is induced by \({\mathbf G}\) on the group \(G\) of sections of \(X/S\). -- In the important case where \(S\) is proper, and \(X\) is quasiprojective over \(\mathbb{C}\), our main theorem is a special case of a representability theorem of \textit{A. Grothendieck} [Séminaire de géométrie algébrique. Bures-Sur-Yvette: IHÉS (1962; Zbl 0159.50402), Exposé 13 and 20]. In this situation, Grothendieck's theorem implies that \({\mathbf G}\) is representable as a functor on \(\ll\mathbb{C}\)-schemes\(\gg\). sections; commutative group scheme David B. Jaffe, On sections of commutative group schemes, Compositio Math. 80 (1991), no. 2, 171 -- 196. Group schemes, Representations of Lie and linear algebraic groups over global fields and adèle rings On sections of commutative 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 The generalized étale homotopy pro-groups \(\pi_1^{\text{ét}}(\dot{c}{C}, x)\) associated to pointed connected small Grothendieck sites \((\dot{c}{C}, x)\) are defined and their relationship to Galois theory and the theory of pointed torsors for discrete groups is explained. Applications include new rigorous proofs of some folklore results around \(\pi_1^{\text{ét}}({\text{ét}}{X}, x)\), a description of Grothendieck's short exact sequence for Galois descent in terms of pointed torsor trivializations, and a new étale van Kampen theorem which gives a simple statement about a pushout square of pro-groups that works for covering families which do not necessarily consist exclusively of monomorphisms. A corresponding van Kampen result for Grothendieck's profinite groups \(\pi_1^{\text{Gal}}\) immediately follows. étale homotopy theory; simplicial sheaves étale homotopy theory; simplicial sheaves Misamore, Michael D.: Nonabelian H1 and the étale Van kampen theorem, Can. J. Math. 63, No. 6, 1388-1415 (2011) Simplicial sets, simplicial objects (in a category) [See also 55U10], Homotopy theory and fundamental groups in algebraic geometry Nonabelian \(H^1\) and the étale van Kampen 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 Let \(W_d\) denote the cycle on the Jacobian of a smooth projective curve of genus \(g\) given by the image of the \(d\)th symmetric product of the curve under the Abel-Jacobi map. Poincaré's formula says that \(W_{g-d}={W^d_{g-1}\over d!}\). The present paper gives an analogue of this formula for a nodal curve \(X\) of arithmetic genus \(g\) with \(k\) nodes in characteristic 0. Let \(\overline J^0(X)\) denote the moduli space of rank-1 torsion free sheaves of degree 0 on \(X\) and \(\widetilde J^0(X)\) its normalization. Using the symmetric powers of the smooth part of \(X\) the autors define cycles \(\widetilde W_d\) in \(\widetilde J^0(X)\) and proved the analogue of Poincaré's formula \(\widetilde W_{g-d}= {\widetilde W^d_{g-1}\over d!}\). Moreover, identifying \(\widetilde J^0(X)\) with the corresponding Brill-Noether locus \(\widetilde B_X(1,d,1)\), also an analogue of the Riemann singularity theorem is shown, namely:
(1) \(\widetilde B_X(1,d,1)\) is a normal projective variety for all \(d\), (2) \(\widetilde B_X(1,d,2)\) is the singular locus of \(\widetilde B_X(1,d,1)\) for all \(d< g\) and (3) for \(d< g\), \(r\geq 2\), \(\widetilde B_X(1,d,r)\) has codimension \(\geq r\) in \(\widetilde B_X(1,d,1)\).
The proofs use some results on compactifications of the Picard scheme as well as on parabolic bundles of the first author. nodal curve; Poincaré's theorem; Riemann's singularity theorem Bhosle Usha, N; Parameswaran, AJ, On the Poincaré formula and Riemann singularity theorem over nodal curves, Math. Ann., 342, 885-902, (2008) Families, moduli of curves (algebraic), Jacobians, Prym varieties, Singularities of curves, local rings On the Poincaré formula and the Riemann singularity theorem over nodal 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 In his previous paper [Math. Res. Lett. 7, 123--132 (2000; Zbl 0959.14013)], the author proved that in characteristic zero the Jacobian \(J(C)\) of a hyperelliptic curve \(C: y^2=f(x)\) has only trivial endomorphisms over an algebraic closure \(K_a\) of the ground field \(K\) if the Galois group \(\text{Gal}(f)\) of the irreducible polynomial \(f(x) \in K[x]\) is either the symmetric group \(S_n\) or the alternating group \(A_n\). Here \(n>4\) is the degree of \(f\). In the next paper [in: Moduli of abelian varieties. Proc. 3rd Texel conf., Netherlands 1999 (Basel: Birkhäuser) Prog. Math. 195, 473--490 (2001; Zbl 1047.14015)], we extended this result to the case of certain``smaller'' Galois groups. In particular, we treated the infinite series \(n=2^r+1, \text{Gal}(f)=L_2(2^r)\). The case of small Mathieu groups \(M_n\) (with \(n=11,12)\) was also treated.
In this paper we do the case of large Mathieu groups \(M_n\) (with \(n=22,23,24\)). We also treat the infinite series \(\text{Gal}(f)=L_m(2^r)\) (with \(m>2\) except the cases \((m,r)=(3,2)\) or \((4,1)\)), assuming that the set \(R\) of roots of \(f\) can be identified with the corresponding projective space \(\mathbb{P}^{m-1}(\mathbb{F}_{2^r})\) over the finite field \(\mathbb{F}_{2^r}\) of characteristic 2 in such a way that the Galois action on \(R\) becomes the natural action of \(L_m(2^r)\) on the projective space. Zarhin Yu.G. (2002). Hyperelliptic jacobians without complex multiplication, doubly transitive permutation groups and projective representations. Contemp. Math. 300: 195--210 Jacobians, Prym varieties, Algebraic theory of abelian varieties, Curves of arbitrary genus or genus \(\ne 1\) over global fields, Abelian varieties of dimension \(> 1\) Hyperelliptic Jacobians without complex multiplication, doubly transitive permutation groups and projective representations. | 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 recalls C. Chevalley's and E. Warning's theorem on the number of rational points of an affine variety defined over a finite field and the related results of \textit{J. Ax} [Am. J. Math. 86, 255--261 (1964; Zbl 0121.02003)] and \textit{N. Katz} [Am. J. Math. 93, 485--499 (1971; Zbl 0237.12012)]; he then briefly describes the Weil conjectures and mentions the different cohomology theories inspired by those conjectures. The author pays particular attention to the rigid cohomology defined by \textit{P. Berthelot} [Mém. Soc. Math. Fr., Nouv. Sér. 23, 7--32 (1986; Zbl 0606.14017)] and describes several applications of that cohomology theory: a theorem of Chevalley-Warning type, a recent theorem of \textit{H. Esnault} [Invent. Math. 151, No. 1, 187--191 (2003, Zbl 1092.14010)] asserting in particular that \(|X({\mathbb F}_q)|\equiv 1\pmod q\) for a Fano variety \(X\) defined over the finite field \({\mathbb F}_q\) of \(q\) elements, a rather technical theorem, attributed by the author to T. Ekedahl, which implies in particular that
\[
|X({\mathbb F}_q)|\equiv|Y({\mathbb F}_q)|\pmod q
\]
for any two smooth proper birationally equivalent to each other geometrically connected varieties \(X\) and \(Y\) over \({\mathbb F}_q\), and a theorem asserting that the fundamental group of a smooth proper rationally chain connected variety over an algebraically closed field of positive characteristic \(p\) is finite, of order not divisible by \(p\). Fano varieties; chain rationally connected varieties; rigid cohomology; Weil cohomologies Chambert-Loir, A., Points rationnels et groupes fondamentaux: applications de la cohomologie \textit{p}-adique, Astérisque, 294, 125-146, (2004), (d'après P. Berthelot, T. Ekedahl, H. Esnault, etc.) \(p\)-adic cohomology, crystalline cohomology, Fano varieties, Finite ground fields in algebraic geometry, Rational points, Coverings of curves, fundamental group, Rigid analytic geometry, Varieties over finite and local fields, Coverings in algebraic geometry, Rational and unirational varieties Rational points and fundamental groups: applications of the \(p\)-adic cohomology (following P. Berthelot, T. Ekedahl, H. Esnault, etc.). | 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 \(V\) be a smooth projective variety, \(D \subset V\) a divisor and \(L\) the corresponding line bundle on \(V\). Let \(s : V \backslash D \to L^\times\) be a section, where \(L^\times = L \backslash \{\) the zero section\}. In this paper new conditions are given under which \(s\) induces an isomorphism of fundamental groups.
As an application, some fundamental groups of singular curve complements are computed. divisor; isomorphism of fundamental groups; singular curve complements DOI: 10.2996/kmj/1138039969 Homotopy theory and fundamental groups in algebraic geometry, Divisors, linear systems, invertible sheaves, Singularities of curves, local rings Remarks on fundamental groups of complements of divisors on algebraic 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 This work, having its origin in a lecture delivered by the first author at the Haverford College in 1991, is centred around the famous three point theorem of \textit{G. V. Belyĭ} [Izv. Akad. Nauk SSSR, Ser. Mat. 43, 267--276 (1979; Zbl 0409.12012)]. The authors first prove that if a smooth complete algebraic curve \(X\) is defined over an algebraic number field, then there is a covering \(X\rightarrow{\mathbb{P}}^{1}\) ramified over a subset the three points set \(\{0, 1,\infty\}\), and remark that their method of proof leads to a short proof of the weak Hironaka resolution of singularities in characteristic zero, see [\textit{F. A. Bogomolov} and \textit{T. G. Pantev}, Math. Res. Lett. 3, No. 3, 299--307 (1996; Zbl 0869.14007)].
The authors give two proofs of the converse assertion in Belyi's theorem: a function-theoretical one and a shorter proof phrased in the language of schemes. In Part II of their work the authors ``discuss a number of ideas and problems which came up about the time that Belyi's original theorem was proved, in 1978''. Here is a list of the topics discussed in Part II: triangulations coming from unramified coverings; modular curves and curves ramified over three points; dominant classes of varieties; ramification over more than three points; faithful action of Gal \((\overline{\mathbb Q}\setminus\mathbb Q)\) on the profree group with two generators. algebraic curve; number field; triangulation Arithmetic ground fields (finite, local, global) and families or fibrations, Ramification problems in algebraic geometry, Arithmetic ground fields for curves, Algebraic functions and function fields in algebraic geometry Geometric properties of curves defined over number 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 [For the entire collection see Zbl 0741.00047.]
Let \(M=\Gamma\backslash D\) be the quotient of a symmetric space \(D\) with a hermitian structure (arising from a semisimple algebraic group \(G\) over \(\mathbb{Q})\) by an arithmetic subgroup \(\Gamma\) of \(G(\mathbb{Q})\). Then \(M\) is an algebraic variety having a canonical compactification \(X\) (the Baily- Borel-Satake compactification) which is a projective variety. Let \(E\) be local system on \(M\) defined by a rational representation of \(G\), and denote by \(H(E)\) the intersection cohomology complex of \(X\) associated to \(E\).
The aim of this paper is two-fold. The first one is to relate various weight filtrations on the fiber of a cohomology sheaf of \(H(E)\) in a point of the boundary. The first named author had proved a conjecture of Zucker saying that the \(L^ 2\)-complex represents \(H(E)\) [see \textit{E. Looijenga}, Compos. Math. 67, No. 1, 3-20 (1988; Zbl 0658.14010)]. This proof was based on a purity theorem involving weights with respect to a ``local Hecke operator''. This purity result used in turn the elaborated theory of M. Sakai.
The second aim of this paper is to give another proof of the Zucker conjecture [which in fact is the third known proof after Looijenga's proof mentioned above and the proof by \textit{L. Saper} and \textit{M. Stern} contained in Ann. Math., II. Ser. 132, No. 1, 1-69 (1990; Zbl 0653.14010)]. This new proof has the advantage of using more elementary results and is based on a purity lemma of Serre (which in fact was the starting point of the theory of weights). quotient of a symmetric space; Baily-Borel-Satake compactification; intersection cohomology; weight filtrations; Zucker conjecture Looijenga, E.; Rapoport, M., \textit{weights in the local cohomology of a baily-Borel compactification}, Complex geometry and Lie theory (Sundance, UT, 1989), 223-260, (1991), American Mathematical Society, Providence, RI Other algebro-geometric (co)homologies (e.g., intersection, equivariant, Lawson, Deligne (co)homologies), Modular and Shimura varieties Weights in the local cohomology of a Baily-Borel compactification | 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 studies the \(\mathbb A^1\)-stable homotopy theory of schemes as introduced by \textit{F. Morel} and \textit{V. Voevodsky} [Publ. Math., Inst. Hautes Étud. Sci. 90, 45-143 (1999; Zbl 0983.14007)]. After reviewing the basic notions of this theory, the author studies base change functors associated to a finite separable extension \(L\) of a field \(k\). He introduces an analogue of the extension of scalars functors from spaces over \(k\) to spaces over \(L\) and explicitly describes the left adjoint and the right adjoint of this functor. The extension of these functors to categories of spectra and the relation to change of groups functors in equivariant homotopy theory are discussed. As an application of these functors, the author next constructs elements in the Picard group of the stable homotopy category over \(k\) from finite dimensional representations of the Galois group of the extension. Finally, the author proves an analogue of the Wirthmüller isomorphism in the \(\mathbb A^1\)-setting, which states that applying the left and right adjoint of the extension of scalars functor to an \(L\)-spectrum leads to \(\mathbb A^1\)-weakly equivalent \(k\)-spectra. \(\mathbb A^1\)-homotopy theory; Verdier duality; Wirthmüller isomorphism; Picard group Hu, P., \textit{base change functors in the A1 -stable homotopy category}, Homology, Homotopy Appl., 3, 417-451, (2001) Equivariant homotopy theory in algebraic topology, Homotopy theory and fundamental groups in algebraic geometry, Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20], Motivic cohomology; motivic homotopy theory Base change functors in the \(\mathbb{A}^1\)-stable homotopy category | 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 generalize the algebraic results of \textit{R. M. Skjelnes} and \textit{D. Laksov} [Compos. Math. 126, 323--334 (2001; Zbl 1056.14500)] and \textit{R. M. Skjelnes} [Ark. Mat. 40, No. 1, 189--200 (2002; Zbl 1022.14003)], and obtain easy and transparent proofs of the representability of the Hilbert functor of points on the affine scheme whose coordinate ring is any localization of the polynomial ring in one variable over an arbitrary base ring. The coordinate ring of the Hilbert scheme is determined. We also make explicit the relation between our methods and the beautiful treatment of the Hilbert scheme of curves via norms, indicated by \textit{A. Grothendieck} [Sem. Bourbaki 13(1960/61), No. 221 (1961; Zbl 0236.14003)], and performed by \textit{P. Deligne} [in: Sémin. Géométrie algébrique, Bois-Marie 1963/64 SGA4, Tome 3, exposé XVII, Lect. Notes Math. 305, Springer-Verlag, New York, 250--480 (1973; Zbl 0255.14011)]. Laksov, D., Skjelnes, R.M., Thorup, A.: Norm on rings and the Hilbert scheme of points on a line. Q. J. Math. 56, 367--375 (2005) Parametrization (Chow and Hilbert schemes), Ideals and multiplicative ideal theory in commutative rings Norms on rings and the Hilbert scheme of points on the 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 A regular morphism of Noetherian rings is a filtered colimit of smooth morphisms. This result was proved by the reviewer [\textit{D. Popescu}, Nagoya Math. J. 100, 97-126 (1985; Zbl 0561.14008); 104, 85-115 (1986; Zbl 0592.14014) and 118, 45-53 (1990; Zbl 0685.14009)], by \textit{M. André} [``Cinq exposés sur la desingularization'', Handwritten manuscript, École Polytechnique Fédérale de Lausanne (1971)], by \textit{T. Ogoma} [J. Algebra 167, No.~1, 57-84 (1994; Zbl 0821.13003)] and by \textit{M. Spivakovsky} [J. Am. Math. Soc. 12, No.~2, 381-444 (1999; Zbl 0919.13009)].
The exposition under review contains the easiest, self contained presentation of the proof which could be used also by beginners: Low dimensional Quillen cohomology groups are introduced directly using an argument of Faltings, local criterion of flatness is given only in a simple form used in the proof, Zariski's main theorem is written after Peskine, etc. A strong motivation of the result is given by its applications to Artin approximation and the Bass-Quillen conjecture. The reviewer used this paper a lot in his exposition ``Artin approximation'' [in ``Handbook of Algebra'', ed. \textit{M. Hazewinkel} (Amsterdam 2000; Zbl 0949.00006), section 3A, 321-356 (2000)]. Néron desingularization; Artin approximation; Bass-Quillen conjecture; Quillen cohomology Swan, R.G.: Néron-Popescu desingularization. In: Algebra and Geometry (Taipei, 1995). Lectures on Algebraic Geometry, vol. 2, pp. 135-192. International Press, Cambridge (1998) Étale and flat extensions; Henselization; Artin approximation, (Co)homology of commutative rings and algebras (e.g., Hochschild, André-Quillen, cyclic, dihedral, etc.), Local deformation theory, Artin approximation, etc. Néron-Popescu desingularization | 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 noetherian scheme defined over an algebraically closed field \(k\) of arbitrary characteristic and \(\pi:X\to S\) a smooth projective morphism of relative dimension \(2\). The object of this paper is to define a functor compactifying the stack \(\mathrm{Bun}(r,d)\) of vector bundles of rank \(r\) with \(c_2=d\) along the fibres of \(\pi\). The author calls this functor the Uhlenbeck moduli functor or functor of quasibundles; it should be noted that the term quasibundle has previously been used with two different meanings, but the author clarifies this and gives a precise definition so there is no risk of confusion.
When \(k=\mathbb{C}\) and \(S=\mathrm{Spec}(\mathbb{C})\), the Uhlenbeck compactness theorem applies giving a compact moduli space with a set theoretic decomposition \(M^U(r,d)=\coprod_{k\geq0}M(r,d-k)\times\mathrm{Sym}^kX\), where \(M(r,d)\) is the usual moduli space of semistable bundles. In this paper, however, the author works with an arbitrary algebraically closed field and defines a functor \(\mathrm{QBun}(r,d)\) which contains \(\mathrm{Bun}(r,d)\) as an open subfunctor and has a compactness property.
The functors \(\mathrm{Pic}(X)\to\mathrm{Pic}(S)\) corresponding to \(c_2\) and families of zero cycles (actually \(\mathrm{PIC}(X/S)\to\mathrm{PIC}(S)\), where the \(\mathrm{PIC}\) are larger categories taking base change into account) have a certain multiplicative property (they are compatible with tensor products of line bundles and also with base change). One then considers quadruples \((Z,E,N,D)\), where \(Z\) is a closed subset of \(X\) which is finite over \(S\), \(E\) is a vector bundle on \(X\setminus Z\), \(N\) is a multiplicative functor and \(D\) is a line bundle on \(X\) extending \(\det E\). There is a concept of \(E\)-localisation of \(N\) at \(Z\), which says essentially that \(N\) is identified with \(c_2^E\) on \(X\setminus Z\). This \(E\)-localisation is effective if a certain rational section of a line bundle is regular. A quasibundle is now defined to be a quadruple as above with an effective \(E\)-localisation and the corresponding functor is the Uhlenbeck functor.
There is an extensive introduction containg much background and motivational material as well as describing the main results of the paper. Multiplicative functors are introduced in Section 2. Section 3 is concerned with localisation and Section 4 with effectiveness of localisations. The Uhlenbeck functor \(\mathrm{QBun}(r,d)\) is constructed in Section 5 and its properties established; in particular, it is shown that the existence part of the valuative criterion for properness holds (of course, it is too much to expect uniqueness). A conjectural local covering by schemes is described as well as a morphism of functors from the Gieseker compactification to \(\mathrm{QBun}(r,d)\). Finally, in Section 6, a similar functor is constructed for torsors over split semisimple simply-connected groups. An indication is also given of how to extend the definition of quasibundle to higher dimensions. vector bundles on surfaces; moduli space; Uhlenbeck compactification Baranovsky, V, Uhlenbeck compactification as a functor, Int. Math. Res. Not., 2015, 12678-12712, (2015) Vector bundles on surfaces and higher-dimensional varieties, and their moduli Uhlenbeck compactification as a functor | 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 \({\mathcal A}_{g,n}\) denote the moduli space of principally polarized abelian varieties over \(\mathbb{C}\) of dimension \(g \geq 2\) with level \(n\) structure, \(n \geq 3\). Suppose \(Y\) is a curve in \({\mathcal A}_{g,n}\). An isogeny correspondence on \(Y\) is an irreducible curve \(Z \subset Y \times Y\) for which there exists a quasi-finite map \(Z' \to Z\) from an irreducible curve \(Z'\) with the property that two abelian schemes over \(Z'\) deduced by base change via \(Z' \to Z \subset Y \times Y @>p_ i>> Y\) \((i=1,2)\) are isogenous. The paper studies the question of how many isogeny correspondences can exist on a sufficiently general curve \(Y\). The main result is the statement that a `general' \(Y\) in \({\mathcal A}_{g,n}\) carries at most finitely may isogeny correspondences. moduli space; principally polarized abelian varieties; isogeny correspondence Isogeny, Algebraic moduli problems, moduli of vector bundles, Algebraic moduli of abelian varieties, classification A finiteness theorem for isogeny correspondences | 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 develops the theory of twisted Galois stratification in order to describe first-order definable sets in the language of difference rings over algebraic closures of finite fields equipped with powers of the Frobenius automorphism. After introducing basic concepts of the theory of difference schemes and their morphisms, as well as the notions of a (normal) Galois stratification \(\mathcal{A}\) on a difference scheme (\(X, \sigma\)) and the Galois formula associated with \(\mathcal{A}\), the author develops difference algebraic geometry (in particular, the theory of generalized difference schemes). The main result of the paper is a direct image theorem stating that the direct image of a Galois formula by a morphism of difference schemes is equivalent to a Galois formula over fields with powers of Frobenius. As a consequence of this theorem, the author obtains an effective quantifier elimination procedure and a precise algebraic-geometric description of definable sets over fields with Frobenii in terms of twisted Galois formulas associated with finite Galois covers of difference schemes. In addition, the paper presents a number of new results on the category of difference schemes, Babbitt's decomposition, and effective difference algebraic geometry. Model-theoretic algebra, Varieties over finite and local fields, Zeta functions and related questions in algebraic geometry (e.g., Birch-Swinnerton-Dyer conjecture), Finite ground fields in algebraic geometry, Difference algebra Twisted Galois stratification | 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{E. Bierstone} and \textit{P. D. Milman} [Invent. Math. 101, No. 2, 411--424 (1990; Zbl 0723.32005)] and \textit{A. Parusinski} [Trans. Am. Math. Soc. 344, No. 2, 583--595 (1994; Zbl 0819.32006)] proved the following important rectilinearization result for continuous subanalytic functions:
Let \(U\) be a real analytic manifold and \(f:U\to\mathbb{R}\) a continuous subanalytic function. Then there is a locally finite covering \((\pi_j:U_j\to U)_j\) such that
(i) each \(\pi_j\) is a composite of finitely many mappings each of which is either a local blowing-up with smooth center or a local power substitution;
(ii) each \(f\circ \pi_j\) is analytic and identically \(0\) or a normal crossing or the inverse of a normal crossing.
In the paper under review, the author proves a rectilinearization theorem for functions definable in an o-minimal structure generated by a convergent Weierstrass system. Convergent Weierstrass systems were introduced in [\textit{L. Van den Dries}, J. Symb. Log. 53, No. 3, 796--808 (1988; Zbl 0698.03023)]. They are induced by certain subrings of the ring of all restricted real analytic functions satisfying similar properties, notably being closed under Weierstrass division. Taking all restricted analytic functions, one obtains the o-minimal structure \(\mathbb{R}_{\mathrm{an}}\) and the above result for globally subanalytic functions in a global form. The proofs rely on the fact that functions definable in an o-minimal structure generated by a convergent Weierstrass system are piecewise given by terms in a reasonable language. rectilinearization; subanalytic functions; Weierstrass systems DOI: 10.4064/ap99-2-2 Semi-analytic sets, subanalytic sets, and generalizations, Modifications; resolution of singularities (complex-analytic aspects), Real-analytic and semi-analytic sets, Quantifier elimination, model completeness, and related topics Rectilinearization of functions definable by a Weierstrass system and its applications | 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 \(f\colon X \to Y\) be a morphism of germs of complex analytic spaces where \(X\) is reduced of pure dimension and \(Y\) is smooth of dimension \(n\).
The text gives four conditions each one sufficient for the following characterisation of flatness to hold: The morphism \(f\) is flat if and only if the \(n\)-th analytic tensor power \(\mathcal O_X \hat \otimes_{\mathcal O_Y} \cdots \hat\otimes_{\mathcal O_Y} \mathcal O_X\) is torsion-free as an \(\mathcal O_Y\)-module. The four conditions given are as follows:
(1) \(n < 3\).
(2) \(f\colon X \to Y\) is a Nash morphism of Nash germs.
(3) The singular locus of \(X\) is mapped into a proper analytic subgerm of \(Y\).
(4) The local ring \(\mathcal O_X\) is Cohen-Macauley.
By (3), the result can be seen as a generalisation to the analytic domain of Auslander's result [\textit{M.~Auslander}, Ill.~J.~Math.~5, 631--647 (1961; Zbl 0104.26202)] saying that a module \(M\) finite over an unramified regular local ring \(R\) of dimension \(n > 0\) is free (which is equivalent to torsion-freeness in the finite case) over \(R\) if and only if its \(n\)-th tensor power \(M^{\otimes_R^n}\) is torsion-free as an \(R\)-module. The theorem proven in this article can also be seen as a step towards the general conjecture [\textit{J.~Adamus}, J.~Pure Appl.~Algebra 193, No.~1--3, 1--9 (2004; Zbl 1054.32004)] saying that a local analytic algebra \(A\) over a local analytic algebra \(R\), which is regular of dimension \(n\), is \(R\)-flat if and only if its \(n\)-th analytic tensor power is torsion-free as an \(R\)-module.
Let \(X^n\) be the \(n\)-fold fibre product of \(X\) over \(Y\) and \(f^n\colon X^n \to Y\) the induced morphism. The hard implication of the theorem is to prove that torsion-freeness of the \(n\)-th analytic power follows from the flatness assumption under either conditions (1) to (4).
In order to handle cases (1) and (2), the author makes use of the concept of algebraic vertical components (a component of \(X\) is algebraic vertical if an arbitrarily small representative is mapped by \(f\) into a proper analytic subset) by using the following equivalence: The morphism \(f^n\) has no algebraic vertical components if and only if \(\mathcal O_{X^n}\) is a torsion-free \(\mathcal O_Y\)-module. In the first two cases, it is shown that restricting \(f^n\) to any geometric vertical component \(W\) (a component is geometric vertical if an arbitrarily small representative is mapped into a nowhere dense subset of a neighbourhood of the basepoint) of \(X^n\) yields a Gabrielov regular morphism, which in turn can be used to deduce that \(W\) is actually algebraic vertical. Thus \(\mathcal O_{X^n}\) is a torsion-free \(O_Y\)-module if and only if \(f^n\) has no geometric vertical component, and this is equivalent to \(f\) being flat by a result of Galligo-Kwieciński [\textit{A.~Galligo} and \textit{M.~Kwieciński}, J.~Algebra 232, No.~1, 48--63 (2000; Zbl 1016.14001)].
The case (3) is handled using techniques of Galligo-Kwieciński and the following equivalence, which holds whenever \(X\) is of pure dimension and \(Y\) reduced and irreducible of dimension \(n\): The morphism \(f\) is open if and only if the reduced \(n\)-th analytic tensor power of \(\mathcal O_X\) over \(\mathcal O_Y\) is torsion-free ans an \(\mathcal O_Y\)-module.
The final case (4) is proven by observing that flatness in this case is the same as openness [\textit{G.~Fischer}, Complex analytic geometry (Lecture Notes in Mathematics 538, Springer-Verlag, Berlin-Heidelberg-New York) (1976; Zbl 0343.32002)].
The article ends with a discussion on Gabrielov regularity of fibre products. The author gives a criterion when a fibre power of an analytic morphism is Gabrielov regular. flatness; torsion freeness; vertical component; algebraic vertical; geometric vertical; Gabrielov regularity; Nash morphism; Nash germ Analytic algebras and generalizations, preparation theorems, Analytical algebras and rings, Complex spaces, Nash functions and manifolds Flatness testing and torsion freeness of analytic tensor powers | 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 aim of this paper is to prove the following:
Theorem. Over an algebraically closed field of characteristic \(\neq 2\) for any \(n \geq 2\) there exists an integer \(a_n\) such that for \(a \geq a_n\) and \(b \geq \{(n^2 + 6n + 11) a/3(n + 3) + 2/(n + 3), a + 3\}\) the generic morphism from \(b{\mathcal O}_{\mathbb{P}^n} (2)\) into \(a{\mathcal O}_{\mathbb{P}^n}(3)\) is a locally free sheaf generated by its global sections.
This extends an analogous result over \(\mathbb{P}^3\) to bundles over \(\mathbb{P}^n\) [see the author, Math. Ann. 294, No, 1, 99-107 (1992; Zbl 0782.14017)]. The arguments are applications of basic ideas of the so- called voie ouest [see \textit{P. Ellia} and \textit{A. Hirschowitz}, J. Algebr. Geom. 1, No. 4, 531-547 (1992; Zbl 0812.14036)]. locally free sheaf generated by global sections Dolcetti, A., On the generation of certain bundles over \({ P}^n\), Ann. Univ. Ferrara Sez. VII (N.S.), 39, 77-92, (1993) Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20] On the generation of certain bundles over \(\mathbb{P}^ n\) | 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 generalizes one of his previous theorems: Suppose \(X=G/H\) is an algebraic variety (over a field), homogeneous under a connected algebraic group G. Suppose further that \(Y\subseteq X\) is a subvariety generating G, that is (if Y contains the origin) the connected component of 1 of the preimage of Y in G generates G. Under these assumptions any formal meromorphic function on the formal completion of X along Y is algebraic.
The proof uses rigid analytic methods, but in fact Grothendieck's algebraisation for formal schemes suffices. The key is the behaviour under formal completion of the K/k-span of the variety: If \(K\supseteq k\) is a field extension, X a variety defined over k, and \(Y\subseteq X_ K\) a K-subvariety, its K/k-span is defined to be the smallest k-subvariety of X containing Y. In a previous result [Ann. Math., II. Ser. 89, 391-403 (1969; Zbl 0184.465)] the author considered complex analytic meromorphic functions. homogeneous algebraic variety; formal meromorphic function; formal completion Chow, W.-L., ``Formal functions on homogeneous spaces,'' Inventiones Mathematicae , vol. 86 (1986), pp. 115-30. Homogeneous spaces and generalizations, Foundations of algebraic geometry Formal functions on homogeneous spaces | 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 \(X^{\bullet}=(X, D_X)\) be a pointed stable curve of topological type \((g_X, n_X)\) over an algebraically closed field of characteristic \(p>0\). Under certain assumptions, we prove that, if \(X^{\bullet}\) is \textit{component-generic}, then the first generalized Hasse-Witt invariant of \textit{every} prime-to-\(p\) cyclic admissible covering of \(X^{\bullet}\) attains maximum. This result generalizes a result of S. Nakajima concerning the ordinariness of prime-to-\(p\) cyclic étale coverings of smooth projective generic curves to the case of (possibly ramified) admissible coverings of (possibly singular) pointed stable curves. Moreover, we prove that, if \(X^{\bullet}\) is an \textit{arbitrary} pointed stable curve, then there \textit{exists} a prime-to-\(p\) cyclic admissible covering of \(X^{\bullet}\) whose first generalized Hasse-Witt invariant attains maximum. This result generalizes a result of M. Raynaud concerning the new-ordinariness of prime-to-\(p\) cyclic étale coverings of smooth projective curves to the case of (possibly ramified) admissible coverings of (possibly singular) pointed stable curves. As applications, we obtain \textit{an anabelian formula for} \((g_X, n_X)\), and prove that the field structures associated to inertia subgroups of marked points can be reconstructed group-theoretically from open continuous homomorphisms of admissible fundamental groups. Those results generalize A. Tamagawa's results concerning an anabelian formula for topological types and reconstructions of field structures associated to inertia subgroups of marked points of smooth pointed stable curves to the case of \textit{arbitrary} pointed stable curves. pointed stable curve; admissible covering; generalized Hasse-Witt invariant; Raynaud-Tamagawa theta divisor; admissible fundamental group; anabelian geometry; positive characteristic Coverings of curves, fundamental group, Positive characteristic ground fields in algebraic geometry, Universal profinite groups (relationship to moduli spaces, projective and moduli towers, Galois theory) Maximum generalized Hasse-Witt invariants and their applications to anabelian 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 This paper describes the relative Hilbert Chow morphism for a flat projective family \(\pi: X \to B\) of generically nonsingular curves which are at worst nodal over an arbitrary irreducible base \(B\). The relative Hilbert Chow morphism is the cycle map \(c_m: X^{[m]}_B \to X^{(m)}_B\), where \(X^{[m]}_B\) is the relative Hilbert scheme of \(m\) points and \(X^{(m)}_B\) is the relative symmetric product. The main theorem states that \(c_m\) is equivalent to the blowing up of the discriminant locus \(D^m \subset X^{(m)}_B\). The author works over the complex numbers and uses Serre's GAGA principle, constructing a local analytic model \(H\) for \(X^{[m]}_B\) and reverse engineering an ideal sheaf \(G\) in \(X^{(m)}_B\) to have syzygies so that the blow up at \(G\) maps to the pullback \(OH\) of \(H\) over the Cartesian product via a map \(\gamma\). A local analysis shows that \(\gamma\) is an isomorphism and that \(G\) defines the ordered diagonal, hence descends to the isomorphism claimed. This provides the details of a proof sketched in the author's earlier paper [in: Projective varieties with unexpected properties. A volume in memory of Giuseppe Veronese. Proceedings of the international conference ``Varieties with unexpected properties'', Siena, Italy, June 8--13, 2004. Berlin: Walter de Gruyter. 361--378 (2005; Zbl 1186.14027)].
In the second half of the paper the author uses the local model \(H\) to glean information about the singularity stratification of \(X^{[m]}_B\), specifically the structure of certain node polyscrolls he used earlier [Asian J. Math. 17, No. 2, 193--264 (2013; Zbl 1282.14097)] to develop an intersection calculus for the Hilbert scheme. This extends the intersection theory and enumerative geometry of a single smooth curve developed by \textit{I. G. Macdonald} [Topology 1, 319--343 (1962; Zbl 0121.38003)] to families with at worst nodal singularities, extending work of \textit{E. Cotterill} [Math. Z. 267, No. 3--4, 549--582 (2011; Zbl 1213.14064)]. nodal curves; relative Hilbert-Chow morphism; enumerative geometry; node scrolls Parametrization (Chow and Hilbert schemes), Families, moduli of curves (algebraic), Singularities of curves, local rings Structure of the cycle map for Hilbert schemes of families of nodal 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 \textit{J.-P. Serre} [C. R. Acad. Sci., Paris, Sér. I 296, 397-401 (1983; Zbl 0538.14015)] used explicit formulas involving certain trigonometric polynomials to achieve improvements of Weil's bound on the number of rational points on a curve of a given genus over a finite field. These formulas were discrete versions of explicit formulas used by \textit{A. Weil} [Meddel. Lunds Univ. Mat. Sem. Suppl.-band M. Riesz, 252-265 (1952; Zbl 0049.03205)] in the number field case. The authors use similar explicit formulas to obtain bounds on the number of rational points of a variety of dimension \(d\) with fixed Betti numbers defined over a finite field. They also obtain asymptotic bounds for the ratio of the number of rational points to the sum of the Betti numbers as this sum goes to infinity. In appendices, the authors give examples of the ``doubly positive'' functions they use in their explicit formulas, give the functions of Oesterlé used to obtain the optimal bounds via explicit formulas in the curve case (no such optimal formulas are known for dimension \(> 1\)), and give tables and graphs that illustrate the various bounds they obtain. Some of the results in this article, as applied to surfaces, were contained in an earlier paper by the second author [Arithmetic, geometry and coding theory (Luminy, 1993) (de Gruyter, Berlin), 209-224 (1996)]. explicit formulas; Weil bound; zeta function; doubly positive function Lachaud, G.; Tsfasman, M. A., Formules explicites pour le nombre de points des variétés sur un corps fini, J. Reine Angew. Math., 493, 1-60, (1997) Varieties over finite and local fields, Finite ground fields in algebraic geometry, Zeta functions and related questions in algebraic geometry (e.g., Birch-Swinnerton-Dyer conjecture) Explicit formulas for the number of points of varieties over a finite 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 This is an expository paper, in which the author describes Gieseker's construction of the (coarse) moduli scheme of stable curves of genus \(g \geq 3\). In the first section of the paper the author discusses the existence of a coarse moduli scheme of smooth curves. In section 2, groupoids and their morphisms are defined and the concept of stack is introduced, followed by the definition, characterization and properties of a Deligne-Mumford stack. The third section is devoted to the study of stable curves and the groupoid of stable curves, which is shown to be a Deligne-Mumford stack over Spec \(\mathbb Z\); the section ends with a discussion of the irreducibility of the moduli stack. In the last section the author defines the moduli space of a Deligne-Mumford stack and proves that a geometric quotient of a scheme by a group is the moduli space of the quotient stack, and after a discussion of the method of geometric invariant theory for constructing geometric quotients for actions of reductive groups, the author presents Gieseker's construction of the coarse moduli space. coarse moduli space; stable curves; Deligne-Mumford stack; geometric invariant theory; geometric quotient D. Edidin, Notes on the construction of the moduli space of curves, Recent progress in intersection theory (Bologna 1997), Trends Math., Birkhäuser, Boston (2000), 85-113. Families, moduli of curves (algebraic), Geometric invariant theory, Fine and coarse moduli spaces, Homogeneous spaces and generalizations Notes on the construction of the moduli space 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 In his paper [Manuscr. Math. 139, No. 1--2, 71--89 (2012; Zbl 1252.11046)], \textit{Luis Dieulefait} gave a proof of Serre's modularity conjecture for the case of odd level and arbitrary weight. By means of an intricate inductive procedure he reduced the issue to the case of Galois representations of level 3 and weight 2, 4 or 6. As explained in his paper, these cases are taken care of by the following three theorems, respectively.
Theorem 1.1. There are no non-zero semi-stable abelian varieties over \(\mathbb Q\) with good reduction outside 3.
Theorem 1.2. There are no non-zero semi-stable abelian varieties over \(\mathbb Q(\sqrt 5)\) with good reduction outside 3.
Theorem 1.3. Every semi-stable abelian variety over \(\mathbb Q\) with good reduction outside 15 is isogenous, over \(\mathbb Q\), to a power of the Jacobian of the modular curve \(X_0(15)\).
Theorem 1.1 is due to \textit{A. Brumer} and \textit{K. Kramer} [Manuscr. Math. 106, No. 3, 291--304 (2001; Zbl 1073.14544)]. In this paper the author proves Theorems
1.2 and 1.3, each of which directly imply Theorem 1.1. First, the author discusses extensions of \(\mu_p\) and \(\mathbb Z/p\mathbb Z\) by one another that play an important role in this paper. powers of Jacobian of modular curve \(X_0(15)\) René Schoof, Semistable abelian varieties with good reduction outside 15, Manuscripta Math. 139 (2012), no. 1-2, 49 -- 70. Abelian varieties of dimension \(> 1\), Arithmetic ground fields for abelian varieties Semistable abelian varieties with good reduction outside 15 | 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 Author's abstract: Let \(X\) be an algebraic curve. We study the problem of parametrizing geometric structures over \(X\) which are only generically defined. For example, parametrizing generically defined maps (rational maps) from \(X\) to a fixed target scheme \(Y\). There are three methods for constructing functors of points for such moduli problems (all originally due to Drinfeld), and we show that the resulting functors are equivalent in the fppf Grothendieck topology. As an application, we obtain three presentations for the category of \(D\)-modules `on' \(B(K)\backslash G(\mathbb{A})/G(\mathbb{O})\), and we combine results about this category coming from the different presentations. \(D\)-modules; rational map; contractibility; geometric Langlands Geometric Langlands program (algebro-geometric aspects) \(D\)-modules on spaces of rational maps | 0 |
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