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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 Since its introduction in 1963 by Alexandre Grothendieck, the theorem of cohomology and base change has played an important role in algebraic geometry; the construction of Hilbert and Quot schemes is an important example. The main result of this paper, Theorem A, is a version of cohomology and base change for a morphism \(f: X \to S\) of locally Noetherian algebraic stacks that is locally of finite type, stated in terms of relative Ext sheaves. It states that, for properly supported objects \(\mathcal{M} \in D^{-}_{\mathrm{Coh}}(X)\) and \(\mathcal{N} \in\mathrm{Coh}(X)\) that are flat over \(S\), for each integer \(q\) and morphism of algebraic stacks \(\tau: T \to S\), there is a natural base change morphism:
\[
b^q(\tau) : \tau^{*} \mathcal{E}xt^q (f; \mathcal{M}, \mathcal{N}) \to \mathcal{E}xt^q(f_T: L(\tau_X)_{qc}^{*} \mathcal{M}, \tau_X^{*} \mathcal{N})
\]
that satisfies the expected properties for cohomology and base change. The key to the proof is Theorem C, which states that for an affine scheme \(S\) and a morphism of algebraic stacks \(X \to S\) that is locally of finite presentation, a certain functor \(\mathrm{QCoh}(S) \to\mathrm{Ab}\) is \textit{coherent}, i.e., corepresentable by a morphism of quasi-coherent \(\mathcal{O}_S\)-modules. algebraic stacks; cohomology; derived categories; Hom space Hall, J., \textit{cohomology and base change for algebraic stacks}, Math. Z., 278, 401-429, (2014) Generalizations (algebraic spaces, stacks), Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20], Derived categories and commutative rings, Algebraic moduli problems, moduli of vector bundles, Stacks and moduli problems Cohomology and base change for algebraic stacks | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians The author concludes his introduction to this version of four lectures he gave at the Arizona Winter School 2000 with the words \textit{``caveat emptor''}; but no such disclaimer is necessary, for it offers a clear and valuable introduction to the deep and beautiful ideas to be found in Deligne's work on the Weil conjectures, concerning the number of solutions of equations over finite fields and the Riemann hypothesis for projective varieties over finite fields.
The lectures cover the ideas in Deligne's papers [\textit{P. Deligne}, La conjecture de Weil I, Publ. Math., Inst. Hautes Étud. Sci. 43, 273--307 (1973; Zbl 0287.14001) and La conjecture de Weil II, Publ. Math., Inst. Hautes Étud. Sci. 52, 137--252 (1980; Zbl 0456.14014)] which the author refers to as Weil I and Weil II, together with other related work, such as Laumon's simplification, based on Fourier transform methods, of Deligne's proof of Weil II. One particularly appealing insight is afforded by the author's comment on Deligne's `stunning transposition to the function field case' of the method of `squaring' due to Rankin and to be found in his paper on Ramanujan's \(\tau(n)\) function [\textit{R. A. Rankin}, Proc. Camb. Philos. Soc. 35, 357--372 (1939; Zbl 0021.39202)]. -- The underlying idea is to put the \(L\)-functions, whose Weil II estimate is required, into a family of \(L\)-functions having `big monodromy' and then to ap ply the Rankin squaring method to that family. One then obtains a proof of the Weil II result in the style of Weil I.
In the first lecture, the author reviews the necessary background from the theory of \(l\)-adic sheaves and \(l\)-adic cohomology and which includes the definition of the \(L\)-function of a constructible \(\overline\mathbb{Q}_l\)-sheaf on a scheme \(X/k\) of finite type (those and later definitions and concepts are clearly explained). The lecture concludes with a formulation of and preliminaries concerning the proof of the target theorem, which is fundamental in what follows and which concerns the structure of \(H^1_c(U\otimes_k\overline k,{\mathfrak F})\), where \(U/k\) is a smooth curve over a finite field, \(l\) is a prime invertible in \(k\) and \({\mathfrak F}\) is a `lisse' \(\overline\mathbb{Q}_l\)-sheaf on \(U\) of weight \(w\). One proves that \(H^1_c(U\times_k\overline k,{\mathfrak F})\) is of weight \(\leq w+1\).
Lecture II proves the target theorem in a special case.
Lecture III continues the theme of the proof of the target theorem and relates it to the proof of the monodromy theorem and the application of Rankin's method. The proof of the monodromy theorem and the appeal to Rankin's method are then completed in Lecture IV, as is the proof of the target theorem and its applications.
The lecture concludes with proofs of Weil I and Weil II and with a nice deduction of the Ramanujan conjecture, derived from Sato's idea that `Weil implies Ramanujan'.
The reviewer found the exposition helpful and stimulating and attractive -- so that one gains understanding and is led to a desire to learn more. The author has succeeded admirably in fulfilling his objectives. Weil conjectures; \(L\)-functions Katz, N. M., \textit{L-functions and monodromy: four lectures on Weil II}, Adv. Math., 160, 81-132, (2001) Zeta functions and related questions in algebraic geometry (e.g., Birch-Swinnerton-Dyer conjecture), Finite ground fields in algebraic geometry, Varieties over finite and local fields, Zeta and \(L\)-functions in characteristic \(p\) \(L\)-functions and monodromy: four lectures on Weil II | 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 \(\bar {\mathcal M}_ g\) be the algebraic stack over Spec(\({\mathbb{Z}})\) of stable curves of genus \(g,\) \({\mathcal M}_ g\) the open part corresponding to smooth curves, \(\Delta\) the normally crossing divisor supported by \(\bar {\mathcal M}_ g\setminus {\mathcal M}_ g\), p: \({\mathcal C}\to {\mathcal M}_ g\) the universal curve and \(K=\omega_{{\mathcal C}/\bar {\mathcal M}_ g}\) the relative canonical sheaf. Using Deligne's pairing \(<, >\) the author presents Mumford's isomorphism [\textit{D. Mumford}, Enseign. Math., II. Sér. 23, 39-100 (1977; Zbl 0363.14003)], uniquely up to sign, in the form \((\det (p_*K))^{\otimes 12}\overset \sim \rightarrow <K,K>\otimes {\mathcal O}_{\bar {\mathcal M}_ g}(\Delta)\) (theorem 2.1) and proves that the natural hermitian norms on both invertible sheaves are connected by multiplication with \((2\pi)^{-4g}\) \(e^{\delta}\) along this isomorphism (theorem 2.2), where \(\delta\) is the real function on \({\mathcal M}_ g({\mathbb{C}})\) defined by \textit{G. Faltings} in Ann. Math., II. Ser. 119, 387-424 (1984; Zbl 0559.14005). - The proof applies the functorial theory of relative curves (X/S,\(\ell)\) with theta characteristics \(\ell\) \((2\ell=\) class of \(\Omega^ 1_{X/S})\), in analogy with the paper by \textit{A. A. Bejlinson} and \textit{Yu. I. Manin} [Commun. Math. Phys. 107, 359-376 (1986; Zbl 0604.14016)].
Clearly, the basic theorem implies the Mumford isomorphism and Faltings' Noether formula for arithmetic surfaces X/B:
\[
12\cdot \deg (\det (Rp_*\omega))=(\omega \cdot \omega)+\sum_{b}\delta_ b(X)\cdot \log (N(b)) +\sum_{\sigma}\delta_{\sigma}(X) -4g[L:{\mathbb{Q}}]\log (2\pi),
\]
where \(B=Spec(R)\), R the ring of integers of a number field L, \(\omega =\omega_{X/B}\), N(b) the absolute norm of \(b\in Spec(R)\), ( \(\cdot)\) is Arakelov's intersection pairing, and the \(\sigma\) denote field embeddings of L into \({\mathbb{C}}\). algebraic stack of stable curves; Mumford isomorphism; Faltings' Noether formula; arithmetic surfaces; Arakelov's intersection pairing L. Moret-Bailly , La formule de Noether pour les surfaces arithmétiques , Invent. Math. 98 (1989), 491-498. Arithmetic varieties and schemes; Arakelov theory; heights, Families, moduli of curves (algebraic), Curves of arbitrary genus or genus \(\ne 1\) over global fields La formule de Noether pour les surfaces arithmétiques. (The Noether formula for 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 The goal of this paper is to take up the approaches used to deal with Jacobians and Kummer surfaces of curves of genus 2 by \textit{J. W. S. Cassels} and \textit{E. V. Flynn} [Prolegomena to a middlebrow arithmetic of curves of genus 2. Cambridge: Cambridge Univ. Press (1996; Zbl 0857.14018)] and by the author [Acta Arith. 90, No. 2, 183--201 (1999; Zbl 0932.11043); ibid. 98, No. 3, 245--277 (2001; Zbl 0972.11058)] and extend them to hyperelliptic curves of genus 3.
The author developed an explicit theory of Kummer varieties associated to Jacobians of hyperelliptic curves of genus 3, over any field \(k\) of characteristic \(\neq 2\). In particular, he provided explicit equations defining the Kummer variety \(\text{K}\) as a subvariety of \(\mathbb{P}^{7}\), together with explicit polynomials giving the duplication map on \(\text{K}\). A careful study of the degenerations of this map then forms the basic for the development of an heights on such Jacobians when \(k\) is a number field. The author used this input to obtain a good bound on the difference between naive and canonical height, which is a necessary ingredient for the explicit determination of the Mordell-Weil group. He Illustrated these results with two examples. Kummer variety; hyperelliptic curve; genus 3; canonical height M. Stoll, An explicit theory of heights for hyperelliptic jacobians of genus three, Preprint. Jacobians, Prym varieties, Abelian varieties of dimension \(> 1\), Heights An explicit theory of heights for hyperelliptic Jacobians of genus three | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians For an abelian variety over a complete local field \(K\), let \(A'\) denote its dual and \(\Gamma(A)\) its ``fundamental group''. Grothendieck defined a pairing between \(\Gamma(A)\) and \(\Gamma(A')\) with values in \(\mathbb{Z}\), if \(A\) has semi-stable reduction, which he called the monodromy pairing. In fact, he proved a \(p\)-adic analogue of the Picard-Lefschetz formula when \(A\) is a Jacobian of a curve with semi-stable reduction.
\textit{M. Raynaud} [Actes Congr. internal. Math. 1970, 1, 473--477 (1971; Zbl 0223.14021)] also wrote down a definition of a pairing between \(\Gamma(A)\) and \(\Gamma(A')\) when \(A\) has semi-stable reduction using bi-extensions and Grothendieck asserted that Raynaud's pairing is the same as his in Chapter IX, \S14.2.5 of SGA7 but gave no details of a proof. We provide a proof in \S2 based on Werner's analysis of Raynaud's pairing.
The remainder of the paper is a rigid analytic proof of the \(p\)-adic Picard-Lefschetz formula based on the aforementioned expression of Raynaud's formula for the monodromy pairing.
The main new technical result is an explicit relationship, proven in \S4, between the rigid residue maps from the regular differentials on a curve with semi-stable reduction over \(K\) to \(K\), defined in [\textit{R. F. Coleman}, Compos. Math. 72, No. 2, 205--235 (1989; Zbl 0706.14013)], and rigid homomorphisms from \(\mathbb{G}_m\) into the Jacobian of the curve.
We first prove, in \S5, the Picard-Lefschetz formula in the case of the Jacobians of Mumford curves. Although this is not essential to our ultimate general proof, it provides motivation and in fact we are able to prove more in this case. Coleman, R.: The monodromy pairing. Asian J. Math. 4, 315--330 (2000) Arithmetic ground fields for abelian varieties, Abelian varieties of dimension \(> 1\), Varieties over finite and local fields, Rigid analytic geometry The monodromy pairing. | 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 two powerful tools, both based on deformation theory, to study the smooth projective pointed curves with a prescribed Weierstrass gap sequence \(\ell_ 1,\ell_ 2,\dots,\ell_ g\). On the one hand, \textit{H. C. Pinkham} [``Deformations of algebraic varieties with \(G_ m\)- action'', Astérisque 20 (1974; Zbl 0304.14006)] constructed the moduli space by deforming irreducible singular affine curves, and on the other hand \textit{D. Eisenbud} and \textit{J. Harris} [Invent. Math. 87, 495-515 (1987; Zbl 0606.14014)], by deforming reducible curves, obtained among many other results the existence of smooth curves whenever the weight \(\sum(\ell_ i-i)\) of the prescribed gap sequence is at most \({g\over 2}\).
In his book ``Curves and their Jacobians'' (1975; Zbl 0316.14010), p. 20, \textit{D. Mumford} proposed to obtain the moduli of the smooth projective curves of given genus \(g\) from the coefficients arising in Petri's analysis of the canonical ideal. In this paper we adapt Mumford's idea to pointed curves with a prescribed gap sequence. We assume that the Weierstrass semigroup is symmetric that is \(\ell_ g=2g-1\). Since the weight is larger than \(g-2\), we are out of the range of the theorem of Eisenbud and Harris. While Pinkham uses infinitesimal deformation theory of Lichtenbaum and Schlessinger, we deform curves canonically embedded in the \((g-1)\)-dimensional projective space, and by determining Gröbner bases and analysing syzygies of their ideals, we get a rather explicit construction of the moduli space of the irreducible projective pointed Gorenstein curves of arithmetical genus \(g\) with gap sequence \(\ell_ 1,\ell_ 2,\dots,\ell_ g\). We show that this space is projective and contains Pinkham's moduli space as an open subspace. In particular the boundary of Pinkham's moduli space is built up by isomorphism classes of pointed projective singular irreducible Gorenstein curves. deformation theory; pointed curves with a prescribed Weierstrass gap sequence; moduli of smooth projective curves; Gorenstein curves Stöhr, K-O, On the moduli spaces of Gorenstein curves with symmetric Weierstrass semigroups, J. Reine Angew. Math., 441, 189-213, (1993) Riemann surfaces; Weierstrass points; gap sequences, Families, moduli of curves (algebraic), Varieties defined by ring conditions (factorial, Cohen-Macaulay, seminormal) On the moduli spaces of Gorenstein curves with symmetric Weierstrass semigroups | 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 consider, in the special case of certain one-parameter families of Jacobians of curves defined over a number field, the problem of how the property that the generic fiber of such a family is absolutely simple 'spreads' to other fibers. We show that this question can be approached using arithmetic geometry or with more analytic methods based on sieve theory. In the first setting, non-trivial group-theoretic information is needed, while the version of the sieve we use is also of independent interest.
There are arithmetic properties for which the classical methods known in the context of the Hilbert Irreducibility Theorem do not seem to be directly applicable. One example is the following question: let \(\mathcal A\to S\) be a family of abelian varieties defined over a global field \(k\), and assume that the generic fibre is simple, or geometrically simple. What can be said about the set of rational points \(s\in S(k)\) for which \(A_s\) remains (geometrically) simple over \(k\)? Note that for each prime number \(\ell\), one can pose in this setting the Galois-theoretic question of understanding how the Galois groups of the \(\ell\)-torsion fields of the fibers vary, which are instances of Hilbert-irreducibility type problems. In fact, it will turn out that understanding this, as \(\ell\) varies, plays an important role in the work that follows.
We develop a variety of techniques to approach this particular problem, especially when \(S = \mathbb P^1_k\), and we expect that they would be suitable for many others with a similar flavor. In fact, the parallel with the known approaches to Hilbert's theorem will be obvious: one set of techniques is built on arithmetic geometry, while the other involves sieve methods, while both require some group-theoretic information. This familiar appearance should not be taken too far, however, as the tools involved are quite subtle. In particular, we appeal to difficult results of group theory which had not yet -- to our knowledge -- been applied to arithmetic problems (for some, the only published proof depends on the classification of finite simple groups). On the sieve side, the method is also quite original, and involves proving a new generalization of Gallagher's larger sieve inequality over number fields which is likely to be of independent interest. (Because of the interest of this sieve statement for analytic number theorists, independently of the problem in arithmetic geometry which is involved, we have summarized in an Appendix enough information
to understand the latter.)
Since the goal of this paper is partly to emphasize the general methods, rather than to prove a specific particular case, and since the tools borrow quite freely from arithmetic algebraic geometry, group theory, and analytic number theory, which may not be equally familiar to the interested readers, we have chosen a fairly expository style of writing. For instance, we discuss informally the characteristic strengths and weaknesses of the two basic
approaches, and for the sake of clarity, we do not always pursue the strongest possible
conclusions.
To give a concrete form to our results, here are prototypical consequences of the more general theorems proved in the main body of the paper. They concern a particular type of families of abelian varieties, namely the family \(A_f \to \mathbb A^1\) of Jacobians of the hyperelliptic curves defined by affine equations
\[
y^2 = f(x)(x-t),\quad t\in \mathbb A^1;
\]
for some fixed squarefree polynomial \(f\in\mathbb Z[X]\) of degree \(2g\), \(g > 1\). It is true (though not obvious) that the generic fiber of this family is geometrically simple, and hence we can ask the question discussed above. We phrase it in a quantitative manner. First, for \(t\in\mathbb Q\) written \(t = a/b\) with coprime integers \(a\) and \(b\neq 0\), let \(H(t) = \max(| a|, | b|)\) be the height of \(t\). Let then \(S(B)\) denote the set
\(S(B) =\{t\in\mathbb Q\mid H(t)\leq B\); and the fiber \(A_{f,t}\) is not geometrically simple.\(\}\) We show that \(S(B)\) is ``small'' in some sense.
Theorem A (Arithmetic geometry method). There exists a constant \(C(f)\), depending on \(f\), such that \(| S(B)|\leq C(f)\) for all \(B > 1\). In other words, there are only finitely many \(t\in\mathbb Q\) for which \(A_{f,t}\) is not geometrically simple.
This is a special case of Theorem 8 in Section 1.
Theorem B (Analytic number theory method). There exist absolute constants \(C > 0\) and \(D > 1\), independent of \(f\), such that we have
\(| S(B)|\leq C(g^2D(\log 2B))^{11g^2}\) for all \(B > 1\).
This is a special case of Theorem 24 in Section 3, where we have simplified the bound by worsening it somewhat. one-parameter families of Jacobians of curves; generic fiber; Hilbert Irreducibility Theorem; sieve methods Ellenberg, J.; Elsholtz, C.; Hall, C.; Kowalski, E., \textit{non-simple abelian varieties in a family: algebraic and analytic approaches}, J. Lond. Math. Soc. (2), 80, 135-154, (2009) Abelian varieties of dimension \(> 1\), Applications of sieve methods, Arithmetic ground fields for abelian varieties Non-simple abelian varieties in a family: geometric and analytic approaches | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians The main result of this book is (roughly): a curve is stable if and only if it can be stably embedded in a canonical way (i.e. it is stable in the sense of geometric invariant theory à la Mumford). The notion of a stable curve was introduced by Mumford et al. as a reduced curve with only double points and without continuous automorphisms (in the case of genus \(\geq 2)\) (if the latter condition is omitted, we obtain the notion of a semistable curve) to construct a canonical compactification of the moduli space of smooth curves.
More precisely there are the following two theorems: (I) (Theorem 1.0.1). Let C be a connected (but possibly reducible) curve in \({\mathbb{P}}^ n\), and consider its m-th Hilbert point (the point in the Hilbert scheme corresponding to the embedding obtained by composing \(C\subset {\mathbb{P}}^ n\) with the Veronese map \({\mathbb{P}}^ n\subset {\mathbb{P}}^ N, N=\left( \begin{matrix} n+m\\ n\end{matrix} \right))\). Then if the m-th Hilbert point is semistable, then C is semi-stable. - (II) (Theorem 2.0.2). By taking the quotient, we obtain the moduli space of stable curves which turns out to be irreducible and projective. As a corollary of the second result we have: the n-canonical embedding (embedding by n-th tensor of the dualizing (canonical) sheaf) of a stable curve is stable if \(n\geq 10.\)
The second result was first obtained by Knudsen by a different method, and by Mumford by a similar method but with the Chow scheme [\textit{D. Mumford}, Enseign. Math., II. Sér. 23, 39-110 (1977; Zbl 0363.14003)]. Hilbert scheme; stable point; moduli space of stable curves Gieseker, D.: Geometric invariant theory and applications to moduli problems. In: Invariant Theory. Proceedings of the 1st 1982 Session of the Centro Internazionale Matematico Estivo (CIME), Montecatini, June 10--18, 1982. Lecture Notes in Mathematics, vol. 996, pages v+159. Springer, Berlin (1983) Families, moduli of curves (algebraic), Research exposition (monographs, survey articles) pertaining to algebraic geometry, Parametrization (Chow and Hilbert schemes), Algebraic moduli problems, moduli of vector bundles, Fine and coarse moduli spaces Lectures on moduli of curves. Notes by D. R. Gokhale | 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 establishes a correspondence between a smooth moduli space of vector bundles over a curve and a self-product of the Jacobian, and proves the nontriviality of the Griffiths group of the moduli space for a general curve. Let \(X\) be a smooth algebraic curve over \(\mathbb{C}\), and \(r\) and \(d\) be two coprime positive integers. Let \(\mathcal N\) be the moduli space of isomorphism classes of stable rank-\(r\) vector bundles over \(X\) with a fixed determinant of degree \(d\). Then, it is well-known that \(\mathcal N\) is smooth and there exists a universal bundle \(\mathcal E\) over \(X \times \mathcal N\). The characteristic classes \(a_k \in H^{2k}(X \times \mathcal N, \mathbb{Q})\) (\(2 \leq k \leq r\)) of \(\mathbb{P}(\mathcal E)\) give rise to homomorphisms from \(H_1(X, \mathbb{Q})\) to \(H^{2k-1}(\mathcal N, \mathbb{Q})\) by the slant product operation. These homomorphisms induce an algebra homomorphism from \(H_*(J^{r-1}, \mathbb{Q})\) to \(H^*(\mathcal N, \mathbb{Q})\), where \(J^{r-1}\) is the \((r-1)\)-fold product of the Jacobian \(J\).
The main result of the paper says that the above algebra homomorphism is induced by an algebraic cycle on the product \(\mathcal N \times J^{r-1}\). Moreover, this algebraic cycle is canonical as an element of the Chow ring (cycles modulo rational equivalence) of \(\mathcal N \times J^{r-1}\).
As an application of the main result, the author constructs non-zero elements in the Griffiths group of the moduli space \(\mathcal N\) for a general curve \(X\). These elements are obtained from the non-zero elements of the Jacobian \(J\) of a general curve discovered by G. Ceresa. moduli space; vector bundles; algebraic cycles; Griffith group; product of the Jacobian; Chow ring Families, moduli of curves (algebraic), Jacobians, Prym varieties, (Equivariant) Chow groups and rings; motives, Vector bundles on curves and their moduli, Determinantal varieties, Algebraic moduli problems, moduli of vector bundles, Algebraic cycles A correspondence between the moduli spaces of vector bundles over a curve | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians In this paper it is shown that a theorem of Andreotti and Mayer, which is related to Schottky problem in the classical case, goes through even in positive characteristics different from 2. Let \(A_ g\) be the coarse moduli scheme for principally polarized abelian varieties of dimension g defined over an algebraically closed field of characteristics \(p\geq 0\), \(p\neq 2\), and let \(\bar J_ g\) be the closure of the Jacobian locus \(J_ g\) in \(A_ g\). Let \(N_{g-4}\) be the locus of principally polarized abelian varieties whose theta divisors have singular loci of dimension \(\geq g-4\). Then \(\bar J_ g\) is an irreducible component of \(N_{g-4}\). The main idea of the proof comes from the fact: Let (X,D) be a pair of a smooth algebraic variety X and an effective divisor D on X. Any symmetric global section of \(\otimes^{2} T_ X\) defines canonically a linear infinitesimal deformation of (X,D), where \(T_ X\) is the tangent sheaf on X. This fact leads to an explicit computation of the first order infinitesimal deformations of a principally polarized abelian variety (X,L) along the directions of the local moduli space of (X,L), and replace the role played by the heat equations in the classical case, which was essentially used by Andreotti and Mayer in their papers. Schottky problem; coarse moduli scheme; Jacobian; principally polarized abelian varieties; theta divisors; effective divisor; heat equations G. Welters,Polarized abelian varieties and the Heat Equation, Compositio Math.49 (1983), 173--194. Theta functions and abelian varieties, Picard schemes, higher Jacobians, Algebraic moduli of abelian varieties, classification, Heat equation, Algebraic moduli problems, moduli of vector bundles Polarized abelian varieties and the heat equations | 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 in complex algebraic geometry makes a contribution to the literature on the Kobayashi conjecture that states that a generic complex hypersurface \(X\) of degree at least \(2n-1\) in the projective space \({\mathbb P}^n\) is hyperbolic in the sense that any entire holomorphic function \(\varphi:{\mathbb C}\to X\) reduces to a constant. The author builds upon the work of Siu and Demailly and their coauthors to prove the following theorem.
Theorem~1. Let \(X\subset{\mathbb P}^3\) be a generic hypersurface of degree at least \(18\). Then there is a divisor \(Y\subset{\mathbb P}(T_X)\) in the projectivization of the tangent bundle of \(X\) such that the projectivized derivative of any holomorphic curve \(\varphi:{\mathbb C}\to X\) lies in \(Y\). In particular, \(X\) is hyperbolic.
The main references for the proof are [Am. J. Math. 122, No.~3, 515--546 (2000; Zbl 0966.32014)] by \textit{J.-P. Demailly} and \textit{J. El Goul}, and [Hyperbolicity in complex geometry. The legacy of Niels Henrik Abel, 543--566 (2004; Zbl 1076.32011)] by \textit{Y. Siu}. In \S\,1 the author constructs some explicit meromorphic vector fields on the manifold of vertical jets of order two of the incidence manifold of degree-\(d\) forms on \({\mathbb P}^3\). Then he applies those vector fields to derive a contradiction from the assumption that an entire holomorphic curve has Zariski dense first derivative \(\varphi_{[1]}:{\mathbb C}\to X_{[1]}={\mathbb P}(T_X)\).
The author announces his hope to use further refinements of his methods presented in the article to prove in a sequel the above mentioned conjecture of Kobayashi for \(n=3\). Kobayashi conjecture for surfaces in projective three-space; Kobayashi hyperbolic surfaces; meromorphic vector fields on projective manifolds Păun, Mihai, Vector fields on the total space of hypersurfaces in the projective space and hyperbolicity, Math. Ann., 340, 4, 875-892, (2008) Hyperbolic and Kobayashi hyperbolic manifolds, Hypersurfaces and algebraic geometry Vector fields on the total space of hypersurfaces in the projective space and hyperbolicity | 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 integrals over Hilbert schemes of points involving tautological bundles and certain ``geometric'' subsets of these Hilbert schemes. For a smooth projective complex variety \(X\) of dimension \(d\), the Hilbert scheme \(X^{[n]}\) parametrizes length-\(n\) \(0\)-dimensional closed subschemes of \(X\). A vector bundle on \(X\) induces a tautological bundle \(E^{[n]}\) on \(X^{[n]}\). Roughly speaking, a geometric subset of \(X^{[n]}\) is a constructible subset \(P\) such that if \(Z, Z' \in X^{[n]}\) satisfying \(Z \cong Z'\) as \(\mathbb C\)-schemes, then either \(Z, Z' \in P\) or \(Z, Z' \not \in P\).
The main theorem of the paper states that the integral over \(X^{[n]}\) involving the Chern classes of \(E^{[n]}\) and the fundamental class (respectively, the Chern-Mather class, the Chern-Schwartz-MacPherson class) of a geometric subset \(P\) can be written as a universal polynomial, depending on the type of \(P\), in the Chern numbers involving \(E\) and the tangent bundle \(T_X\) of \(X\). When \(X^{[n]}\) is smooth, the integral is also allowed to contain the Chern classes of \(T_{X^{[n]}}\).
The main idea in proving this theorem is to use Jun Li's concept of Hilbert scheme \(X^{[[\alpha]]}\) of \(\alpha\)-points [\textit{J. Li}, Geom. Topol. 10, 2117--2171 (2006; Zbl 1140.14012)]. The main theorem generalizes many known results when \(X\) is a surface.
As an application, the author obtains a generalized Göttsche's conjecture for all isolated singularity types and in all dimensions. More precisely, if \(L\) is a sufficiently ample line bundle on a smooth projective variety \(X\), then in a general subsystem \(\mathbb P^m \subset |L|\) of appropriate dimension \(m\), the number of hypersurfaces with given isolated singularity types is a polynomial in the Chern numbers involving \(T_X\) and \(L\). Another application is to obtain similar results, when \(X\) is a surface, for the locus of curves with fixed ``BPS spectrum'' in the sense of stable pairs theory.
Section~2 is devoted to the preliminaries such as the definition of the tautological bundle \(E^{[n]}\) on the Hilbert scheme \(X^{[n]}\), the construction of the Chern-Mather and Chern-Schwartz-MacPherson classes, the Hilbert scheme \(X^{[[\alpha]]}\) of \(\alpha\)-points, and the definition of geometric subsets in \(X^{[n]}\) and \(X^{[[\alpha]]}\). Section~3 contains an outline of the proof of the main theorem, while the formal proof of the main theorem is presented in Section~4. In Section~5, the author verifies a technical lemma which is used in Section~4. Section~6 deals with the generating series of the above-mentioned integrals over all the Hilbert schemes \(X^{[n]}\), \(n \geq 0\). In Section~7, The precise definition of sufficiently ample is given, and the main theorem is applied to the problem of counting geometric objects with prescribed singularities. Hilbert schemes; tautological bundles; Göttsche's conjecture; counting singular divisors; BPS spectrum J. V. Rennemo, Universal polynomials for tautological integrals on Hilbert schemes , preprint, [math.AG]. arXiv:1205.1851v1 Parametrization (Chow and Hilbert schemes), Enumerative problems (combinatorial problems) in algebraic geometry, Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects) Universal polynomials for tautological integrals on Hilbert 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 aim of this paper is to construct a link ranging from a class of schemes on curves over some base schemes via infinite Grassmannians to commutative algebras of differential operators and evolution equations.''
In his papers ``Cohomological structure in soliton equations and Jacobian varieties'', J. Differ. Geom. 19, 403-430 (1984; Zbl 0559.35076) and ``Category of vector bundles on algebraic curves and infinite dimensional Grassmannians'', Int. J. Math. 1, No. 3, 293-342 (1990; Zbl 0723.14010), \textit{M. Mulase} related vector bundles over curves which are defined over a field to elements of infinite Grassmannians and used this correspondence to give a complete classification of elliptic commutative algebras of ordinary differential operators.
The aim of the present author is twofold: First, she generalizes the correspondence between vector fields over curves and infinite Grassmannians to the case where the curves are defined over a locally noetherian base scheme. Thus, in section 1 she defines the notation of relative infinite Grassmannian, in section 2 she studies sheaves over families of curves defined over a locally noetherian scheme, and in section 3 she relates this objects via the Krichever functors which is a bijective contravariant functor from the category of Schur pairs -- a Schur pair is a pair of elements of infinite Grassmannians -- to the category of geometric data -- a category modeled on schemes over family of curves. In section 4 the author gives some examples of geometric data, in particular, a family of elliptic curves. The second aim of the author was the relation of the category of geometric data to families of commutative algebras of differential operators. This is done in section \(5\). Here she gives a classification of commutative algebras of ordinary differential operators with coefficients in the ring \( R[[x]] \) of formal power series over a commutative noetherian \(k\)-algebra \(R\), where \(k\) is a field of characteristic zero. algebra of differential operators; Krichever operator; family of curves; relative infinite Grassmannian I. Quandt, On a relative version of the Krichever correspondence,Bayreuth. Math. Schr. 52 (1997), 1--74. Differentials and other special sheaves; D-modules; Bernstein-Sato ideals and polynomials, General theory of ordinary differential operators, Commutative rings of differential operators and their modules On a relative version of the Krichever correspondence | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians Let \(C/\mathbb{Q}\) be a hyperelliptic curve given by a model \(y^2 = f(x)\), with \(f(x) \in \mathbb{Q}[x]\), and let \(J/\mathbb{Q}\) be its Jacobian. The Mordell-Weil theorem states that \(J(\mathbb{Q})\) is a finitely generated abelian group and hence \(J(\mathbb{Q})\) decomposes as a direct sum \(J(\mathbb{Q})_{\text{tors}} \oplus \mathbb{Z}^{R_{J(\mathbb{Q})}}\), where \(J(\mathbb{Q})_{\text{tors}}\) is the subgroup of torsion elements and \(R_{J(\mathbb{Q})}\) is the rank of \(J(\mathbb{Q})\). During the last decades a great amount of research has gone into finding bounds of \(R_{J(\mathbb{Q})}\) in terms of invariants of \(C\). In this article the authors give families of examples of hyperelliptic curves \(C \colon y^2 = f(x)\) defined over \(\mathbb{Q}\), with \(f(x)\) of degree \(p\), where \(p\) is a Sophie Germain prime, such that \(R_{J(\mathbb{Q})}\) is bounded by the genus of \(C\) and the two-rank of the class group of the cyclic field defined by \(f(x)\). They further exhibit examples where the given bound is sharp. This extends work of \textit{D. Shanks} [Math. Comput. 28, 1137--1152 (1974; Zbl 0307.12005)] and
\textit{L. C. Washington} [Math. Comput. 48, 371--384 (1987; Zbl 0613.12002)] where a similar bound is given for the rank of certain elliptic curves. Jacobian; hyperelliptic curve; Mordell-Weil; rank; Selmer; descent Abelian varieties of dimension \(> 1\), Arithmetic ground fields for abelian varieties Bounds of the rank of the Mordell-Weil group of Jacobians of hyperelliptic 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 \(\mathcal M_{g,n}(S)\) be the groupoid of smooth, proper curves over a scheme \(S\) with geometrically connected fibres of genus \(g\) and \(n\) ordered sections. In case \(S\) is a normal, excellent scheme and \(U\subset S\) an open dense subset, the author gives two monodromy criteria for extending a curve in \( \mathcal M_{g,n}(U)\) to a curve in \(\mathcal M_{g,n}(S)\).
The first criterion states that the extension property is equivalent to the factorization of the natural map from the fundamental group of \(U\) to that of \(\mathcal M_{g,n}\) through the fundamental group of \(S\) (over \(\mathbb Z[1/N]\) for all \(N\in \mathbb N\)). The second (and stronger) criterion gives the analogous result by requiring the factorization of the outer pro-\(\mathbb L\) representation of the fundamental group of \(U\) with respect to a certain set of prime numbers \(\mathbb L\). The result is also generalized to the case of morphisms of finite type \(U\rightarrow S\).
If \(S\) is regular along the boundary \(S\backslash U\) the two criteria follow from previous results by \textit{L. Moret-Bailly} [C. R. Acad. Sci., Paris, Sér. I. 300, 489--492 (1985; Zbl 0591.14017)] and \textit{A. Tamagawa} [Compos. Math. 109, No. 2, 135--194 (1997; Zbl 0899.14007)]. extension; monodromy representation Stix J.: A monodromy criterion for extending curves. Int. Math. Res. Not. 29, 1787--1802 (2005) Families, moduli of curves (algebraic), Structure of families (Picard-Lefschetz, monodromy, etc.) A monodromy criterion for extending curves | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians The Jacobian of a genus g hyperelliptic curve outside a theta divisor is explicitly isomorphic to an affine subvariety of \({\mathbb{C}}^{3g+1}\) [\textit{C. G. T. Jacobi}, J. Reine Angew. Math. 32, 220-226 (1846); cf. \textit{D. Mumford}, ``Tata lectures on theta. II: Jacobian theta functions and differential equations'', Prog. Math. 43 (1984; Zbl 0549.14014)]. This paper presents an analogous construction for general curves. The construction uses the interpretation of a generic line-bundle as a commutative ring of differential operators and is thus suited to a solution of the KP equation. KP flows; Jacobian of a genus g hyperelliptic curve; theta divisor; differential operators; KP equation Previato, E.: Generalized Weierstrass \wp-functions and KP flows in affine space. Comment. math. Helvetici 62, 292-310 (1987) Theta functions and abelian varieties, Partial differential equations of mathematical physics and other areas of application, Dynamics induced by flows and semiflows, Jacobians, Prym varieties, Completely integrable finite-dimensional Hamiltonian systems, integration methods, integrability tests, Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.) Generalized Weierstrass \(\wp\)-functions and KP flows in affine space | 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 Grothendieck standard conjecture \(B(X)\) of Lefschetz type on the algebraicity of the operators * and \(\Lambda\) of Hodge theory is shown to hold for the following two types of smooth complex projective varieties \(X\):
(i) \(X\) is a compactification of the Néron minimal model of an abelian scheme of relative dimension three over an affine curve, and its generic scheme fibre has reductions of multiplicative type at all infinite places;
(ii) \(X\) is an irreducible holomorphic symplectic four-dimensional variety which coincides with the Altman-Kleiman compactification of the relative Jacobian of a family of hyperelliptic curves of genus two over \(\mathbb{P}^2\) with weak degenerations such that the canonical projection \(X\rightarrow\mathbb{P}^2\) is a Lagrangian fibration.
The first result provides us with a generalization of a result in [\textit{S. G. Tankeev}, Izv. Math. 78, No. 1, 169--200 (2014); translation from Izv. Ross. Akad. Nauk, Ser. Mat. 78, No. 1, 181--214 (2014; Zbl 1290.14007)] which assumes the generic scheme fibre is an absolutely simple abelian variety possessing reductions of multiplicative type at all infinite places. Grothendieck standard conjecture; fourfolds; abelian schemes; standard conjecture of Lefschetz type; Néron minimal model; reduction of multiplicative type; \(K3\) surface; hyperkähler variety; Chow-Lefschetz decomposition; Abel-Jacobi map 10.1070/IM2015v079n01ABEH002738 Algebraic cycles, Classical real and complex (co)homology in algebraic geometry, \(3\)-folds, \(4\)-folds On the standard conjecture and the existence of a Chow-Lefschetz decomposition for complex projective varieties | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians In the paper under review, the authors considerably generalize several of their previously established methods and results in arithmetic geometry, and that in various directions. Namely, in a previous paper [\textit{J. Denef} and \textit{F. Loeser}, Invent. Math. 135, 201--232 (1999; Zbl 0928.14004)], they had developed a general theory of integration -- called motivic integration -- on the space of arcs of an algebraic variety \(X\) over a field \(k\) of characteristic zero. This theory, which they now refer to as ``geometric'' motivic integration, is here replaced by a suitable arithmetic analogue which appears to be much better adapted to the study of certain rationality questions concerning Poincaré series of arithmetic schemes. More precisely, the authors develop a different kind of motivic integration theory, the so-called ``arithmetic'' motivic integration, which is largely based on methods from general field arithmetic and first-order model theory in mathematical logic. On the other hand, the Poincaré series \(P_p(T)\) associated to any pair \((X,p)\),where \(X\) is a reduced and separated scheme of finite type over \(\mathbb{Z}\) and \(p\) is a prime number, is known to be a rational function. This fact was proved by the first author [\textit{J. Denef}, Invent. Math. 77, 1--23 (1984; Zbl 0537.12011)]. Then, just a little later, J. Denef, J. Pas, and A. Macintyre proved, independently, that the degrees of the nominator and denominator of the rational function \(P_p(T)\) are bounded independently of the prime number \(p\). In view of these deep results, the basic task of the paper under review was to derive a much stronger uniformity result by constructing a canonical rational function \(P_{\text{ar}}(T)\) which specializes to \(P_p(T)\) for almost all prime numbers \(p\). In fact, it is precisely to this end that the authors develop their new ``arithmetic'' motivic integration theory, in the first part of the paper, which then turns out to be the right tool for constructing the universal rational function \(P_{\text{ar}}(T)\) by means of the Grothendieck ring of certain Chow motives. The paper consists of ten sections whose contents are as follows:
After a first section devoted to preliminaries on Grothendieck rings of Chow motives, the authors develop in section 2 what is needed, in the sequel, from the theory of Galois stratifications in field arithmetic. This and the construction of virtual motives from Galois stratifications associated to first-order logic formulas, as carried out in section 3, relies on the fundamental work of \textit{M. D. Fried} and \textit{M. Jarden} [``Field arithmetic'', Ergebnisse der Mathematik und ihrer Grenzgebiete, 3. Folge, Bd. 11, Berlin (1986; Zbl 0625.12001)]. Here the authors also introduce a new class of objects, the so-called ``definable subassignments'', which are more geometric in nature than first-order Galois formulas. Section 4 discusses definable subassignments for rings, and section 5 does the same for power series rings. These constructions are then used to define a measure on the set of definable subassignments of the space of germs of arcs of an arithmetic scheme \(X\) and this leads to the basic theory of arithmetic motivic integration developed in section 6. Applications to general rationality results for power series associated to algebraic varieties, in the spirit of the earlier works of J. Denef and F. Loeser mentioned above, are presented in section 7. The general results obtained here strikingly demonstrate the enormous power of the authors' new arithmetic motivic integration theory, and this insight is even strengthened by the results exhibited in the remaining three sections. Namely, in section 8, the authors show that arithmetic motivic integration indeed specializes to \(p\)-adic integration.
Finally, the arithmetical Poincaré series \(P_{\text{ar}}(T)\) for an algebraic variety \(X\) over a field of characteristic zero is established in section 9 where it is also proved that \(P_{\text{ar}}(T)\) factually specializes to the usual \(p\)-adic Poincaré series when \(k\) is a number field.
As the authors point out, the arithmetical Poincaré series seems to contain much more information about the underlying variety \(X\) than the authors' formerly studied geometric counterpart \(P_{\text{geom}}(T)\) defined by geometric motivic integration. This observation is demonstrated by means of a concrete example discussed in the concluding section 10. Actually, the authors compute the two different Poincaré series \(P_{\text{ar}}(T)\) and \(P_{\text{geom}}(T)\) for branches of plane algebraic curves, thereby showing that the poles of \(P_{\text{ar}}(T)\) completely determine the Puiseux pairs of the branch, whereas the series \(P_{\text{geom}}(T)\) only encodes the multiplicity at the origin.
All in all, the significance of the methods and results exhibited in this important paper can barely be overestimated, since these achievements unquestionably signify a major step forward in the theory of motivic integration within arithmetic geometry. Moreover, the various contributions of the two authors in this realm, over many years and in their entirety, must be seen as being pioneering and epoch-making in the field. arithmetic geometry; motives; arithmetic motivic integration; field arithmetic; first-order model theory; arithmetical Poincaré series; zeta functions; rationality questions; Galois stratifications; definable subassignments; \(p\)-adic integration Denef, Jan; Loeser, François, Definable sets, motives and \(p\)-adic integrals, J. Amer. Math. Soc., 0894-0347, 14, 2, 429-469, (2001) Arithmetic varieties and schemes; Arakelov theory; heights, Local ground fields in algebraic geometry, Quantifier elimination, model completeness, and related topics, Applications of model theory, Other classical first-order model theory, Abstract model theory, Field arithmetic, Model theory of fields, Finite ground fields in algebraic geometry, Other nonalgebraically closed ground fields in algebraic geometry, Varieties over finite and local fields, Zeta functions and \(L\)-functions, Ultraproducts and field theory, Étale and other Grothendieck topologies and (co)homologies, Rational points, Zeta functions and related questions in algebraic geometry (e.g., Birch-Swinnerton-Dyer conjecture), Arithmetic ground fields for surfaces or higher-dimensional varieties, Motivic cohomology; motivic homotopy theory Definable sets, motives and \(p\)-adic integrals | 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 deals with an algorithmic proof of the General Néron-Popescu Desingularization. Let us recall this statement:
Let \(u:A \longrightarrow A'\) be a regular morphism of Noetherian rings and let \(B\) be a finitely generated \(A\)-algebra. Then any morphism \(v: B\longrightarrow A'\) factors through a smooth \(A\)-algebra \(C\), i.e. \(v\) is the composition of two \(A\)-morphisms \(B\longrightarrow C \longrightarrow A'\).
This theorem has first been proven in this general form by the second author [Nagoya Math. J. 104, 85--115 (1986; Zbl 0592.14014)] and several other proofs have been provided afterwards. This theorem is a very important result in commutative algebra that appears to have a lot of deep applications (see for instance [\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)], [\textit{M. Hazewinkel} (ed.), Handbook of algebra. Volume 2. Amsterdam: North-Holland (2000; Zbl 0949.00006)] or [\textit{G. Rond}, ``Artin approximation'', \url{arXiv:1506.04717}] for an account of this).
In the present paper the authors construct in an effective and algorithmic way the smooth \(A\)-algebra along with the morphisms \(B\longrightarrow C\longrightarrow A'\) in the case where \(A\) and \(A'\) are one-dimensional local domains. Moreover their proof has been implemented in the computer algebra system SINGULAR. The end of the paper is devoted to the presentation of explicit examples.
This algorithm is important in view of the applications of the General Néron-Popescu Desingularization Theorem. regular morphisms; smooth morphisms; smoothing ring morphisms Popescu, A.; Popescu, D., A method to compute the general neron desingularization in the frame of one dimensional local domains, In: Decker Wolfram, Pfister Gerhard, Schulze Mathias (Eds.), Singularities and Computer Algebra - Festschrift for Gert-Martin Greuel, On the Occasion of his 70th Birthday, Springer Monograph. Étale and flat extensions; Henselization; Artin approximation, Local structure of morphisms in algebraic geometry: étale, flat, etc., Regular local rings, Henselian rings A method to compute the general Neron desingularization in the frame of one-dimensional local domains | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians For a smooth projective variety \(X\), \(Z^s(X)\) denotes the group of algebraic cycles on \(X\) of codimension \(s\), and \(C^s(X)= (Z^s(X)/ \sim) \otimes \mathbb{Q}\) for any adequate equivalence relation \(\sim\) on algebraic cycles. Let \({\mathcal D}(X)\) denote the \(\mathbb{Q}\)-subalgebra of \(\bigoplus C^s(X)\) generated by the divisor classes. The main theorem of this paper states that, for any Weil cohomology theory \(X\mapsto H^*(X)\) and any abelian variety \(A\) over an algebraically closed field, there is a reductive algebraic group \(L(A)\), called the Lefschetz group, such that the cycle class map induces an isomorphism \({\mathcal D}_{\text{hom}}^s(A^r)\otimes_\mathbb{Q} k\to H^{2s} (A^r) (s)^{L(A)}\) for all integers \(r,s\geq 0\), with \(k\) the coefficient field for the cohomology theory.
The Lefschetz group \(L(A)\) is defined to be the largest algebraic subgroup of \(GL(V(A)) \times {\mathbb{G}}_{m/k}\) fixing the elements of \({\mathcal D}_{\text{hom}}^s(A^r) \otimes_\mathbb{Q} k\subset H^{2s}(A^r)(s)\), thus it is a natural generalization of the corresponding notion introduced by \textit{V. Kumar Murty} [Math. Ann. 268, 197-206 (1984; Zbl 0543.14028)]. algebraic cycles; Weil cohomology; abelian variety; Lefschetz group; cycle class map Milne, J. S., Lefschetz classes on abelian varieties, Duke Math. J., 96, 3, 639-670, (1999) Algebraic cycles, Algebraic theory of abelian varieties, Abelian varieties of dimension \(> 1\), Transcendental methods, Hodge theory (algebro-geometric aspects) Lefschetz classes on abelian varieties | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians The articles of this volume will be reviewed individually.
Publisher's description: Robin Hartshorne's classical 1966 book ''Residues and Duality'' [RD] developed Alexandre Grothendieck's ideas for a pseudofunctorial variance theory of residual complexes and duality for maps of noetherian schemes.
The three articles in this volume rework the main parts of the last two chapters in [RD], in greater generality--for Cousin complexes on formal schemes, not just residual complexes on ordinary schemes--and by more concrete local methods which clarify the relation between local properties of residues and global properties of dualizing pseudofunctors. A new approach to pasting pseudofunctors is applied in using residual complexes to construct a dualizing pseudofunctor over a fairly general category of formal schemes, where compactifications of maps may not be available.
A theory of traces and duality with respect to pseudo-proper maps is then developed for Cousin complexes. For composites of compactifiable maps of formal schemes, this, together with the above pasting technique, enables integration of the variance theory for Cousin complexes with the very different approach to duality initiated by Deligne in the appendix to [RD].
The book is suitable for advanced graduate students and researchers in algebraic geometry.
Table of contents: J. Lipman, S. Nayak, and P. Sastry, Part 1. Pseudofunctorial behavior of Cousin complexes on formal schemes (3--133); P. Sastry, Part 2. Duality for Cousin complexes (137-192); S. Nayak, Part 3. Pasting pseudofunctors (195--271); Index (273--276) Lipman, J., Nayak, S., Sastry P.: Variance and duality for Cousin complexes on formal schemes. In: Pseudofunctorial Behavior of Cousin Complexes on Formal Schemes. Contemp. Math., vol. 375, pp. 3--133. American Mathematical Society, Providence (2005) Proceedings, conferences, collections, etc. pertaining to algebraic geometry, Collections of articles of miscellaneous specific interest Variance and duality for Cousin complexes on 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 The book contains an introduction to some applications of the theory of formal functions on subvarieties \(Y\) of projective varieties \(X\). Formal functions and formal schemes were introduced by Zariski and Grothendieck as the algebraic counterpart of the analytic concept of functions over tubular neighbourhoods of \(Y\) inside \(X\). Technically, formal schemes are defined by taking the algebraic completion of the structure sheaf of \(X\) modulo the ideal sheaf defining \(Y\), i.e. the direct limit of successive infinitesimal neighbourhoods of \(Y\) in \(X\). Formal functions are thus a tool towards the study of the extensions of functions on \(Y\) to functions defined in the ambient variety \(X\).
From this point of view, formal functions have a natural application to the study of subvarieties of \(X\) which contain \(Y\) (an approach which can be used in any characteristic). The author points out mainly the relations of formal geometry with the problem of extending a variety \(Y\subset \mathbb P^n\) as a hyperplane section of a subvariety \(Y'\subset \mathbb P^{n+1}\), extending bundles from \(Y\) to \(X\) and Hartshorne's conjecture for varieties of low codimension. A first motivation for this analysis is the remark that the first proof of the celebrated Zak extension principle was obtained by the machinery of formal extensions. A simplified proof, by means of the Fulton--Hansen connectedness principle, was introduced later.
The author exploits carefully the interplay between formal geometry and the connectedness principle. For instance, \(Y\) is defined of type G3 in \(X\) if the field \(K(X_Y)\) of meromorphic formal functions of \(Y\) in \(X\) is isomorphic to \(K(X)\). Then it is a general fact that subvarieties of low codimension are likely to be of type G3. The author shows that (in any characteristic), for any closed irreducible subvariety \(X\subset \mathbb P^n\times\mathbb P^n\) of dimension \(>n\), the intersection of \(X\) with the diagonal is G3 in \(X\). This result, an improvement for the connectedness principle, can be used to determine the geometry of varieties of low codimension and their deformations. For instance, it is applied in the book to prove restrictions on subvarieties of \(X\) which are \(0\)-loci of sections of vector bundles. Further applications to generating subvarieties of homogeneous spaces and the geometry of quasi--lines (rational curves with many infinitesimal deformations) are discussed. Bădescu, L., IMPAN Monogr. Mat. (N. S.), 65, (2004), Birkhäuser Verlag: Birkhäuser Verlag, Basel Research exposition (monographs, survey articles) pertaining to algebraic geometry, Formal neighborhoods in algebraic geometry, Low codimension problems in algebraic geometry Projective geometry and formal 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 \(A\) be a complete discrete valuation ring with quotient field \(K\), and algebraically closed residue field \(k\). Let \(X\) be a complete non- singular curve over \(K\), \({\mathcal X}\) be the minimal model of \(X\) over \(A\), and \(J\) be the Jacobian variety of \(X\). Moreover let \({\mathcal J}\) be the Néron model of \(J\). In this paper, the author considers the order \(\varphi\) of the étale group scheme consisting of the components of the special fibre of \({\mathcal J}\), and he gives an estimation of \(\varphi\) using the difference between the Euler-Poincaré characteristics of the special fibre \({\mathcal X}_s\) and the generic fibre \({\mathcal X}_\eta\) of \({\mathcal X}\), under the condition that the gcd of the multiplicities of the irreducible components of \({\mathcal X}_s\) is equal to one. In the proof, he uses a result of \textit{M. Raynaud} [Publ. Math., Inst. Hautes Étud. Sci. 38, 27--76 (1970; Zbl 0207.51602)]. Moreover, he gives an improvement for a result of \textit{H. W. Lenstra} jun. and \textit{F. Oort} [J. Pure Appl. Algebra 36, 281--298 (1985; Zbl 0557.14022)]. minimal model; Jacobian variety; Néron model Lorenzini, D, Groups of components of Néron models of Jacobians, Compositio Math., 73, 145-160, (1990) Jacobians, Prym varieties, Arithmetic ground fields for curves, Minimal model program (Mori theory, extremal rays) Groups of components of Néron models 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 Let \(C\) be a projective Gorenstein curve over \(\mathbb{C}\). First pure sheaves on \(C\) together with a quadratic form with values in the dualizing sheaf \(\omega_ C\), called \(\omega_ C\)-quadratic sheaves, are studied. Then a coarse moduli space \({\mathcal Q}_ C(r)\) of semi-stable \(\omega_ C\)- quadratic sheaves of multiplicity \(r\) is constructed using \(GIT\) (geometric invariant theory) and the underlying set of closed points is determined. It turns out that \({\mathcal Q}_ C(r)\) is projective. Also a relative version \({\mathcal Q}_{C/S}(r)\), relative to a Gorenstein morphism \(C\to S\) of dimension 1, is constructed.
Now let \(X\) be a smooth projective surface over \(\mathbb{C}\). The notion of theta-characteristic of a (possible singular) curve lying on \(X\) is defined and the variety \({\mathcal Q}_{{\mathcal D}/{\mathcal C}}(d)\), where \({\mathcal D}\to{\mathcal C}\) is the universal curve of degree \(d\) on \(X\), is used to construct a coarse moduli space \(\Theta_ X(d)\) of semi-stable theta- characteristics of degree \(d\). The variety \(\Theta_ X(d)\) is projective and there is a morphism \(\sigma\) to the space of curves of degree \(d\) by associating to a theta-characteristic its schematic support.
Generalizing the invariance \(\bmod 2\) theorem of Atiyah and Mumford to this situation it is shown that \(\Theta_ X(d)\) has at least 2 (possibly reducible) components corresponding to even and odd theta- characteristics. From a study of the local structure of \(\Theta_ X(d)\) it is then deduced that \(\Theta_{\mathbb{P}^ 2}(d)\) has only rational singularities and is of dimension \(d(d+3)/2\). This in turn is used to prove that the introduction of singular curves does not introduce an irreducible component of \(\Theta_{\mathbb{P}^ 2}\) other than those known from Beauville and Catanese for the open set of theta-characteristics having smooth schematic support. Over the open set of plane curves having finite fibers for \(\sigma\) the morphism \(\sigma\) will be flat and so there will be, counting with multiplicities, \(2^{2g}\) theta- characteristics, \(2^{g-1}(2^ g+1)\) even and \(2^{g-1}(2^ g-1)\) odd, over every such curve as in the smooth case. theta-characteristic of singular curve on smooth surface; Gorenstein curve Sorger C., Thêta-caractéristiques des courbes tracées sur une surface lisse, J. Reine Angew. Math., 1993, 435, 83--118 Theta functions and abelian varieties, Theta functions and curves; Schottky problem, Surfaces and higher-dimensional varieties Theta characteristics of curves on smooth 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 paper is a survey about generalized theta functions as sections of ample line bundles on moduli spaces [see also \textit{G. Faltings}, J. Algebr. Geom. 18, No. 2, 309--369 (2009; Zbl 1161.14025)]. It has seven sections:
\textit{1. Introduction}. Here the frame of the paper is introduced: one considers moduli of \(G\)-bundles over a curve, where \(G\) is a semisimple simply connected algebraic group and the main results are described: the Picard group of \({\mathcal M}_G\) is infinite cyclic, (the sections of these vector bundles are generalized theta functions) and the dimension of the vector space of sections is given (via Verlinde formula).
\textit{2. Moduli spaces.} Here the construction of the stack \(\mathcal{M}_{r,d}\) of vector bundles of rank \(r\) and degree \(d\) over a curve is sketched. Then one describes in few words the changes if \(GL_r\) is replaced with a group as above.
\textit{3. The double quotient.} Here the following setting is considered: take \(C\) a smooth projective curve over an algebraic closed field \(k\), \(x\in C\) a point and \(C^0=C\setminus \{x\}\). Then the moduli space is a double quotient:
\[
\mathcal{M}_G(k)=G(C^0)\setminus G(k((t))) /G(k[[t]]) .
\]
One studies first the right quotient, which turns out to be an affine Grassmanian \({\mathbb D}_G\). Using \(\text{Pic}({\mathbb D}_G)\) one reduces the study of \(\text{Pic}({\mathcal M}_G)\) to \(G(C^0)\)-equivariant line bundles on \({\mathbb D}_G\).
\textit{4. Line bundles on \({\mathcal M}_G\).} Two important examples corresponding to \(G=SL_r\) and \(G=Spin(r)\) (spingroup of \(SO(r)\)) are sketched.
\textit{5. Construction of a line bundle} of invariant \(1\).
\textit{6. The Verlinde formula.} Here the use of Verlinde formula to compute, in characteristic zero, \(\text{dim }\Gamma ({\mathcal M}_G,{\mathcal L}_c)\) for a line bundle of invariant \(c\) is explained in a simplified way. Some comments about ``das große offene Problem'' of describing geometrically these sections are given.
\textit{7. The Hitchin fibration.} Here the use of the Hitchin fibration to obtain results in positive characteristic is described, in a simplified version. vector bundles; moduli spaces; theta functions; algebraic group Vector bundles on curves and their moduli, Theta functions and curves; Schottky problem Theta functions on moduli spaces of vector bundles | 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 gives a inductive generalisation of the notion of an elementary modification of a vector bundle along a divisor. As input, this generalisation takes a vector bundle \(E\) on a scheme, and a descending chain of full-rank locally free subsheaves of \(E\), along with a divisor filtration satisfying a certain compatibility. The main theorem constructs an associated \textit{echelon modification} of \(E\), satisfying an appropriate universal property. This construction is used in the author's study of the stable hyperelliptic locus in the moduli space of stable curves [\textit{Z. Ran}, ``Modifications of Hodge bundles and enumerative geometry I: the stable hyperelliptic locus'', \url{arXiv:1011.0406}]. elementary modification; filtration; vector bundle Ran, Z, Echelon modifications of vector bundles, Commun. Alg., 41, 1846-1853, (2013) Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20] Echelon modifications of vector bundles | 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: The paper proposes a nonabelian version of the Jacobian for a smooth complex projective surface \(X\). Our version possesses all the classical features: it is the parameter space for a canonical family of torsion-free sheaves over \(X\) having fixed Chern invariants and rank 2, it carries a distinguished divisor (a theta-divisor), a package of nonabelian theta-functions. But it also has a new feature: our Jacobian carries a distinguished family of Higgs bundles. The parameter space \(H\), called (nonabelian) Albanese, of this family is a projective toric (singular) Fano variety whose hyperplane sections are (singular) Calabi-Yau varieties. In particular, it comes with a distinguished degenerate hyperplane section \(H_o\) equipped with degenerate symplectic structure, i.e., \(H_o\) is the union of projectivized Lagrangian subspaces of a certain symplectic vector space naturally associated with \(H_o\).
Our Jacobian and its Albanese \(H\) are related by two correspondences: (i) a geometric correspondence which sends points of the nonabelian Jacobian to a cycle of Calabi-Yau varieties, (ii) a cohomological correspondence, which is a Fourier-Mukai functor from the Higgs category on the Jacobian (algebraic/holomorphic side) to the so called F-category on \(H\) (algebraic/symplectic side).
Furthermore, there is a quantum correspondence which associates an operator-valued series with points of our Jacobian. The operator coefficients of this series are most naturally considered as elements of the universal enveloping algebra of a certain Lie algebra canonically associated to every point of the Jacobian. This gives a sheaf of Lie algebras on our Jacobian which could be viewed as a natural analogue of the Lie algebraic structure of the classical Jacobian. The basic Lie algebraic properties of this sheaf are established and a dictionary between its representation theory and geometry of the underlying points is given. nonabelian Jacobian; nonabelian theta functions; nonabelian Albanese Reider I. Nonabelian Jacobian of smooth projective surfaces. J Differential Geometry, 2006, 74: 425--505 Vector bundles on surfaces and higher-dimensional varieties, and their moduli, Transcendental methods, Hodge theory (algebro-geometric aspects), Calabi-Yau manifolds (algebro-geometric aspects), Abelian varieties and schemes, Theta functions and curves; Schottky problem Nonabelian Jacobian of smooth projective 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 \(f:X\rightarrow Y\) be a smooth morphism between compact complex varieties. Let
\[
C(X)_*: {\mathcal F}(X) \rightarrow A_*(X)
\]
denote the Chern-Schwartz-MacPherson transformation from the group \({\mathcal F}(X)\) of constructible functions on \(X\) to the Chow group \(A_*(X)\). The author shows that the Chern-Schwartz-MacPherson transformation commutes with pull-back in the following way: \(C(X)_* \circ f^* = (c(T_f) \cap f^*) \circ C(Y)_*\); here \(c(T_f)\) denotes the total Chern class of the relative tangential bundle \(T_f\). This formula may be viewed as an analogue of a formula in Riemann-Roch theory proved by Verdier. Furthermore, the paper contains an example which shows that the corresponding formula does not hold, if \(f\) is only a local complete intersection morphism. Finally, the author gives a necessary condition and a sufficient condition for the existence of a map \(f^G: A_*(Y) \rightarrow A_*(X)\) such that \(C(X)_* \circ f^* = f^G \circ C(Y)_*\), if \(f\) is an arbitrary morphism. constructible functions; Chern-Schwartz-MacPherson class; pull-back; Euler morphism; Chern-Mather class; Fulton-Johnson class; Milnor number; Riemann-Roch S. Yokura: ''On a Verdier-type Riemann-Roch for Chern-Schwartz-MacPherson class'', Topology and Its Applications, Vol. 94, (1999), pp. 315--327. Riemann-Roch theorems, Characteristic classes and numbers in differential topology On a Verdier-type Riemann-Roch for Chern-Schwartz-MacPherson class | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians Let \(C\) be a hyperelliptic curve of genus \(g \geq 2\) or an elliptic curve over a number field \(K\). A model of \(C\) over \(S=\mathrm{Spec}\, O_K\), where \(O_K\) is the ring of integers of \(K\), is a normal projective scheme \(f : \mathcal{C} \rightarrow S\) whose generic fiber \(\mathcal{C}_K\) is isomorphic to \(C\). A model \(\mathcal{C}\) is a Weierstrass model if the hyperelliptic involution on \(C\) extends to an \(S\)-involution \(\sigma\) on \(\mathcal{C}\) and the quotient \(\mathcal{L} = \mathcal{C} /\langle\sigma\rangle\) is smooth over \(S\). A Weierstrass equation of \(C\) is an equation of the form \(y^2 + Q(x)y = P (x)\), where \(P (x), Q(x)\in K[x]\) with \(\deg Q(x) \leq g + 1\), \(\deg P (x) \leq 2g + 2\) such that the corresponding affine curve is isomorphic to an open subscheme of \(C\). If \(P (x), Q(x) \in R[x]\), then the above equation is said to be integral over \(O_K\). If an integral Weierstrass equation of \(C\) defines a normal affine scheme \(\mathrm{Spec}\, O_K[x, y]/(y^2 + Q(x)y - P (x))\), then a Weierstrass model \(\mathcal{C}\) is provided by embedding \(\mathrm{Spec}\,O_K[x]\) in \(\mathbb{P}^1_S\) and taking the normalization of \(\mathbb{P}^1_S\) in the degree 2 morphism \( C \rightarrow \mathbb{P}^1_K\). Such a Weierstrass model is said to be defined by the integral Weierstrass equation.
Let \(\Delta_{\mathcal{S}}\) be the discriminant ideal of \(\mathcal{C}\) and \(\omega_{\mathcal{C}/S}\) the dualizing sheaf of \(\mathcal{C}\) over \(S\). In this paper, it is proved that a Weierstrass model \(\mathcal{C}\) of \(C\) over \(S\) is defined by an integral Weierstrass equation under any of the following conditions:
\begin{itemize}
\item[1.] \(\Delta_{\mathcal{S}}\) is principal, \(\det f^*\omega_{\mathcal{C}/S}\) is free and either \(g\) is odd, and \(\mathcal{L} \cong \mathbb{P}^1_S\) or \(g \equiv 2 \ (\bmod\ 4)\);
\item[2.] \(\mathrm{Pic}(S)\) is finite of odd order, \(\Delta_{\mathcal{S}}\) is principal and \(\det f^*\omega_{\mathcal{C}/S}\) is free;
\item[3.] \(\mathrm{Pic}(S)\) is finite of order prime to \(2(2g + 1)\) and \(\Delta_{\mathcal{S}}\) is principal;
\item[4.] \(\mathrm{Pic}(S)\) is finite of order prime to \(2g\) and \(\det f^*\omega_{\mathcal{C}/S}\) is free.
\end{itemize}
Furthermore, if \( C\) is a hyperelliptic curve of genus \(g\) over a number field \(K\) having everywhere good reduction and the class number of \(K\) is prime to \(2(2g + 1)\), then it is proved that \(C\) is defined by an integral Weierstrass equation with invertible discriminant. This result answers positively a conjecture of Mohammad Sadek. hyperelliptic curves; discriminant; Weierstrass equation; good reduction Curves of arbitrary genus or genus \(\ne 1\) over global fields, Elliptic curves over global fields, Arithmetic ground fields (finite, local, global) and families or fibrations, Arithmetic ground fields for curves Global Weierstrass equations of hyperelliptic 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 A generalized étale cohomology theory is a representable cohomology theory for presheaves of spectra on an étale site of an algebraic variety. These cohomology theories simultaneously generalize the homotopy-theoretic cohomologies of algebraic topology and the algebraic theories (for example: étale and crystalline) of Grothendieck.
Consequently this volume, in developing the techniques of the subject, introduces the reader to the stable homotopy category of simplicial presheaves. This is an extremely delicate development, obstructed by the need for coherent constructions involving very ``large'' objects such as limits of Čech constructions involving presheaves of spectra. The development of an adequate theory, particularly in respect of its applications to algebraic \(K\)-theory, was held up by difficulties with smash-products of spectra and with transfer constructions. This book provides the user with the first complete account which is sensitive enough to be compatible with the sort of closed model category necessary in \(K\)-theory applications [i.e., the closed model structure of \textit{A. K. Bousfield} and \textit{E. M. Friedlander}, Lect. Notes Math. 658, 80-130 (1978; Zbl 0405.55021)].
As an application of the techniques the author gives proofs of the descent theorems of \textit{R. W. Thomason} and \textit{Y. A. Nisnevich}. In particular, this implies the celebrated result of \textit{R. W. Thomason}, ``Algebraic \(K\)-theory and étale cohomology'', Ann. Sci. Éc. Norm. Supér., IV. Sér. 18, 437-552 (1985; Zbl 0596.14012)] which identifies \(\text{mod }p\) \(K\)-theory, after being inflicted with Bott periodicity in the manner introduced by the reviewer, with \(\text{mod }p\) étale \(K\)-theory.
The book concludes with a discussion of the Lichtenbaum-Quillen conjecture (an approximation to Thomason's theorem without Bott periodicity). The recent proof of this conjecture, by \textit{V. Voevodsky}, when \(p=2\) for fields of characteristic zero makes this volume compulsory reading for all who want to be au fait with current trends in algebraic \(K\)-theory! étale site of algebraic variety; generalized étale cohomology; presheaves of spectra; closed model category; Lichtenbaum-Quillen conjecture Jardine, J. F., Generalized Étale Cohomology Theories, Progress in Mathematics, vol. 146, (1997), Birkhäuser Verlag: Birkhäuser Verlag Basel Étale cohomology, higher regulators, zeta and \(L\)-functions (\(K\)-theoretic aspects), Stable homotopy theory, spectra, Research exposition (monographs, survey articles) pertaining to category theory, Research exposition (monographs, survey articles) pertaining to algebraic topology, Étale and other Grothendieck topologies and (co)homologies, Algebraic \(K\)-theory and \(L\)-theory (category-theoretic aspects), Presheaves and sheaves, stacks, descent conditions (category-theoretic aspects) Generalized étale cohomology theories | 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 to the ``secant'' case the methods previously developed [cf. \textit{A. Treibich}, Duke Math. J. 59, No. 3, 611--627 (1989; Zbl 0698.14029) and \textit{A. Treibich} and \textit{J.-L. Verdier}, in: The Grothendieck Festschrift, Collect. Artic. in Honor of the 60th Birthday of A. Grothendieck. Vol. III, Prog. Math. 88, 437--480 (1990; Zbl 0726.14024)] for studying the tangential covers. We construct, in particular, all complete and integral curves over an algebraically closed field of characteristic 0, which are secant to an elliptic curve in their generalized Jacobians. secant curve to an elliptic curve; secant to an ellipuc curve; generalized Jacobians Jacobians, Prym varieties, Coverings in algebraic geometry Discrete analogs of polynomials and tangential covers | 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 some notes [see ``Mordell-Weil lattices and Galois representation'', I-III, Proc. Japan Acad., Ser. A 65, No.7, 268-271; No.8, 296-299 and 300-303 (1989; Zbl 0715.14015-14017)], the author announced new results on the Mordell-Weil group E(K), \(K=k(C)\) the function field of a smooth projective curve C over an algebraically closed field k and E an elliptic curve over K. It is understood as general fibre of an elliptic surface S/C. The present article is dedicated to basic proofs. It is subdivided into two parts.
Part I. The Mordell-Weil group and the Néron-Severi group of an elliptic surface (1. The basic theorems; 2. Intersection theory on an elliptic surface; 3. Algebraic and numerical equivalence; 4. The Picard variety of an elliptic surface; 5. Proof of theorem 1.3).
Three fundamental theorems are proved by geometric meethods based on the intersection theory of algebraic surfaces: 1.1 \(E(K)\) is finitely generated; 1.2 The Néron-Severi group NS(S) is finitely generated and torsionfree; 1.3 \(E(K)=NS(S)/T,\) T the subgroup generated by the algebraic equivalence classes of vertical curves and the 0-section.
Part II. The Mordell-Weil lattice of an elliptic surface (6. Preliminaries on lattices; 7. The Néron-Severi lattice; 8. The Mordell- Weil lattices; 9. The unimodular case; 10. Rational elliptic surfaces).
The lattice structure of \(E(K)/E(K)_{tor}\) is introduced coming from the intersection product on NS(S). An explicit formula for the height pairing is proved (theorem 8.6). Looking at the fibres the Mordell-Weil lattices are complementary to sublattices of sums of some of the standard lattices \((E_ 6, E_ 7, E_ 8, A_ n, D_ n\) and their duals). This is the starting point for finer classifications. For example, the determination of the Mordell-Weil groups of rational elliptic surfaces is reduced to the study of root sublattices of \(E_ 8\). Mordell-Weil group; Néron-Severi group of an elliptic surface; intersection theory; Mordell-Weil lattice of an elliptic surface; height pairing Tetsuji Shioda, ``On the Mordell-Weil lattices'', Comment. Math. Univ. St. Pauli39 (1990) no. 2, p. 211-240 Rational points, Elliptic curves, Elliptic surfaces, elliptic or Calabi-Yau fibrations, Elliptic curves over global fields, Arithmetic varieties and schemes; Arakelov theory; heights On the Mordell-Weil lattices | 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 Denote by \(M_{g,n}\) the moduli space of \(n\)-pointed smooth algebraic curves over the field of complex numbers, and by \(\overline M_{g,n}\) its compactification via stable curves of this type. Let \({\mathcal M}_{g,n}\) and \(\overline{\mathcal M}_{g,n}\) be the respective moduli stacks.
Over the past thirty years, the study of the enumerative geometry of the spaces \(M_{g,n}\) and \(\overline M_{g,n}\) has played a central role in the classification theory of algebraic curves, in connection with which the calculation of the rational (co)homology of these spaces proved to be of crucial significance. In this regard, the computation by \textit{J. Harer} [Invent. Math. 72, 221--239 (1983; Zbl 0533.57003)] of the second homology groups of \(M_{g,0}\), accomplished already in the early 1980s by using rather intricate topological and analytical methods of Teichmüller theory, was certainly a major step forward in the moduli theory of algebraic curves and compact Riemann surfaces. However, Harer's approach relies entirely on transcendental concepts and tools, without reflecting the algebro-geometric nature of his result as distinct as it would be desirable.
In view of this peculiar fact, the authors of the article under review explain the possibility to reduce the transcendental part of Harer's involved calculation to another important result of his, thereby providing an alternative proof of Harer's theorem on the second homology groups of \(M_{g,0}\), which appears much more algebro-geometric in nature. More precisely, \textit{J. L. Harer's} so-called homological vanishing theorem [Invent. Math. 84, 157--176 (1986; Zbl 0592.57009)] asserts that the homology of \(M_{g,n}\) vanishes above a certain explicit degree, \(c(g,n)\). This result, a proof of which is outlined in the present survey, is combined with a series of additional algebro-geometric arguments to obtain an explicit calculation of the cohomology groups \(H^1(M_{g,n};\mathbb{Q})\) and \(H^2(M_{g,n};\mathbb{Q})\) in various cases with respect to the parameters \(g\) and \(n\). Although these are meanwhile classical results [\textit{D. Mumford}, J. Anal. Math. 18, 227--244 (1967; Zbl 0173.22903); \textit{J. Harer}, Invent. Math. 72, 221--239 (1983; Zbl 0533.57003)], the authors' approach sheds some important new light on the overall picture, first and foremost from the viewpoint of algebraic geometry.
In addition, the authors also describe how the first and second rational cohomology groups of the moduli spaces \(\overline M_{g,n}\) of stable \(n\)-pointed curves, of genus \(g\) can be calculated. Here they follow the presentation of an earlier work of theirs [Publ. Math., Inst. Hautes Étud. Sci. 88, 97--127 (1998; Zbl 0991.14012)], in which the third and the fifth cohomology groups were also exhibited and shown to always vanish.
As for the basic, largely new methods described in the paper under review, the authors investigate the boundary strata in \(\overline M_{g,n}\) by means of the corresponding moduli stack \(\overline{\mathcal M}_{g,n}\) and certain graphs attached to stable \(n\)-pointed curves, study then some natural (or tautological) classes in the cohomology ring of \(\overline{\mathcal M}_{g,n}\) and various relations satisfied by them, and finally establish a version of Deligne's ``Gysin spectral squence'' in Hodge theory for the pair consisting of \(\overline M_{g,n}\) and its boundary \(\partial M_{g,n}\) to calculate the cohomology of the open variety \(M_{g,n}=\overline M_{g,n}\setminus M_{g,n}\) in terms of the cohomology of the boundary, strata described before. In the special case of genus \(g= 0\), S. Keel's approach to calculate the Chow ring \(A^*(\overline M_{0,n})= H^*(\overline M_{0,n}; \mathbb{Z})\) is suitably modified (and simplified) in order to deal with the relevant divisor classes in the cotext of the present article.
It should be pointed out that most of the material discussed in the present, highly enlightening survey article can also be found, in the meantime, in the recent monograph ``Geometry of algebraic curves. Volume II'' by \textit{E. Arbarello}, \textit{M. Cornalba} and \textit{P. A. Griffiths} [Grundlehren der Mathematischen Wissenschaften 268. Berlin: Springer (2011; Zbl 1235.14002)], where Chapter XIX is particularly devoted to the (co)homology of moduli spaces of curves, though in a much wider context, in greater generality and in a more elaborate presentation. algebraic curves; Riemann surfaces; moduli spaces; cohomology rings; Chow rings; divisor classes; homology groups; Picard groups; Teichmüller theory; Gysin spectral sequence Arbarello E., Grundlehren der Mathematischen Wissenschaften 268, in: Geometry of Algebraic Curves (2010) Families, moduli of curves (algebraic), (Equivariant) Chow groups and rings; motives, Divisors, linear systems, invertible sheaves, Picard groups, Other algebro-geometric (co)homologies (e.g., intersection, equivariant, Lawson, Deligne (co)homologies), Teichmüller theory for Riemann surfaces Divisors in the moduli spaces of curves | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians Let \(C\) be a hyperelliptic curve over a field \(k\) of characteristic \(0\), and let \(P \in C(k)\) be a marked Weierstrass point. As \textit{M. Bhargava} and \textit{B. H. Gross} [Prog. Math. 257, 33--54 (2014; Zbl 1377.11045)] have observed, the 2-descent on the Jacobian of \(C\) can be rephrased in terms of the language of arithmetic invariant theory, using the geometry of pencils of quadrics. We give a simple reinterpretation of their construction using instead the geometry of the curve \(C\). hyperelliptic curves; Jacobian; arithmetic invariant theory; line bundles Curves of arbitrary genus or genus \(\ne 1\) over global fields, Arithmetic ground fields for curves A remark on the arithmetic invariant theory of hyperelliptic 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 author offers an alternative to his own compactification of moduli of abelian varieties given in [Invent. Math. 136, No.3, 659--715 (1999; Zbl 0982.14026]. The starting point is the same: a polarised abelian variety \((G_\eta,{\mathcal L}_\eta)\) over \(k(\eta)\), the fraction field of a complete DVR \(R\). To fill in the central fibre one needs degeneration data in the sense of Faltings and Chai, for which a finite bae change may be necessary. Already in [loc. cit.] there was a choice about this, depending on whether one was content with a general fibre \(Q\) giving a Kempf-stable central fibre \(Q_0\) (called a projectively stable quasi-abelian scheme or PSQAS) or insisted on a reduced central fibre \(P_0\), the general fibre then being \(P\). The author gave a standard way to achieve the latter by a further base change: the result is called a torically stable quasi-abelian scheme or TSQAS. The challenge then is to globalise these constructions. In [loc. cit.] that was done for PSQASs and here it is done, by GIT methods again, for TSQASs.
As the aim is to produce moduli schemes or at least algebraic spaces, rather than stacks, it is necessary to work with level structures (which is likely to be quite helpful for applications anyway). For this the author starts with a finite abelian group \(H\) of exponent \(N\) and works throughout over \({\mathcal O}={\mathbb Z}[{{1}\over{N}},\zeta_N]\). In this context one can define a finite (and an infinite) Heisenberg group scheme \(G(K)\) (where \(K\) is the direct sum of \(H\) with its Cartier dual), and one must develop enough of its representation theory to define a level-\(G(K)\) structure on \(P_0\) (and then globalise). Finally, since \(Q_0\) determines \(P_0\) and \(P\) determines \(Q\), one expects the two moduli spaces to be close.
The paper works through the technical difficulties of this. To give a good moduli space, \(K\) needs to be not too small: it is a finite symplectic abelian group consisting of \(g\) pieces of \(e_i\)-torsion, and one needs the smallest of these \(e_i\) to be at least \(3\) (the largest is \(N\)). With this much level structure, however, the functor of \(g\)-dimension TSQASs with level-\(G(K)\) structure, over reduced algebraic spaces, has a complete (separated, reduced) moduli space over \({\mathcal O}\), which comes with a canonical bijective finite birational morphism to the moduli of PSQASs. In particular, the two moduli spaces constructed here and in [loc. cit.] have the same normalisation. abelian variety; moduli; compactification; Heisenberg group; geometric invariant theory Nakamura, Iku, Another canonical compactification of the moduli space of abelian varieties.Algebraic and arithmetic structures of moduli spaces (Sapporo 2007), Adv. Stud. Pure Math. 58, 69-135, (2010), Math. Soc. Japan, Tokyo Algebraic moduli of abelian varieties, classification, Families, moduli, classification: algebraic theory, Theta functions and abelian varieties Another canonical compactification of the moduli space of abelian varieties | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians The present book is the second edition of an English translation of the original text. The first edition appeared in 1989 (Zbl 0696.14012). The text on its own has been left completely intact and unchanged. However, having put it in a modern type-setting, the editors have taken the occasion to eliminate the numerous (trivial) misprints in the original and, to such a degree, improve the outward aesthetical appearance of this great book. Also, being now available as a relatively inexpensive paperback edition, this very special introductory text on the theory of complex algebraic curves has finally become affordable for less wealthy prospective customers.
Now as before, Griffiths's elementary crash course on the geometry of complex curves, which was taught at the University of Beijing (China) in the summer of 1982 and first published in Chinese, represents one of the most efficient modern introductions to this fascinating classical subject. Keeping the technical requirements to an absolute minimum, and even relinquishing sheaf theory and cohomological methods, the author leads the beginner to the deep classical theorems of Abel, Jacobi, Riemann-Hurwitz, Riemann-Roch, and others, basically so by laying emphasis on the purely geometric and complex-analytic aspects of the theory. In regard of this particularly beginner-friendly approach, Griffiths's textbook will certainly maintain its timelessly unique character of being an excellent and thorough guide to the more advanced topics in algebraic curve theory, such as they are comprehensively treated in the two-volume monograph ``Geometry of algebraic curves'' by \textit{E. Arbarello}, \textit{M. Cornalba}, \textit{P. A. Griffiths} and \textit{J. Harris} [Vol. I: Grundlehren Math. Wiss. 267 (1985; Zbl 0559.14017); Vol. II: to appear]. -- As to a related, however more recent and somewhat deeper-going introduction to complex curve theory, the interested reader is referred to \textit{R. Miranda}'s textbook ``Algebraic curves and Riemann surfaces'' [Graduate Stud. Math. 5 (1995; Zbl 0820.14022)], whereas the more algebraic and arithmetic aspects of algebraic curve theory are excellently (and complementarily) provided by \textit{D. Lorenzini}'s ``An invitation to arithmetic geometry'' [Graduate Stud. Math. 9 (1995; Zbl 0847.14013)]. Riemann surfaces; holomorphic differentials; meromorphic differentials; Riemann-Roch theorem; Abel's theorem; Jacobi inversion theorem; elliptic curves; ramification; complex algebraic curves GRIFFITHS P. A.: Introduction to Algebraic Curves. (2nd. Transl. Math. Monographs 76, Amer. Math. Soc, Providence, RI, 1996 Curves in algebraic geometry, Introductory exposition (textbooks, tutorial papers, etc.) pertaining to algebraic geometry, Compact Riemann surfaces and uniformization, Research exposition (monographs, survey articles) pertaining to algebraic geometry Introduction to algebraic curves. Transl. from the Chinese by Kuniko Weltin. | 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 describes two approaches to the Jacobian conjecture in ``semi expository'' style (its novelty is a new approach to the known results of the authors). The first is through the following generalized variant of the Jacobian conjecture: when is an étale endomorphism of a complex affine variety \(X\) a finite morphism? A positive answer to this question in the case \(X=\mathbb{C}^{n}\) would solve the Jacobian conjecture. The authors give positive and negative examples to this question in various classes of varieties.
The second approach is through infinite dimensional algebraic varieties called ind-affine varieties. In particular, they endow the sets: \(\mathcal{ E}^{1}\) -- the principal (i.e. inducing the identity map on the tangent space at the origin) endomorphisms of the affine \(n\)-space \(\mathbb{A}^{n}\), \(\mathcal{J}^{1}\) -- the endomorphisms of \(\mathbb{A}^{n}\) of the Jacobian determinant \(1\) and \(\mathcal{G}^{1}\) -- the automorphisms of \( \mathbb{A}^{n}\), with the structure of ind-affine varieties and prove (over an algebraically closed field \(k\) of characteristic zero) that:
1. \(\mathcal{E}^{1}\) is isomorphic to the ind-affine space \(\mathbb{A} ^{\infty },\) 2. \(\mathcal{J}^{1}\) is canonically closed embedded in \(\mathcal{E}^{1},\) 3. \(\mathcal{G}^{1}\) is canonically weakly-closed embedded in \(\mathcal{J} ^{1}.\) étale endomorphism; finite morphism; ind-affine variety Kambayashi, T.; Miyanishi, M., On two recent views of the Jacobian conjecture, (Affine Algebraic Geometry, Contemp. Math., vol. 369, (2005), Amer. Math. Soc. Providence, RI), 113-138 Jacobian problem On the recent views of the Jacobian conjecture | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians In a series of notes the author announces new results on the Mordell-Weil group \(E(K)\), where \(K=k(C)\) is the function field of a smooth projective curve C over an algebraically closed field k and E an elliptic curve over K. The K-rational points of E are identified with the global sections of the associated elliptic surface \(f: S\to C\), the Kodaira-Néron model of E/K.
Under the assumption that f is not smooth the Néron-Severi group NS(S) of S is torsion-free, hence a lattice with respect to the intersection product of curves on S. Let T be the sublattice generated by the zero section and all fibre components of f. It holds that \(E(K)\cong NS(S)/T\) [theorem 1.1, proved by the author, J. Math. Soc. Japan 24, 20-59 (1972; Zbl 0226.14013)].
A finer investigation yields a commutative diagram
\[
\begin{alignedat}{3} 0\rightarrow E(K)_{\text{tor}} \rightarrow & E(K) && \rightarrow && L^* \\ & \cup &&&& \cup \\ & E(K)^ 0 && \tilde\rightarrow && L \end{alignedat}
\]
with exact rows (theorem 1.2). here L is the orthogonal complement of T in \(NS(S)\), \(L^*\) its dual lattice and \(E(K)^ 0\) the subgroup of E(K) consisting of all sections passing through the same fibre components as the zero section. Changing the sign \(E(K)/E(K)_{tor}\) is endowed with a positive-definite lattice structure. It is called the Mordell-Weil lattice of E/K. If NS(S) is unimodular, then \(E(K)/E(K)_{tor}\cong L^*\) (up to sign, theorem 1.4).
Up to a trivial factor the product \(<.,.>\) of the Mordell-Weil lattice above is the same as the canonical height pairing due to Néron, Tate and Manin. From the explicit knowledge of the special fibres of f the author deduces explicit formulas for the height pairing and for the relation between \(\rho (S)=rk(NS(S))\) and the rank \(r=rk(E(K))\) of the Mordell-Weil group.
[See also part II and III of this paper, see the following reviews.] elliptic curve over function field; Mordell-Weil group; rational points; elliptic surface; Kodaira-Néron model; Néron-Severi group; Mordell- Weil lattice; height pairing F. Denef, \textit{Les Houches lectures on constructing string vacua}, arXiv:0803.1194 [INSPIRE]. Arithmetic varieties and schemes; Arakelov theory; heights, Rational points, Elliptic surfaces, elliptic or Calabi-Yau fibrations Mordell-Weil lattices and Galois representation. I | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians Let \(X\) be a smooth, proper and geometrically irreducible curve over a finite field \(\mathbb F_q\), \(\infty\) a rational point of \(X\) and \(A=H^0 (X-\infty,{\mathcal O}_X)\). This paper is developed to study the moduli spaces of \(k\)-elliptic sheaves, \(A\)-motives and \(t\)-modules over \(A\) with formal \(D\)-level structures along a divisor \(D\) on \(\text{Spec} A\subset X\). The author's main idea is to apply the algebraic theory of Sato's infinite Grassmannian [developed by the author, \textit{J. M. Muñoz Porras} and \textit{F. J. Plaza Marín}, Workshop on abelian varieties and theta functions, Morelia 1996, Proc., Aportaciones Mat., Investig. 13, 3--40 (1998; Zbl 0995.14021)] to study these moduli spaces. The author constructs a Krichever morphism for elliptic sheaves with \(x\)-formal level structure (\(x\) being a rational point) and proves that the functor of moduli of elliptic sheaves with \(x\)-formal level structure is representable by a locally closed subscheme of the infinite Grassmannian \(\text{Gr} (\mathbb F_q ((t))^n\), \(\mathbb F_q [[t]]^n\)).
This result allows the author to apply the methods of soliton theory to study the moduli of elliptic sheaves: He defines the elliptic Baker functions, which are the counterpart of the classical Baker-Akhiezer functions of the KdV-hierarchy; these elliptic Baker functions characterize, up to isomorphisms, the elliptic sheaves. Finally, the author applies these results to study the determinant of elliptic sheaves and gives some explicit computations in the cases \(g=0\) and \(g=1\) (\(g=\) genus of \(X\)). elliptic sheaves; Drinfeld modules; Baker functions; determinants; \(t\)-modules; finite field; infinite Grassmannian; moduli spaces; soliton theory; KdV-hierarchy A. Álvarez, Uniformizers for elliptic sheaves, Internat. J. Math. 11 (2000), no. 7, 949 -- 968. Drinfel'd modules; higher-dimensional motives, etc., Algebraic moduli problems, moduli of vector bundles, Relationships between algebraic curves and integrable systems, Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.), Relations of infinite-dimensional Hamiltonian and Lagrangian dynamical systems with algebraic geometry, complex analysis, and special functions Uniformizers for elliptic sheaves | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians The paper begins with two genericity results for rigid analytic morphisms (in the adic setting) analogous to classical results in algebraic geometry. Let \(K\) be a non-Archimedean complete valued field. Let \(f \colon X \to Y\) be a quasi-compact map of rigid spaces over \(K\), with \(Y\) geometrically reduced. Then \(f\) is flat over a dense open subset of~\(Y\). If \(K\) is of characteristic 0, the result holds with smooth instead of flat. Note that similar results were proved by \textit{A. Ducros} in the setting of Berkovich spaces in [Families of Berkovich spaces. Paris: Société Mathématique de France (SMF) (2018; Zbl 1460.14001)]. Even though there is no good notion of generic point in rigid geometry, the proof presented in the paper makes use of weakly Shilov points (introduced by the authors), which behave similarly in some aspects. For instance, a morphism as above is always flat (and smooth in characteristic 0) over such a point, and the property extends to a neighborhood.
The authors then apply those results to the theory of Zariski-constructible étale sheaves on rigid spaces, as defined by \textit{D. Hansen} [Compos. Math. 156, No. 2, 299--324 (2020; Zbl 1441.14085)]. Under the assumption that the base field \(K\) is of characteristic 0, they prove that the notion of Zariski-constructibility is local for the étale topology and develop a six-functor formalism in this setting.
As further applications of the theory of constructible sheaves, the authors introduce perverse sheaves and intersection cohomology on rigid spaces in characteristic 0, and prove that those notions are well-behaved. rigid analytic spaces; étale cohomology; generic smoothness; six functors; constructible sheaves; perverse sheaves; intersection cohomology Rigid analytic geometry, Étale and other Grothendieck topologies and (co)homologies, Sheaves in algebraic geometry The six functors for Zariski-constructible sheaves in rigid 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 As a generalization of the known Jacobian conjecture, Kulikov posed the following conjecture: if \(F:X\rightarrow \mathbb{C}^{n}\) is an étale morphism which is surjective modulo codimension two with \(X\) simple connected then \(F\) is birational. He also found a counter-example to this conjecture. The authors solve this problem in positive under some additional assumptions.
The main theorem is the following. If \(F:X\rightarrow \mathbb{C}^{n}\) is a local diffeomorphism (not necessary étale morphism) such that 1. \(X\) is a simple connected manifold, 2. there exists a hypersurface \(D\subset \mathbb{C}^{n}\) such that the restriction \(X\setminus F^{-1}(D)\rightarrow \mathbb{C}^{n}\setminus D\) of \( F \) is a \(d\)-fold covering mapping, 3. \(D\) has at worst normal crossing singularities away from a set of codimension \(3\), 4. the closure \(\overline{D}\subset \mathbb{C}^{n}\) cuts the hypersurface \(H\) at infinity transversely, then \(d=1\) or \(d=\infty .\)
In an algebraic geometric setting this implies. If \(F:X\rightarrow \mathbb{C} ^{n} \) is an étale morphism with \(X\) simple connected variety and \(D\) is as in the main theorem then \(F\) is injective.
Reviewer's remark (to the proof): It is well-known that properness of the Jacobian mapping \(F\) implies the bijectivity of \(F\) (by the monodromy theorem). Jacobian conjecture; étale morphism; regular cover Nollet, S.; Xavier, F.: On kulikov's problem, Arch. math. 89, No. 3, 385-389 (2007) Jacobian problem On Kulikov's problem | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians Let \(X\) be a smooth projective curve of genus \(g>1\) over \(\mathbb{C}\) and \({\mathcal U}(r,d)\) the moduli space of semistable vector bundles of rank \(r\) and degree \(d\) on \(X\). The Picard Sheaf \({\mathcal E}_{r,d}\) on \({\mathcal U}(r,d)\) is the direct image sheaf of a universal Poincaré sheaf on \(X\times {\mathcal U}(r,d)\). Drezet and Narasimhan constructed a generalized theta divisor \(\Theta_{{\mathcal U}(r,d)}\) on \({\mathcal U}(r,d)\). The main theorem of the paper under review states that for relatively prime integers \(r\) and \(d\) with \(d>2gr\) the Picard Sheaf \({\mathcal E}_{r,d}\) is \(\Theta_{{\mathcal U}(r,d)}\)-stable. For the proof the author chooses a spectral curve \(Y\) over \(X\). An open subset \(T\) of the Jacobian \(J(Y)\) dominates \({\mathcal U}(r,d)\). It is shown that the pull back of \(\Theta_{{\mathcal U}(r,d)}\) to \(T\) is a power of the usual theta divisor on \(J^ (Y)\) restricted to \(T\). spectral curves; stability of Picard sheaf; generalized theta divisor Li, Y., Spectral curves, theta divisors and Picard bundles, Int. J. Math., 2, 525-550, (1991) Vector bundles on curves and their moduli, Theta functions and abelian varieties, Picard groups, Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20] Spectral curves, theta divisors and Picard bundles | 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 is the second edition of the author's [Geometric modular forms and elliptic curves, ibid. 361 p. (2000; Zbl 0960.11032)]. The contents (of the first edition) of the book has been described by the reviewer in Zbl 0960.11032. Here we only indicate the main changes done in the second edition.
A detailed description of Barsotti-Tate groups (including formal Lie groups) is added in Chapter 1. As an application, a down-to-earth description of formal deformation theory of elliptic curves is incorporated in Section 2.10.
In Chapter 3, two subsections (3.2.6 and 3.2.7) are added to facilitate good transition from horizontal/vertical control results in the earlier part of Section 3.2 for modular forms to ring/scheme theoretic control results on the side of Hecke algebra.
The newly added Section 4.3 contains Ribet's theorem of full image of modular \(p\)-adic Galois representation and its generalization to `big' \(\Lambda\)-adic Galois representations under middle assumptions (a new result of the author).
The newly added Section 5.3 discusses modularity of abelian \(\mathbb Q\)-varieties. Modularity of abelian \(\mathbb Q\)-varieties of \(\text{GL}(2)\)-type was predicted by Ribet, and finally proved in 2009 by \textit{Khare} and \textit{Winterberger} as a special case of modularity of strict compatible systems of odd two-dimensional Galois representations. The author gives a proof of special cases of the modularity directly based on the theorem of Wiles-Taylor-Diamond-Skinner (Theorem 5.2.1).
The bibliography has been extended and updated.
The book, addressed to graduate students and experts working in number theory and arithmetic-geometry, is a welcome addition to this beautiful and difficult subject. control theorems; Shimura-Taniyama-Weil conjecture; elliptic curve; modular curve; deformation rings; Hecke algebras; modular Galois representations; moduli spaces of elliptic curves; modular forms; Abelian \(\mathbb{Q}\)-curves Hida, H.: Geometric Modular Forms and Elliptic Curves, 2nd edn. World Scientific, Singapore (2012) Galois representations, Elliptic curves over global fields, Research exposition (monographs, survey articles) pertaining to number theory, \(p\)-adic theory, local fields, Holomorphic modular forms of integral weight, Local ground fields in algebraic geometry, Research exposition (monographs, survey articles) pertaining to algebraic geometry Geometric modular forms and elliptic 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 a smooth scheme \(X_0\) over a perfect field \(k\) of characteristic \(p>0\) there is a particular sheaf \({\mathcal D}^{(\infty)}_{X_0}\) introduced by A. Grothendieck, namely the so-called sheaf of differential operators on \(X_0\) relative to the scheme \(S_0:= \text{Spec}(k)\). In this context, a classical result of N. Katz states that there is an equivalence between the category of vector bundles on \(X_0\) equipped with a left action of \({\mathcal D}^{(\infty)}_{X_0}\), on the one hand, and the category of families of vector bundles \({\mathcal E}_i\) on \(X_0\), \(i\geq 0\), endowed with \({\mathcal O}_{X_0}\)-linear isomorphisms \(\alpha_i:= F^*_{X_0}{\mathcal E}_{i+ 1}\tilde{\rightarrow}{\mathcal E}_i\), \(i\geq 0\), where \(F_{X_0}\) denotes the absolute Frobenius endomorphism of \(X\), on the other hand.
In the present paper, the author shows how his theory of arithmetic \({\mathcal D}\)-modules, developed by the author [I: Ann. Sci. Éc. Norm. Supér. (4) 29, No. 2, 185--272 (1996; Zbl 0886.14004) and II: Mém. Soc. Math. Fr., Nouv. Sér. 81, 138 p. (2000; Zbl 0948.14017)], can be used to generalize this classical result from the 1970s to any infinitesimal deformation \(f: K\to S\) of the above set-up, which is endowed with Frobenius liftings. Also, a similar generalization is provided for separated and complete modules over a formal scheme in mixed characteristics. The first step of these generalizations is based on the concept of Frobenius-divided modules on the scheme \(X\), which is both introduced and analyzed in Section 1 of the current paper. The main result of these investigations is Theorem 1.2. assertig that the forgetful functor from the category of Frobenius-divided \({\mathcal D}^{(\infty)}_X\)-modules to the category of Frobenius-divided \({\mathcal O}_X\)-modules is an equivalence.
In Section 2, this crucial new result, together with an application of the author's theory of Frobenius descent for arithmetic \({\mathcal D}\)-modules, is used to derive the above-mentioned generalization of N. Katz's classical result in the following form (Theorem 2.4.): Under suitable assumptions, there exists an equivalence between the category of Frobenius-divided \({\mathcal D}^{(\infty)}_X\)-modules, on the one hand, and the category of ordinary \({\mathcal D}^{(\infty)}_X\)-modules on the other.
Moreover, this explicitely established categorical equivalence admits a number of special properties as exhibited in the course of the proof. Finally, the author points out how a similar result (Theorem 2.6.) can be obtained for adic formal schemes. In a special case, this general result is applied to recover a correspondence constructed earlier by \textit{B. H. Matzat} in the context of local differential modules over one-dimensional local differential rings [Integral \(p\)-adic differential modules. Groupes de Galois arithmétique et différentiels. Paris: Société Mathématique de France. Séminaires et Congrès 13, 263--292 (2006; Zbl 1158.13009)]. \(D\)-modules; differential sheaves; sheaves of differential operators; Frobenius morphism; descent theory; deformation theory; Frobenius descent; formal schemes Pierre Berthelot, ``A note on Frobenius divided modules in mixed characteristics'', Bull. Soc. Math. Fr.140 (2012) no. 3, p. 441-458 Differentials and other special sheaves; D-modules; Bernstein-Sato ideals and polynomials, \(p\)-adic cohomology, crystalline cohomology, Rings of differential operators (associative algebraic aspects), Differential algebra, \(p\)-adic differential equations, Commutative rings of differential operators and their modules, Differential and difference algebra A note on Frobenius divided modules in mixed characteristics | 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 Consider the moduli stack \(\mathcal{M}_{g,n}\) of genus \(g\) curves with \(n\) marked points and denote its Deligne-Mumford compactification by \(\overline{\mathcal{M}}_{g,n}\). In the article under review, the author's main result establishes existence, as an algebraic space, of a universal Neron model admitting morphism for \(\overline{\mathcal{M}}_{g,n}\). The author is motivated by the problem of extending sections of the universal Jacobian \(\mathcal{J}_{g,n}\) to the universal semi-abelian scheme \(\mathcal{J}^0_{g,n}\) over \(\overline{\mathcal{M}}_{g,n}\).
In particular, the author constructs a regular integral stack \(\widetilde{\mathcal{M}}_{g,n}\) together with a separated representable map \[\widetilde{\mathcal{M}}_{g,n} \rightarrow \overline{\mathcal{M}}_{g,n}\text{,}\] which is locally of finite presentation and which has the following three properties:
(i) the structure morphism \(\widetilde{\mathcal{M}}_{g,n} \rightarrow \overline{\mathcal{M}}_{g,n}\) is an isomorphism over \(\mathcal{M}_{g,n}\);
(ii) the universal Jacobian \(\mathcal{J}_{g,n}\) of the universal curve of \(\mathcal{M}_{g,n}\) admits a Neron model over \(\widetilde{\mathcal{M}}_{g,n}\); and
(iii) if \(t \colon T \rightarrow \overline{\mathcal{M}}_{g,n}\) is a morphism which is such that \(t^* \mathcal{J}_{g,n}\) admits a Neron model over \(T\), then the map \(t\) factors uniquely via the structure morphism \(\widetilde{\mathcal{M}}_{g,n} \rightarrow \overline{\mathcal{M}}_{g,n}\).
A key point in the author's proof of this result is the concept of aligned prestable curve and its relation to the question of existence of Neron models for Jacobians.
The author's approach builds on his previous work \textit{D. Holmes} [J. Reine Angew. Math. 747, 109--145 (2019; Zbl 1423.14203)] and the author explains how it fits into the related work of \textit{L. Caporaso} [Am. J. Math. 130, No. 1, 1--47 (2008; Zbl 1155.14023)] and others.
Finally, to indicate the key points that arise in the proof of the main result, the author discusses, in detail, the special case that \(g = 1\) and \(n = 2\). moduli of curves; Jacobians; Néron models Picard schemes, higher Jacobians, Families, moduli of curves (algebraic), Jacobians, Prym varieties A Néron model of the universal 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 In the present paper, we discuss the Grothendieck conjecture for hyperbolic curves over Kummer-faithful fields. In particular, we prove that every point-theoretic and Galois-preserving outer isomorphism between the étale/tame fundamental groups of affine hyperbolic curves over Kummer-faithful fields arises from a uniquely determined isomorphism between the original hyperbolic curves. This result generalizes results of \textit{A. Tamagawa} [Compos. Math. 109, No. 2, 135--194 (1997; Zbl 0899.14007)], i.e., our main result in the case where the basefields are either finite fields or mixed-characteristic local fields. Grothendieck conjecture; hyperbolic curve; Kummer-faithful field Coverings of curves, fundamental group, Universal profinite groups (relationship to moduli spaces, projective and moduli towers, Galois theory) On the Grothendieck conjecture for affine hyperbolic curves over Kummer-faithful fields | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians Let \(f: X \to Z\) be a surjective morphism between normal complex projective varieties with connected fibers. A real Cartier divisor on \(X\) is said \(f\)-numerically trivial if its intersection with every vertical curve class vanishes. (Recall that a curve class is vertical if its intersection with \(f^*H\) vanishes for some ample \(H\) on \(Z\).) The main theorem in this paper (see Thm. 1.2) states the equivalence, on a real Cartier divisor of \(X\), of being \(f\)-numerically trivial and numerically equivalent to the pullback of a real Cartier divisor on \(Z\), \(Z\) being \(\mathbb{Q}\)-factorial. Moreover, if the numerical triviality is asked only for the general fiber, then one can find a birational model of the setup (see Thm. 1.3) for which the \textit{movable part} in the (Nakayama) Zariski decomposition of the divisor is that of a pullback, \(Z\) is asked to be integral. numerical triviality; surjective morphisms; pullbacks Divisors, linear systems, invertible sheaves Numerical triviality and pullbacks | 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 subject of the paper under review is the cohomology of the moduli space \({\mathcal H}_{g,n}\) of smooth hyperelliptic curves of genus \(g\) with \(n\) distinct marked points. The action of the symmetric group \(S_n\) by permuting the marked points endows both the Betti and the \(\ell\)-adic cohomology with a natural structure of \(S_n\)-representation that respects mixed Hodge structures, respectively, the structure as Galois representation. In this paper, the \(S_n\)-equivariant Euler characteristic of the (étale, resp., Betti) cohomology of \({\mathcal H}_{g,n}\) in the Grothendieck group of rational Hodge structures (respectively, Galois representations) is computed for all \(g\) and for \(n\leq 7\).
This result is achieved by performing an \(S_n\)-equivariant count of the number of points of the moduli space \({\mathcal H}_{g,n}\) defined over finite fields. This approach leads to the discovery that the number of points of \({\mathcal H}_{g,n}\) satisfies recurrence relations, so that the \(S_n\)-equivariant count of points of \({\mathcal H}_{g,n}\) for \(n\) fixed and \(g\) small determines the \(S_n\)-equivariant count of points for all \(g\). In particular, for \(n\leq 7\) the formulas for the \(S_n\)-count of points of \({\mathcal H}_{g,n}\) are obtained starting from known results for genus \(0\) and \(1\). In all these cases, the count of points gives a polynomial in the number elements of the field. For \(n\) small, these polynomials are independent of the characteristic of the finite field; the dependence starts for \(n=6\). The results on the count of points determine the \(S_n\)-equivariant Euler characteristic of \(\ell\)-adic cohomology by the Lefschetz trace formula. The corresponding result for the Betti cohomology of the complex moduli space follows from comparison theorems.
As an application, the author computes the cohomology of the moduli space of stable curves of genus \(2\) and \(n\) marked points with \(n\leq 7\). This extends the results of \textit{E. Getzler} [in: Integrable systems and algebraic geometry. Proceedings of the 41st Taniguchi symposium, Kobe, Japan, June 30-July 4, 1997, and in Kyoto, Japan, July 7-11, 1997. Singapore: World Scientific. 73--106 (1998; Zbl 1021.81056)] for \(n\leq 3\). Furthermore, if \(g\) is sufficiently large, there is a uniform description of the part of sufficiently high weight of the Euler characteristic of the \(\ell\)-adic cohomology of \({\mathcal H}_g\) with coefficients in certain local systems, which suggests the existence of stabilization phenomena in the cohomology. cohomology of moduli spaces of curves; curves over finite fields; hyperelliptic curves J. Bergström, Equivariant counts of points of the moduli spaces of pointed hyperelliptic curves, Doc. Math. 14 (2009), 259-296. Families, moduli of curves (algebraic), Curves over finite and local fields, Transcendental methods, Hodge theory (algebro-geometric aspects) Equivariant counts of points of the moduli spaces of pointed hyperelliptic 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 author shows that in the ring of correspondences of a Fermat (resp. Artin-Schreier) curve, Frobenius can be identified, in a suitable sense, with a Jacobi (resp. a Gauss) sum. This gives a strikingly simple interpretation of Weil's results on the eigenvalues of Frobenius on the \(\ell\)-adic cohomology of those curves.
The author then deduces from the same identification a very short proof, in the case of Fermat curves of the Brumer-Stark conjecture for function fields, proven in general by P. Deligne [Cf. \textit{J. Tate}, ``Les conjectures de Stark sur les fonctions L d'Artin en \(s=0\)'', Prog. Math. 47 (1984; Zbl 0545.12009)].
The case of Artin-Schreier curves can be treated similarly. The proof is analogous to Stickelberger's proof of Stickelberger's theorem.
This paper is a jewel of clarity.
We should point out a few misprints on page 465 of the text:
line 19, \(``(u,v)\to u^ m\)'' instead of ``(u,v)\(\to u\)''; line 19 and 24, \(``G_ m\)'' instead of ``G''; line 20, \(``G_ m\)'' instead of \(``F_ m\)''; line 25, ``image'' instead of ``kernel''. Fermat curves; Brumer-Stark conjecture for function fields; Artin- Schreier curves Coleman, R.: On the Frobenius endomorphisms of the Fermat and Artin--Schreier curves. In: The Arithmetic of Function Fields. Proc. Amer. Math Soc, vol. 102, pp. 463--466 (1988) Arithmetic ground fields for curves, Finite ground fields in algebraic geometry, Zeta functions and \(L\)-functions of number fields, Local ground fields in algebraic geometry, Zeta functions and related questions in algebraic geometry (e.g., Birch-Swinnerton-Dyer conjecture) On the Frobenius endomorphisms of Fermat and Artin-Schreier 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 This is a very well written survey of the article ``Families of rationally simply connected varieties over surfaces and torsors for semisimple groups'', by \textit{A. J. de Jong} et al. [Publ. Math., Inst. Hautes Étud. Sci. 114, 1--85 (2011; Zbl 1285.14053)].
Let \(f: X\to B\) be a morphism of finite type between two schemes with \(B\) integral, and let \(\eta\in B\) be the generic point. Then there is a one to one correspondence between the set of rational sections of \(f\) and the set of rational points of the generic fibre \(X_{\eta}\) over the function field \(K(B)\). This provides a link between the geometric problem of finding rational sections for a surjective morphism of smooth projective varieties and the arithmetic problem of finding solutions for ploynomial equations in a non-algebraically closed field.
The central theme of this paper is to find rational sections for surjective morphism \(f: X\to B\) between smooth projective varieties over an algebraically closed field \(k\). When \(B\) is a curve and the generic fibre \(X_{\eta}\) is separably rationally connected, then a section exists (when \(\dim(B)=1\), a rational section extends automatically to a section). This is a theorem which was first proved by \textit{T. Graber} et al. [J. Am. Math. Soc. 16, No. 1, 57--67 (2003; Zbl 1092.14063)] for characteristic 0 and then by \textit{A. J. de Jong} and \textit{J. Starr} [Am. J. Math. 125, No. 3, 567--580 (2003; Zbl 1063.14025)] for positive characteristic. When \(B\) is surface and \(k\) is of characteristic 0, the main result of [Zbl 1285.14053] provides conditions on the generic fibre of \(f: X\to B\), where \(X\) is equipped with a relatively ample line bundle, to guarantee the existence of a rational section of \(f\).
Because of the link between rational sections and rational points of the generic fibre, the main theorem of [Zbl 1285.14053] is applied to Serre's conjecture on the vanishing of the first Galois cohomology for simply connected semi-simple algberaic groups over a field of cohomological dimension less or equal to 2. The conjecture is proved in the case when the field is the function field of a surface. Although the link between rational sections and rational points is quite natural, it is still very surprising to see that a highly geometric result in characteristic 0 could be applied to such an algebraic problem in arbitrary characteristic. Serre's conjecture; rational sections; rational points Rational points, Families, fibrations in algebraic geometry, Surfaces and higher-dimensional varieties Sections rationnelles de fibrations sur les surfaces et conjecture de 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 The Brill-Noether-Petri theory governs the behavior of line bundles on algebraic curves. In the last three decades there have been a number of proofs for the Brill-Noether theorem and Petri theorem, using degeneration [\textit{D. Eisenbud} and \textit{J. Harris}, Invent. Math. 74, 269--280 (1983; Zbl 0533.14012); \textit{D. Gieseker}, Invent. Math. 66, 251--275 (1982; Zbl 0522.14015); \textit{P. Griffiths} and \textit{J. Harris}, Duke Math. J. 47, 233--272 (1980; Zbl 0446.14011)], curves on \(K3\) surfaces [\textit{R. Lazarsfeld}, J. Differ. Geom. 23, 299--307 (1986; Zbl 0608.14026)], or tropical geometry [\textit{F. Cools} et al., Adv. Math. 230, No. 2, 759--776 (2012; Zbl 1325.14080); \textit{D. Jensen} and \textit{S. Payne}, Algebra Number Theory 8, No. 9, 2043--2066 (2014; Zbl 1317.14139)].
In this paper, the authors study du Val curves on the blowup of \(\mathbb P^2\) which arise from a plane curve of degree \(3g\) having a \(g\)-tuple point at eight points, a \((g-1)\)-tuple at another point, and no other singularities. They show that a general such curve is a Brill-Noether-Petri general curve of genus \(g\). The authors provide two proofs. The first one uses \textit{M. Nagata}'s results [Mem. Coll. Sci., Univ. Kyoto, Ser. A 33, 271--293 (1960; Zbl 0100.16801)] and the second one uses \textit{D. Eisenbud} and \textit{J. Harris}' limit linear series [Invent. Math. 74, 269--280 (1983; Zbl 0533.14012)]. As a consequence, the authors give explicitly smooth Brill-Noether-Petri general curves of genus \(g\) defined over \(\mathbb Q\) for every \(g\). Brill-Noether theory; moduli of curves; surfaces with canonical sections Arbarello, E; Bruno, A; Farkas, G; Saccà, G, Explicit brill-Noether-Petri general curves, Comment. Math. Helv., 91, 477-491, (2016) Plane and space curves, Families, moduli of curves (algebraic), Special divisors on curves (gonality, Brill-Noether theory) Explicit Brill-Noether-Petri general 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 This paper is an expanded version of the note with the same title in [C. R., Math., Acad. Sci. Paris 346, No. 9--10, 491--494 (2008; Zbl 1197.11066)]. The main theorem asserts that there are at most finitely many complex numbers \(\lambda\neq0,1\) such that the points \((2, \sqrt{2(2-\lambda)})\) and \((3,\sqrt{6(3-\lambda)})\) have both finite order in the elliptic curve \(Y^2=X(X-1)(X-\lambda)\). The outline of the proof has already been described in the review of the mentioned note. Besides the detailed proof of the theorem, the paper includes a final section devoted to describe the connections of scheme-theoretic versions of the theorem with a conjecture due to Zilber regarding intersections on semiabelian schemes over \(\mathbb{C}\). The conjecture implies the theorem in the article, and combined with the techniques introduced by the authors should allow to prove similar results or even some Manin-Mumford type results for abelian varieties over curves. torsion points; elliptic curves; Mumford-Manin theorem; Zilber conjecture D. Masser and U. Zannier, ''Torsion anomalous points and families of elliptic curves,'' Amer. J. Math., vol. 132, iss. 6, pp. 1677-1691, 2010. Elliptic curves over global fields, Rational points, Real-analytic and semi-analytic sets Torsion anomalous points and families of elliptic 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 paper details a new construction of the quot scheme of Grothendieck. Quotient schemes are a fundamental classification tool in algebraic geometry. Several constructions of the scheme have been given (e.g. Altman and Kleiman) but typically one must rely on embeddings into Grassmannians of impractically large dimensions dependent on (Castelnuovo-Mumford) regularity. This paper instead gives (local) equations in relatively low degree. This generalizes an earlier construction of the Hilbert scheme of \(n\) points in a fixed scheme \(X\) by the authors
[J. Pure Appl. Algebra 210, No. 3, 705--720 (2007; Zbl 1122.14004)] as well as the standard description of the Grassmann scheme. As an example a description of quot schemes over the affine line is worked out. Hilbert schemes; quotient schemes Gustavsen, Trond Stølen; Laksov, Dan; Skjelnes, Roy Mikael: An elementary, explicit, proof of the existence of quot schemes of points. Pacific J. Math. 231, No. 2, 401-415 (2007) Algebraic moduli problems, moduli of vector bundles, Schemes and morphisms An elementary, explicit, proof of the existence of Quot schemes of 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 Let \(p:Z\to X\) and \(q:Z\to Y\) be two morphisms in the category of schemes over a base scheme \(S\). Whereas a pull-back (or fiber product) diagram for the pair \((p,q)\) in this category always exists, the dual construction of a push-out diagram
\[
\begin{tikzcd} Z\rar["p"]\dar["q" '] & X\dar["f"]\\Y \rar["g" '] & T \end{tikzcd}
\]
for \((p,q)\), with the according universal property, may not exist at all. In the paper under review, the author studies the question of under which particular conditions, with respect to the given data \((p,q)\), the existence of a push-out diagram can still be established. Quotients by groupoids (or more specifically, quotient spaces modulo group scheme actions) often provide special cases of existing push-out diagrams, and these play a crucial role in both the construction and the geometric study of various moduli spaces in algebraic geometry. This very fact not only motivates the author's more general investigation concerning the existence of push-out diagrams for morphisms of schemes, but it also determines his strategic approach to the problem. Indeed, using the construction of quotients by groupoids [cf.: \textit{S. Keel} and \textit{S. Mori}, Ann. Math. (2) 145, No. 1, 193--213 (1997; Zbl 0881.14018)], he establishes necessary and sufficient conditions for the existence of a push-out in several special cases of morphisms over a noetherian base scheme \(S\), including the flat protective case, the finite normal case, and the so-called ``partially flat case''. This very subtle and detailed analysis is enhanced by numerous examples illustrating the existence or non-existence of a push-out in a concrete situation, on the one hand, and the connection of existing push-out schemes with other basic constructions on the other. Also, the underlying categorical framework is thoroughly developed in the first part, which makes the entire paper totally self-contained. schemes and morphisms; group actions on schemes; quotients; moduli spaces; push-out diagrams; groupoids; categories and geometry Schemes and morphisms, Group actions on varieties or schemes (quotients), Fine and coarse moduli spaces, Groupoids, semigroupoids, semigroups, groups (viewed as categories), Limits and colimits (products, sums, directed limits, pushouts, fiber products, equalizers, kernels, ends and coends, etc.), Categories in geometry and topology Push-out of 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 the theory of elliptic curves \(C\) over a field \(K\), the so-called minimal discriminant ideal \({\mathfrak D}_{C/K}\), associated to a minimal Weierstrass equation for \(C\), is an important (classical) tool for geometric and arithmetic investigations.
In the present paper the author constructs an analogue of the minimal discriminant for a wider class of algebraic curves, namely for pointed hyperelliptic curves of arbitrary genus \(g \geq 1\). More precisely, let \(C\) be a hyperelliptic curve over a field \(K\), and let \(P\) be one of the \(2g + 2\) \(K\)-rational Weierstrass points of \(C\) (provided that \(\text{char} (K) \neq 2)\). Then the pair \((C,P)\) is called a pointed hyperelliptic curve over \(K\).
In the first section of the paper, the author defines the notion of a Weierstrass equation for such an object and the so-called discriminant polynomial of that Weierstrass equation. It turns out that this naturally defined discriminant (polynomial) gives the usual discriminant in the case of \(g = 1\) and, in general, that the hyperelliptic curve \(C\) is singular if and only if its discriminant vanishes.
In Section 2 the discriminant is studied for pointed hyperelliptic curves over local fields and global fields. The results obtained here are essentially natural generalizations of those well-known for arithmetic elliptic curves, as far as the group structure is not involved. In particular, the concept of minimal discriminants of elliptic curves is generalized to the hyperelliptic case, and a reduction theory with respect to finite sets of places of \(K\) is developed.
Section 3 deals with the analytic case, i.e., with hyperelliptic curves over \(\mathbb{C}\). The main result here shows that the hyperelliptic discriminant can be explicitly expressed in terms of Siegel modular forms, again like in the classical case of elliptic discriminants.
The concluding Section 4 is devoted to examining a hyperelliptic analogue of a conjecture by \textit{L. Szpiro} concerning elliptic curves over number fields and their discriminants [cf. Astérisque 86, 44-78 (1981; Zbl 0517.14006)]. The author shows, after having generalized Szpiro's conjecture to minimal hyperelliptic discriminants, that the famous ABC Conjecture of \textit{D. W. Masser} and \textit{J. Oesterlé} [cf., e.g. \textit{P. Vojta}, Diophantine approximations and value distribution theory (Lect. Notes Math. 1239), pp. 84-88 (Springer 1980; Zbl 0609.14011)] implies the author's generalization of Szpiro's conjecture to hyperelliptic discriminants.
Altogether, this paper provides an important generalization of some significant parts of the extensive theory of elliptic curves. minimal discriminant ideal; pointed hyperelliptic curves; Weierstrass points; Weierstrass equation; discriminant polynomial; local fields; global fields; Szpiro's conjecture; ABC Conjecture Lockhart, P., \textit{on the discriminant of a hyperelliptic curve}, Trans. Amer. Math. Soc., 342, 729-752, (1994) Curves of arbitrary genus or genus \(\ne 1\) over global fields, Arithmetic ground fields for curves, Elliptic curves, Global ground fields in algebraic geometry On the discriminant of a hyperelliptic curve | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians Let \(U\) be a smooth and geometrically connected hyperbolic curve over a finite field \(k\). Grothen\-dieck's philosophy of anabelian geometry states that the isomorphism type of \(U\) as a scheme should be encoded in the isomorphism type of \(\pi^{\text{ét}}_1(U)\) as a profinite group. This was proved to be true by Tamagawa (in the affine case) and Mochizuki (in the proper case). However, at present there is no example of a curve \(U\) for which the structure of \(\pi_1(U)\) is known. Hence, it is natural to search for better understood quotients of \(\pi_1(U)\) that still encode the isomorphy type of \(U\). The first half of the present paper is a survey of recent results concerning this search, while the final section presents a new proof of the authors' birational prime-to-characteristic result, which they established by different means in their previous paper [[1]: Publ. Res. Inst. Math. Sci. 45, No. 1, 135--186 (2009; Zbl 1188.14016)].
Section 1 of the paper reviews the Isom-form of Grothendieck's anabelian conjecture, which states that any isomorphism between étale fundamental groups of hyperbolic curves over finite fields arises from a unique isomorphism of schemes. As mentioned above, this conjecture was proved by \textit{A. Tamagawa} [Compos. Math. 109, No. 2, 135--194 (1997; Zbl 0899.14007)] and \textit{S. Mochizuki} [J. Math. Kyoto Univ. 47, No. 3, 451--539 (2007; Zbl 1143.14305)] in the affine and proper cases, respectively. This result implies the birational analogue concerning isomorphisms between Galois groups of function fields, due originally to \textit{K. Uchida} [Ann. Math. (2) 106, 589--598 (1977; Zbl 0372.12017)].
After stating these older results, the authors describe their improvements in [1], where the full fundamental groups (resp. Galois groups) are replaced by the geometrically prime-to-characteristic quotients. Finally, they present a further refinement of these theorems to the case of geometrically pro-\(\Sigma\) fundamental groups (resp. Galois groups), where \(\Sigma\) is a sufficiently large set of primes distinct from the characteristic. Here ``sufficiently large'' is defined by a technical condition involving the non-injectivity of certain Galois-representations, but as a simple example, all cofinite sets of primes are ``sufficiently large''. At the other extreme, the authors stress the importance of investigating the case of \(\Sigma=\{l\}\), i.e. the question of whether the Isom-form of the Grothendieck conjecture holds for the geometric pro-\(l\) quotients, where \(l\) is a prime distinct from the characteristic.
Section 2 is devoted to a discussion of the difficulties in proving a Hom-form of the Grothendieck conjecture, which states that any continuous open homomorphism between étale fundamental groups of hyperbolic curves over finite fields arises from a unique generically étale morphism of schemes. These difficulties center on the lack of a good ``local theory'' for such homomorphisms. Motivated by these reflections, the authors describe a restricted class of homomorphisms with good local properties and then a recently obtained theorem in the birational case (resp. conjecture in the proper case) asserting the existence of a Hom-form for this restricted class of homomorphisms between Galois groups (resp. between fundamental groups).
The final section of the paper (Section 3) gives a new proof of the prime-to-characteristic version of Uchida's birational theorem mentioned in Section 1. In [1], this result was derived as a corollary to the prime-to-characteristic version of the Tamagawa-Mochizuki Isom-form for anabelian curves; here the result is proved directly using class field theory in a manner inspired by Uchida's original proof. anabelian geometry; fundamental group; hyperbolic curve Saïdi, Mohamed; Tamagawa, Akio, On the anabelian geometry of hyperbolic curves over finite fields.Algebraic number theory and related topics 2007, RIMS Kôkyûroku Bessatsu, B12, 67-89, (2009), Res. Inst. Math. Sci. (RIMS), Kyoto Curves over finite and local fields, Finite ground fields in algebraic geometry, Coverings of curves, fundamental group, Arithmetic ground fields for curves On the anabelian geometry of hyperbolic curves over finite 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 The authors generalize the classical Albanese map from smooth projective to smooth quasi-projective varieties and they extend a theorem of \textit{A. A. Rojtman} [Ann.\ Math.~(2)~111, 553--569 (1980; Zbl 0504.14006)] and a theorem of \textit{K. Kato} and \textit{S. Saito} [Ann.\ Math.~(2)~118, 241--275 (1983; Zbl 0562.14011)] to their generalized Albanese map.
Let \(X\) be a quasi-projective smooth variety over an algebraically closed field \(k\) of characteristic \(p \geq 0\). While the authors use the degree zero part \(h_0(X)^0\) of A. Suslin's \(0\)th algebraic singular homology group [\textit{A. Suslin} and \textit{V. Voevodsky}, Invent. Math.~123, No.~1, 61--94 (1996; Zbl 0896.55002)] as source for their generalized Albanese map, they use the generalized Albanese variety \(\text{Alb}_X\) introduced by \textit{J.-P. Serre} [in: Variétés de Picard, Sem.\ C.\ Chevalley~3 (1958/59), No. 10 (1960; Zbl 0123.13903)] as target. If \(X\) admits a smooth compactification, they show that the generalized Albanese map induces an isomorphism on prime-to-\(p\) torsion subgroups (proved by Rojtman if \(X\) is projective) and, if moreover \(k\) is the algebraic closure of a finite field, that the generalized Albanese map is an isomorphism of torsion groups (proved by Kato and Saito again if \(X\) is projective).
The authors' approach is more conceptual than the classical one. Their proof exploits on the one hand the comparison isomorphisms \(h^i(X,{\mathbb Z}/n{\mathbb Z}) \rightarrow H^i_{\text{ét}}(X,{\mathbb Z}/n{\mathbb Z})\) according to \textit{A. Suslin} and \textit{V. Voevodsky} [loc. cit.] and on the other hand the ``tamely ramified class field theory'' developed by \textit{A. M. Schmidt} and \textit{M. Spieß} [J. Reine Angew. Math. 527, 13--36 (2000; Zbl 0961.14013)]. algebraic singular homology; comparison isomorphisms; tamely ramified class field theory Spieß, Michael; Szamuely, Tamás, On the Albanese map for smooth quasi-projective varieties, Math. Ann., 325, 1, 1-17, (2003) Motivic cohomology; motivic homotopy theory, Algebraic cycles, Geometric class field theory On the Albanese map for smooth quasi-projective 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 1972, \textit{J.-P. Serre} [Invent. Math. 15, 259--331 (1972; Zbl 0235.14012)] proved that given a non CM elliptic curve \(E\) defined over a number field \(K\), the Galois group of the field of rationality of the \(l\)-torsion points of \(E\) \((l\) prime) is as big as possible for \(l>l_ 0 (K)\) large enough, i.e. \(\simeq \text{GL}_ 2 (\mathbb F_ l)\). However, except in the case \(K = \mathbb Q\), no effective upper bound for \(l_ 0(K)\) was known (although Serre did announce such an estimate in an oral talk in Paris, 1988).
The object of the paper under review is to provide such a bound. The authors rely on a series of previous papers (in fact a real program), they started with a crucial though rather technical estimate on periods of abelian varieties [Ann. Math. (2) 137, No. 2, 407--458 (1993; Zbl 0796.11023)]. This estimate, which controls the degree of the smallest abelian subvariety containing a prescribed period of a given abelian variety \(A\) in its tangent space in terms of the Faltings height of \(A\), enables them to get a bound on the degree of a minimal isogeny linking two abelian varieties \(A\) and \(B\) (say principally polarized) in terms of the Faltings height of, say \(A\) [Ann. Math. (2) 137, No. 3, 459--472 (1993; Zbl 0804.14019)]. The first step is then to use Zarkhin's trick to get a more technical but partially unpolarized isogeny estimate. The two following sections of the paper make use of the list of possible subgroups for \(\text{GL}_ 2 (\mathbb F_ l)\) to systematically eliminate all subgroups not containing \(\text{SL}_ 2 (\mathbb F_ l)\). The isogeny estimates take care of Borels and unsplit Cartans. The proof can then be completed following Serre's paper (loc. cit.) and one gets a bound of the form \(l_ 0<c \max \{d, h(j_ E)\}^ \gamma\), where \(c\) and \(\gamma\) are universal constants, \(d\) is the degree of \(K/ \mathbb Q\), and \(h(j_ E)\) is the Weil height of the \(j\)-invariant of the curve \(E\). The exponent \(\gamma\) is not given, and is certainly large. However, it is expected that a ``customized'' approach would give a reasonable exponent. On the other hand, the constant \(c\) is ineffective. This comes from the fact that the main period estimate itself involves some constants in the dimension of the abelian variety whose existence is given by compactness arguments. Until these are replaced, this theorem wouldn't yield an algorithm usable on a computer.
The two last sections of the paper are devoted to analogs of the main result on products of (non CM) elliptic curves and to Kummer theory, i.e. Galois properties of division points of a finite set of linearly independent points of infinite order of a (non CM) elliptic curve, extending results of \textit{M. I. Bashmakov} [Russ. Math. Surv. 27(1972), 25--70 (1973); translation from Usp. Mat. Nauk 27, No. 6(168), 25--66 (1972; Zbl 0256.14016)] and \textit{D. Bertrand} [Glasg. Math. J. 22, 83--88 (1981; Zbl 0453.14019)]. effectivity; \(l\)-torsion points; non CM elliptic curve; Galois group; field of rationality; periods of abelian varieties; Faltings height; minimal isogeny; Kummer theory Masser, D.; Wüstholz, G., Galois properties of division fields of elliptic curves, Bull. Lond. Math. Soc., 25, 247-254, (1993) Elliptic curves, Isogeny, Rational points, Elliptic curves over global fields, Abelian varieties of dimension \(> 1\) Galois properties of division fields of elliptic 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 author gives a readable survey of recent progress of so-called anabelian algebraic geometry, proposed by \textit{A. Grothendieck} in his mimeographed note [1984, to appear in ``Geometric Galois actions: Around Grothendieck's esquisse d'un programme'', LMS]. After reviewing Grothendieck's original ideas and conjectures (Section 1), the author discusses the relation between anabelian geometry and hyperbolic geometry (Section 2), reviews the `coordinate' method, due to Y. Ihara and others, of analyzing pro-\(\ell\) fundamental groups and pro-\(\ell\) mapping class groups of algebraic curves, and its applications (Section 3), and summarizes recent developments on the Galois theory of moduli spaces of algebraic curves, mainly due to the author (Section 4).
The English translation by the author will appear in Sugaku Exposition and contains added notes on various developments after the original manuscript was submitted. Among other things, the above conjectures of Grothendieck (for hyperbolic curves) were finally settled by S. Mochizuki. coordinate method; pro-\(\ell\) fundamental groups; pro-\(\ell\) mapping class groups; anabelian algebraic geometry; hyperbolic geometry; algebraic curves; Galois theory of moduli spaces Hiroaki Nakamura, Galois rigidity of profinite fundamental groups, Sūgaku 47 (1995), no. 1, 1 -- 17 (Japanese). Separable extensions, Galois theory, Coverings of curves, fundamental group, Galois theory, Families, moduli of curves (algebraic) Galois rigidity of profinite fundamental groups | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians This paper is concerned with relative Brauer groups \(\text{Br}(X/Y)\), which is defined as the kernel of the pullback map from the Brauer group of \(Y\) to the Brauer group of an \(Y\)-scheme \(X\).
The case of primary interest is that \(Y\) is the spectrum of a field \(k\) and \(X\) is a smooth projective curve of genus one, which thus is a homogeneous space over an elliptic curve \(E\), the jacobian of \(X\). The main result is then that if the homogeneous space is nontrivial, that is \(X(k)=\emptyset\), then the relative Brauer group \(\text{Br}(X_{k(E)}/k(E))\) is nontrivial as well. In other words, the absence of rational points is detected by relative Brauer groups after suitable field extensions.
The bulk of the paper contains a careful discussion of various exact sequences and pairings relating groups of units, divisors, invertible sheaves and homogeneous spaces with Brauer groups. The last section also gives a computational description of the relative Brauer groups for suitable curves of genus one. Relative Brauer groups Çiperiani, Mirela; Krashen, Daniel: Relative Brauer groups of genus 1 curves. Israel J. Math. 192, No. 2, 921-949 (2012) Brauer groups of schemes, Elliptic curves over global fields, Finite-dimensional division rings, Brauer groups (algebraic aspects) Relative Brauer groups of genus 1 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 Denote by \(M_g\) the coarse moduli space of smooth complex curves of genus \(g\geq 2\). Let \(\overline M_g\) be its Deligne-Mumford compactification, i.e., the moduli space of stable genus-\(g\) curves over \(\mathbb{C}\). The construction and investigation of these moduli spaces can be done from different viewpoints, as the long and fascinating history of moduli of curves has shown. Essentially, geometric invariant theory and complex-analytic Teichmüller theory are the fundamental methods to approach them, and the combination of these frameworks has been proved particularly efficient. In the present paper, the author gives alternative proofs of six well-known and very important results concerning families of stable curves and the geometry of the moduli spaces \(M_g\) and \(\overline M_g\). These results, obtained in the past by several people and by quite different methods, are here derived by a unified approach, which makes their interrelation within the theory as a whole more apparent. More precisely, using more recent, very deep and general results on the differential geometry of the Teichmüller spaces \(T_g\) and the moduli spaces \(M_g\), the author obtains fairly easy and elementary proofs of the following well-known theorems:
(1) For any family \(f:X\to B\) of stable curves of genus \(g >1\), the Hodge bundle is strictly positive if the family is non-isotrivial. Here \(B\) is a projective curve by assumption.
(2) For such a family, the relative dualizing sheaf \(\omega_{X/B}\) satisfies the self-intersection inequality \((\omega^2_{X/B}) \leq(2g-2) (2q-2+s)\), where \(q\) is the genus of the base curve \(B\) and \(s\) is the number of singular fibres.
(3) If the family is non-isotrivial and \(B= \mathbb{P}^1\), then \(s\geq 3\); if \(B\) is an elliptic curve, then \(s\geq 1\).
(4) The Petersson-Weil metric on \(\overline M_g\) is a Kähler metric and its Kähler class is rational.
(5) The Deligne-Mumford moduli space \(\overline M_g\) is a projective variety.
(6) The Teichmüller space \(T_g\) is a domain of holomorphy.
The author's method of reproving these theorems is based upon S.-T. Yau's generalization of the classical Schwarz lemma to Kähler manifolds [\textit{S.-T. Yau}, Am. J. Math. 100, 197-203 (1978; Zbl 0424.53040)] and upon Quillen's metric theory for determinant bundles. As to the properties of Quillen metrics, the author utilizes the most general results obtained by \textit{J. M. Bismut} and \textit{J.-B. Bost} [Acta Math. 165, No. 1/2, 1-103 (1990; Zbl 0709.32019)], whereas the fundamental formulae for the Petersson-Weil form are basically due to \textit{S. A. Wolpert} [J. Differ. Geom. 31, No. 2, 417-427 (1990; Zbl 0698.53002)].
Altogether, this work is a nice contribution towards the systematization in complex geometry. As the author points out, his method can be also applied to the study of other moduli spaces, e.g., to moduli spaces of principally polarized abelian varieties. moduli of curves; stable curves; Teichmüller spaces; Hodge bundle; dualizing sheaf; Petersson-Weil metric; Schwarz lemma; determinant bundles; Quillen metrics Liu, K., Remarks on the geometry of moduli spaces, Proc. Amer. Math. Soc., 1996, 124(3): 689--695. Families, moduli of curves (analytic), Moduli of Riemann surfaces, Teichmüller theory (complex-analytic aspects in several variables), Families, moduli of curves (algebraic), Algebraic moduli problems, moduli of vector bundles Remarks on the geometry 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 For a connected reduced scheme X over an arbitrary field k the fundamental group scheme is constructed and studied (which is a group scheme over k) and its connection with principal G-fiberings, where G is a finite (or nilpotent) group scheme, and finite linear fiberings on X is investigated.
For X and its fixed k-point \(\chi_ 0:\quad Spec k\to X\) a triple \((P,\pi (X,\chi_ 0),*)\) is constructed (where \(\pi (X,\chi_ 0)\) is the group k-scheme which is the projective limit of finite group schemes, P is the principal \(\pi (X,\chi_ 0)\) fibering over X with marked point * lying over \(\chi_ 0)\) which has the universality property with respect to principal G-fiberings on X, where G is a finite group k-scheme with marked point lying over \(\chi_ 0\). The group k-scheme \(\pi (X,\chi_ 0)\) is called the fundamental group scheme of the scheme X and possesses many natural properties.
The case of a complete scheme X over a perfect field k was considered by the author earlier and is connected with finite and essentially finite vector bundles over X which are multidimensional generalizations of one- dimensional bundles corresponding to the finite order points on the Jacobian of a curve [the author, Compos. Math. 33, 29-41 (1976; Zbl 0337.14016)]. Owing to the consideration of non-complete schemes the author carries over these results to linear, parabolic (defined by Seshadri) essentially finite bundles over a smooth projective curve X with a finite set S of points deleted. Such bundles are constructed by k- linear representations of the group \(\pi (X-S,\chi_ 0)\), where \(\chi_ 0\not\in S\), which are passed through representations of finite group schemes over k.
The last chapter is devoted to the study of a nilpotent fundamental group-scheme \(U(X,\chi_ 0)\) (it is constructed if \(\Gamma\) (X,\({\mathcal O}_ X)=k)\). If char k\(>0\) and \(\dim H^ 1(X,{\mathcal O}_ X)<\infty\) then \(U(X,\chi_ 0)\) is a factor of \(\pi (X,\chi_ 0)\). The connection of \(U(X,\chi_ 0)\) with Pic X is studied. In the case of complete reduced curves and \(p=char k>0\), the computation of \(U(X,\chi_ 0)\) leads to non-commutative formal groups which are computed by the author for rational curves with the simplest singularities. As a corollary, an old result of I. R. Shafarevich is reproved: For a complete curve X with \(\Gamma\) (X,\({\mathcal O}_ X)=k\), the maximal p-factor of the étale fundamental group is a free pro-p-group in characteristic p.
The main means used by the author are the equivalence between the Tannaka category and the category of finite-dimensional representations of a certain affine group scheme [see \textit{N. R. Saavedra}, ''Catégories tannakiennes'', Lect. Notes Math. 265 (1972; Zbl 0241.14008)]. finite-dimensional representations of affine group scheme; fundamental group scheme; non-commutative formal groups; characteristic p Nori, M. V., \textit{the fundamental group-scheme}, Proc. Indian Acad. Sci. Math. Sci., 91, 73-122, (1982) Coverings in algebraic geometry, Group schemes, Homotopy theory and fundamental groups in algebraic geometry, Schemes and morphisms 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 \(k\) be a perfect field of characteristic \(p\) and let \(Y\) be a curve with function-field \(k(Y)\) of genus \(g_ Y\geq 1\). \textit{H. Hasse} and \textit{E. Witt} [Monatsh. Math. Phys. 43, 477--492 (1936; Zbl 0013.34102)] showed that the number of geometric, unramified extensions of \(k(Y)\) of degree divisible by \(p\) is \(p^{\gamma(Y)}\) where the integer \(\gamma(Y)\) is the rank of the `Hasse-Witt matrix'. \textit{J.-P. Serre} showed that that matrix is the Cartier operator belonging to the holomorphic differentials \(\Omega'(k(Y))\) [see Sympos. internac. Topologia algebraica, 24--53 (1958; Zbl 0098.13103)]. For \(k\) finite, \textit{Yu. I. Manin} [Transl., II. Ser., Am. Math. Soc. 45, 245--264 (1965); translation of Izv. Akad. Nauk SSSR, Ser. Mat. 25, 153--172 (1961; Zbl 0102.27802)] showed that the characteristic polynomial of the Hasse-Witt matrix is essentially the zeta-function of \(Y\), reduced mod \(p\). It follows that \(\gamma (Y)\) is the degree of that polynomial and, by class field theory, it is also the rank of the group \(C^ 0(k(Y))(p)\) of points of the group of classes of divisors of degree 0, whose orders are divisible by \(p\).
In this interesting dissertation, the author investigates a generalisation of the foregoing, as follows. Let \(Y/X\) be a Galois covering of curves, \(G=\mathrm{Gal}(Y/X)\) and suppose that \(p\) does not divide the order of \(G\). The decomposition of the group-ring \(k[G]\) by characters \(\chi\) yields a decomposition of \(C^ 0(k(Y))(p)\) in terms of \(\chi\)-eigenspaces. The generalised Hasse-Witt invariants \(\gamma (Y,\chi)\) are the dimensions of the \(\chi\)-eigenspaces and \(\sum_{\chi}(Y,\chi)=\gamma(Y)\).
The other features of the foregoing theory are appropriately generalised: the zeta function of \(Y\) is replaced by the \(L\)-functions \(L(t,Y/X,\chi)\) of the covering \(Y/X\), expressed in terms of the \(p\)-adic representation of the endomorphism ring of the Jacobian \(J_ Y\) and the sheaf of Witt vectors. arithmetic theorem of algebraic function fields; L-function of Galois covering of curves; function-field; characteristic polynomial of the Hasse-Witt matrix; generalised Hasse-Witt invariants Cyclotomic function fields (class groups, Bernoulli objects, etc.), Arithmetic theory of algebraic function fields, Algebraic functions and function fields in algebraic geometry, Zeta functions and related questions in algebraic geometry (e.g., Birch-Swinnerton-Dyer conjecture), Galois theory Class groups and \(L\)-series of congruence function fields | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians [For the entire collection see Zbl 0611.00006.]
The author generalizes in the frame of coherent sheaves his earlier construction for vector bundles [in Number Theory, Algebraic Geometry, Commut. Algebra, in Honor of Y. Akizuki, 95-146 (1973; Zbl 0282.14002)]. Let X be a locally noetherian scheme, Y an effective Cartier divisor, E a coherent sheaf on X and F a coherent sheaf on Y. Assume we have an epimorphism \(\Psi: E\to F.\) Then the sheaf \(E'=Ker(\psi)\) is denoted \(elm_ F(E)\) and is called the elementary transformation of E along F. Let denote \(F'=\ker (\psi | Y)\) and \(\psi ': E'\to F'\) the induced morphism. Using an exact commutative diagram (which is called the display of the elementary transformation) we get the inverse transformation \(E(- Y)=elm_{F'}(E')\). Main result: P(E) and P(E') are (in a precise manner) birationally equivalent (under suitable conditions on the above sheaves). The case of vector bundles is just theorem 1.1. (loc.cit.) birational equivalence; Cartier divisor; coherent sheaf; display of the elementary transformation; vector bundles Maruyama, M., On a generalization of elementary transformations of algebraic vector bundles, Rend. Sem. Mat. Univ. Politec. Torino, 44, 1-13, (1986) Rational and birational maps, Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20] On a generalization of elementary transformations of algebraic vector bundles | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians Let C be a complete smooth curve of genus g over an algebraically closed field of characteristic zero and let \(Q(n,r,d)^ 0\) be the open subscheme parametrizing quotient bundles \(0_ c^ n\twoheadrightarrow E\) of rank r and degree d with \(h^ 1(E)=0\). The aim of the paper is to prove the following theorem: The Harder-Narasimhan strata [\textit{S. S. Shatz}, Compos. Math. 35, 163-187 (1977; Zbl 0371.14010)] in \(Q(n,r,d)^ 0\) given by the universal quotient bundle are irreducible and smooth.
This generalizes partially a previous work of \textit{J. L. Verdier} for the case \(g=0\) [in Group theoretical methods in physics, Proc. XIth internat. Colloq., Istanbul 1982; Lect. Notes Phys. 180, 136-141 (1983; Zbl 0528.58008)] and \textit{A. Bruguières} [''Le schéma des morphismes d'une courbe elliptique dans une grassmannienne'' (These, Paris 1984)]. The smoothness part of the statement is also a consequence of a theorem of \textit{A. Bruguières} [in Module des fibres stables sur les courbes algébriques, Notes Éc. Norm. Super., Paris 1983, Prog. Math. 54, 81- 104 (1984; Zbl 0577.14012)]. complete smooth curve; Harder-Narasimhan strata; universal quotient bundle R. Hernández, On Harder-Narasimhan stratification over Quot schemes , J. Reine Angew. Math. 371 (1986), 114-124. Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20], Homogeneous spaces and generalizations, Stratifications in topological manifolds On Harder-Narasimhan stratification over 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 Let \(M_g\) be the coarse moduli space of proper, smooth curves of genus \(g\) in characteristic \(p>0\). For \(x\in M_g\), let \(\pi_1(x)\) denote the étale fundamental group of the corresponding curve. Let \(\eta\) be the generic point of \(M_g\). For every \(x\in M_g\), fix a specialization homomorphism \(Sp_x: \pi_1(\eta) \to \pi_1(x)\). Denote by \(S \subset M_g\) the Zariski dense subset corresponding to curves having absolutely simple Jacobian and by \(T\subset S\) the set corresponding to curves with \(p\)-rank \(g\) or \(g-1\). The main results of the paper are the following.
Theorem. (1) For all \(s\in S\), the homomorphism \(Sp_s\) is not an isomorphism.
(2) If \(x\in M_g\) specializes to \(t\in T\) with \(x\neq t\), then the specialization homomorphism \(Sp_{x,t}: \pi_1(x) \to \pi_1(t)\) is not an isomorphism.
In particular, for a given \(t\in T\) , there exist only finitely many \(t'\in T\) such that \(\pi_1(t) \cong \pi_1(t')\).
Corollary. There is no nonempty open subset \(U \subset M_g\) such that the isomorphy type of the geometric fundamental group \(\pi_1(x)\) is constant on \(U\). étale fundamental group; curves in characteristic p F. Pop and M. Saïdi, On the specialization homomorphism of fundamental groups of curves in positive characteristic, in Galois groups and fundamental groups, Math. Sci. Res. Inst. Publ., 41 , Cambridge University Press, 2003, pp. 107-118. Coverings of curves, fundamental group, Arithmetic ground fields for curves, Matrices, determinants in number theory On the specialization homorphism of fundamental groups of curves 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 In the introduction, ``some of the numerous references'' for the past 4 decades came under criticism of the author because ``most of the techniques have focused mainly on detecting families of examples and their features...'' By contrast, the author's ``goal is to report the following surprising result, which had gone unnoticed and in fact turns out to be a new module-theoretic characterization of hypersurfaces''. More precisely, his main result is formulated as follows: the module of tangent vector fields to an affine algebraic variety over the field of characteristic zero is reflexive if and only if the variety is a hypersurface.
Reviewer's remark: It should be noted that this fact is a very particular case of the well-known statement concerning the behavior of reflexive sheaves on normal varieties (see, e.g., Corollary 1.5 in [\textit{R. Hartshorne}, Math. Ann. 254, 121--176 (1980; Zbl 0431.14004)]). More precisely, if \(\mathcal G\) is a subsheaf of a reflexive sheaf \(\mathcal F\) given on a normal variety, then \(\mathcal G\) is reflexive if and only if the set of associated primes of the quotient \(\mathcal F/\mathcal G\) consists of points of codimensions 0 and 1 only. reflexive modules; hypersurfaces; tangent vector fields; logarithmic vector fields; free divisors Hypersurfaces and algebraic geometry, Derivations and commutative rings, Relations with arrangements of hyperplanes, Structure, classification theorems for modules and ideals in commutative rings, Projective and free modules and ideals in commutative rings, Configurations and arrangements of linear subspaces, Divisors, linear systems, invertible sheaves, Complex vector fields, holomorphic foliations, \(\mathbb{C}\)-actions A module-theoretic characterization of algebraic hypersurfaces | 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 some generalisations of the classical Verdier hypercovering theorem. This theorem approximates the morphisms \([X,Y]\) in the category of simplicial sheaves and presheaves by simplicial homotopy classes of maps. In its standard form this approximation is given by the comparison function \(\lim_{[p] : Z \rightarrow X} {\pi}(Z,Y) \rightarrow [X,Y]\) defined by mapping the element
\[
\begin{tikzcd} X & Z \lar["{[p]}"]\rar["f"] & Y \end{tikzcd}
\]
to the morphism \(f\cdot p^{-1}\) . Here, \({\pi}(Z,Y )\) denotes simplicial homotopy classes of maps corresponding to a hypercover \(p : Z\rightarrow X\). The theorem asserts that the comparison function is an isomorphism provided \(X\) an \(Y\) are locally fibrant. Based on results developed by him in [Algebraic topology. The Abel symposium 2007. Proceedings of the fourth Abel symposium, Oslo, Norway, August 5--10, 2007. Berlin: Springer. Abel Symposia 4, 185--218 (2009; Zbl 1182.55006)] the author gives a new easier proof of the theorem. This new proof yields a pointed generalisation which does not require the assumption that \(X\) is fibrant. simplicial presheaf; hypercover; cocycle Jardine, The Verdier hypercovering theorem, Canad. Math. Bull. 55 pp 319-- (2012) Homotopy theory and fundamental groups in algebraic geometry, Simplicial sets, simplicial objects (in a category) [See also 55U10], Abstract and axiomatic homotopy theory in algebraic topology, Coverings of curves, fundamental group, Simplicial sets and complexes in algebraic topology The Verdier hypercovering 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 In this work, we give an algebro-geometric proof of Hrushovski's generalization of the Lang-Weil estimates on the number of points in the intersection of a correspondence with the graph of the Frobenius map. This result has numerous applications to various areas of mathematics, including model theory, algebraic dynamics, group theory and arithmetic algebraic geometry. intersection theory; finite fields; cohomological methods Positive characteristic ground fields in algebraic geometry, Intersection theory, characteristic classes, intersection multiplicities in algebraic geometry The Hrushovski-Lang-Weil estimates | 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 presents the elements of the singularity theory of analytic spaces with applications; it consists of a preface, two main parts, three appendices and a bibliography including 158 items among which are 18 references on works written by the authors with collaborators.
The first part deals with complex spaces and germs. It contains basic notions and results of the general theory such as the Weierstraß preparation theorem, the finite coherence theorem with applications, finite and flat morphisms, normalization, singular locus and relations with differential calculus. In addition, the cases of isolated hypersurface and plane curve singularities are treated. Thus the authors describe some well-known invariants of hypersurface singularities including the Milnor and Tjurina numbers and methods of their computation. They also discuss the concept of finite determinacy, the property of quasihomogeneity, algebraic group actions, the classification of simple singularities, the parameterization and resolution of plane curve singularities, the intersection multiplicity and the semigroup of values associated with a plane curve singularity, the conductor and other classical topological and analytic invariants.
The second part is concerned with local deformation theory of complex space germs. First the authors describe the general deformation theory of isolated singularities of complex spaces. Then the notions of versality, infinitesimal deformations and obstructions are considered in detail. The final section contains a new treatment of equisingular deformations of plane curve singularities including a proof for the smoothness of the \(\mu\)-constant stratum which is based on properties of deformations of the parametrization. This result is obtained, in fact, as a further development of ideas by \textit{J. M. Wahl} [Trans. Am. Math. Soc. 193, 143--170 (1974; Zbl 0294.14007)]. Three appendices include a detail description of basic notions and results from sheaf theory, commutative algebra and formal deformation theory.
The book is written in a clear style, almost all key topics are followed by carefully chosen computational examples together with algorithms implemented in the computer algebra system \textsl{Singular} [see \textit{G.-M. Greuel, G. Pfister} and \textit{H. Schönemann}, Singular 3. A computer algebra system for polynomial computations. Centre for Computer Algebra, Univ. Kaiserslautern (2005), \url{http://www.singular.uni-kl.de}]. Moreover, the exposition contains many non-formal comments, remarks and good exercises illustrated by nice pictures. Without a doubt this book is comprehensible, interesting and useful for graduate students; it is also very valuable for advanced researchers, lecturers, and practicians working in singularity theory, algebraic geometry, complex analysis, commutative algebra, topology, and in other fields of pure mathematics. complex spaces and germs; isolated hypersurface singularities; equisingular deformations; \(\mu\)-constant stratum; embedded deformations; plane curve singularities; parametrization; resolution; normalization; versal deformations; obstructions; cotangent complex G.-M. Greuel, C. Lossen, E. Shustin, \(Introduction to Singularities and Deformations\) (Springer, Berlin, 2007) Deformations of complex singularities; vanishing cycles, Research exposition (monographs, survey articles) pertaining to several complex variables and analytic spaces, Invariants of analytic local rings, Equisingularity (topological and analytic), Complex surface and hypersurface singularities, Modifications; resolution of singularities (complex-analytic aspects), Deformations of singularities, Singularities of surfaces or higher-dimensional varieties, Global theory and resolution of singularities (algebro-geometric aspects), Research exposition (monographs, survey articles) pertaining to algebraic geometry Introduction to singularities and deformations | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians Let \(J(C)\) be the Jacobian variety of a curve \(C\) of genus 2. The author defines a certain basic cycle \(Z(C)\) for the Quillen group \(H^1(J(C), {\mathcal K}_2)\) (depending on certain choices). This cycle is a higher \(K\)-theoretic analogue of the Ceresa cycle [\textit{G. Ceresa}, Ann. Math., II. Ser. 117, 285-291 (1983; Zbl 0538.14024)]. In section 3 an adjunction map associated to a certain (first-order) hyperelliptic deformation is studied. It is then used in the computation of the Griffiths' infinitesimal invariant (theorem 4.7). The first consequence is the nontriviality of \(Z(C)\). Another corollary is the Torelli-like property of the associated Griffiths' infinitesimal invariant \(\delta \nu(C)\), where \(\nu(C)\) denotes the class of \(Z(C)\) in the primitive intermediate Jacobian \(P(C)\) of \(J(C)\). higher \(K\)-theory; hyperelliptic Jacobian; Jacobian variety; Quillen group; Griffiths' infinitesimal invariant Collino, A., Griffiths' infinitesimal invariant and higher \(K\)-theory on hyperelliptic Jacobians, J. Alg. Geom. 6 (1997), 393-415. Algebraic cycles and motivic cohomology (\(K\)-theoretic aspects), Algebraic cycles, Applications of methods of algebraic \(K\)-theory in algebraic geometry, Jacobians, Prym varieties, Riemann surfaces; Weierstrass points; gap sequences Griffiths' infinitesimal invariant and higher \(K\)-theory on hyperelliptic 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 Let \(X\) denote a smooth scheme and \(\mathcal{L}\) be an invertible sheaf on \(X\). In this very interesting paper, the author defines a filtration on the Gersten-Witt complex of any pair \((X, \mathcal{L})\) and studies the cohomology of the corresponding filtered complex, called \(\mathbf{I}^j\)-cohomology. The cohomology of the associated graded complex is the motivic cohomology of \(X\) with \(\mathbb{Z}/2\mathbb{Z}\)-coefficients (see \textit{V. Voevodsky}'s paper [Publ. Math., Inst. Hautes Étud. Sci. 98, 59--104 (2003; Zbl 1057.14028)]). The \(\mathbf{I}^j\)-cohomology of a smooth scheme \(X\) is an algebraic analog of the singular cohomology group \(\text{H}^*(X(\mathbb{R}), \mathbb{Z})\) as \(\mathbb{Z}/2\mathbb{Z}\)-motivic cohomology is an algebraic analog of \(\text{H}^*(X(\mathbb{R}), \mathbb{Z}/2 \mathbb{Z})\). The principal aim of the author is to understand which characteristic classes can be constructed in \(\mathbf{I}^j\)-cohomology. This is done by a thorough description of the \(\mathbf{I}^j\)-cohomology of a projective bundle. The corresponding result for Grothendieck-Witt groups has already been proved in a preprint by \textit{Ch. Walter} [``Grothendieck-Witt groups of projective bundles'', \url{http://www.math.uiuc.edu/K-theory/0644/} (2003)], and a new proof of it is provided here. It is important to notice that when \(E\) has even rank, the pull-back morphism from the \(\mathbf{I}^j\)-cohomology of the base to the \(\mathbf{I}^j\)-cohomology of \(\mathbb{P}(E)\) is not always injective. This issue, already observed by Walter for Grothendieck-Witt groups, introduces many conceptual and technical problems in this theory. Nevertheless, for any algebraic vector bundle \(E\) on \(X\) of odd rank and every odd integer \(m\), the author succeeds in constructing a Stiefel-Whitney class \(c_m(E)\) in the \(\mathbf{I}^j\)-cohomology of \(X\). Lastly, the cohomology of the Milnor-Witt sheaves for projective bundles is given. Gersten-Witt complex; cohomology theory; projective bundle Fasel, J., The projective bundle theorem for \textit{j}-cohomology, J. K-Theory, I, 2, 413-464, (2013) Other algebro-geometric (co)homologies (e.g., intersection, equivariant, Lawson, Deligne (co)homologies), Hermitian \(K\)-theory, relations with \(K\)-theory of rings, Algebraic cycles The projective bundle theorem for \(I^j\)-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 development of algebraic geometry in the 20th century is characterized by several radical changes in style and language. The process that led to algebraic geometry at its present stage started in the later 1940s, when sheaf theory and cohomology theories emerged as a basic toolkit in various branches of mathematics, and it culminated in the 1960s with A. Grothendieck's new foundation of algebraic geometry via the abstract-algebraic theory of schemes.
The first textbooks on algebraic geometry, which reflected these new concepts and methods in a systematic and (more or less) comprehensive way, appeared in the 1970s. After \textit{I. R. Shafarevich}'s ``Basic algebraic geometry'' (in Russian 1972; Zbl 0258.14001; English edition 1974), \textit{D. Mumford}'s ``Algebraic geometry. I'' (1976; Zbl 0356.14002) and \textit{R. Hartshorne}'s ``Algebraic geometry'' (1977; Zbl 0367.14001) had appeared, the authors published their ``Principles of algebraic geometry'' first in 1978 (cf. Zbl 0408.14001). Their treatise differed from the others in many regards and offered many particular features.
First of all, the book under review is a text on complex algebraic geometry. It does not stress the most general abstract-algebraic approach via schemes and their categorical sheaf theory. Instead, it focuses on the transcendental aspects of complex projective varieties, that is on their underlying Kähler geometry, Hodge theory and Kodaira-Lefschetz theory. Secondly, this transcendental approach is organically combined with the classical projective geometry of algebraic varieties, including various classical topics such as Grassmannians, enumerative formulae, varieties of lines, and others. -- Finally, apart from the presentation of the wide spectrum of modern analytic and algebraic methods, together with their application to the study of complex projective manifolds, the book offers a particularly nice and comprehensive discussion of the classical theories of algebraic curves and surfaces, and that from both the modern and the classical viewpoint.
These features give the book under review its unique character among the (meanwhile) many existing textbooks on modern algebraic geometry. Now as before, it represents the by far most valuable and complete supplement to the more general and astract textbooks on algebraic geometry, with regard to the transcendental and classic-projective topics, and it contains a wealth of geometric-intuitive ideas, ingenious arguments and particular results that cannot be found in any other text For both researchers and students this book is still an inexhaustible and indispensible source.
The present edition is a reprint of the original text, without any changes. Being one of the really great classics of algebraic geometry, this text is timelessly outstanding and does not need any additional enlargement or any modifications. complex manifolds; Kähler manifolds; complex algebraic varieties; algebraic curves; Riemann surfaces; algebraic surfaces; Hodge theory Phillip Griffiths and Joseph Harris. {\em Principles of algebraic geometry}. Wiley Classics Library. John Wiley \& Sons, Inc., New York, 1994. Reprint of the 1978 original. zbl 0836.14001; MR1288523 Research exposition (monographs, survey articles) pertaining to algebraic geometry, Surfaces and higher-dimensional varieties, Compact analytic spaces, Curves in algebraic geometry, Transcendental methods, Hodge theory (algebro-geometric aspects) Principles of algebraic geometry. | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians In a classical paper (``A class of algebraic curves with cyclic group and their Jacobian varieties''), \textit{S. Lefschetz} studied the Jacobian varieties of a family of algebraic curves of genus \(g=(p-1)/2\) \((p\) a prime number \(\geq 5)\) with an automorphism of order \(p\). \textit{A. Adler} [J. Algebra 72, 115-145 (1981; Zbl 0479.20017)] and \textit{G. Riera} and \textit{R. E. Rodríguez} [Duke Math. J. 69, No. 1, 199-217 (1993; Zbl 0790.14039)] studied more generally and principally polarized abelian varieties of dimension \(g=(p-1)/2\) admitting an automorphism of order \(p\). The aim of the present paper is to give a more precise description of these abelian varieties. As a consequence of this the authors obtain a formula for the number of isomorphism classes of these varieties for a fixed dimension \(g\).
The main tool for the proof is a classical result of Brauer on the characterisation of the linear groups in two variables over the field with \(p\) elements. There is an exceptional case, namely if \(p\equiv -1\bmod 4\), where the group of all automorphisms of the polarized abelian variety is nonsolvable. In this case there is a contribution to the number of isomorphism classes coming from a class of lattices studied by Adler. In the last section an explicit description of these lattices is given. abelian variety; principally polarized abelian varieties; automorphism of order \(p\); number of isomorphism classes H. Bennama and J. Bertin, Remarques sur les varietés abéliennes avec un automorphisme d'ordre premier, Manuscripta Math. 94 (1997), no. 4, 409 -- 425 (French). Algebraic theory of abelian varieties, Group actions on varieties or schemes (quotients), Birational automorphisms, Cremona group and generalizations Remarks on abelian varieties with an automorphism of prime order | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians Let \(X\) be a smooth proper curve over \(\mathbb{F}_p\). The (unramified) geometric class field theory establishes an association from rank one étale \(\overline{\mathbb{Q}}_{\ell}\)-local systems on \(X\) to \(\overline{\mathbb{Q}}_{\ell}\)-sheaves on \(\mathrm{Pic}_X\) satisfying a certain Hecke property. When \(X\) is a smooth proper curve over the complex number \(\mathbb{C}\), an analogous result for algebraic \(\mathscr{D}\)-modules is obtained by Laumon in a different way. A key construction in Laumon's approach is now called Fourier-Mukai-Laumon transform.
The paper under review studies an analogous question with \(p\)-torsion coefficients objects on a smooth proper curve \(X\) over \(\mathbb{F}_q\). In this situation, rank one étale \(\overline{\mathbb{Q}}_{\ell}\)-local systems (resp. \(\overline{\mathbb{Q}}_{\ell}\)-sheaves) are replaced by rank one étale \(W_n(\mathbb{F}_q)\)-local systems (resp. arithmetic \(\mathscr{D}\)-modules in the sense of Berthelot).
Let \(X\) be a smooth projective curve over \(W_n(=W_n(\mathbb{F}_q))\), \({Jac}\) its Jacobian and \(\widetilde{{Jac}}\) the universal extension of \({Jac}\). The author defines a Verschiebung endomorphism \({Ver}:\widetilde{{Jac}}\to \widetilde{{Jac}}\) using Berthelot's Frobenius pull-back functor and \(\widetilde{{Jac}}^{\sharp}\) to be the fixed point subscheme of \({Ver}\). A \(W_n\)-point of \(\widetilde{{Jac}}^{\sharp}\) can be viewed as an line bundle, equipped with a connection and a Frobenius structure, called \textit{unit F-crystal}. The author extends Fourier-Mukai-Laumon transform to unit \(F\)-crystals (more generally to \(p\)-torsion arithmetic \(\mathscr{D}\)-modules). Emerton-Kisin established a Riemann-Hilbert correspondence between unit \(F\)-crystals and certain constructible \(W_n\)-étale sheaves. Based on these results, the author constructs a functor \(\mathbb{L}_n\) from the derived category of certain coherent sheaves on \(\widetilde{{Jac}}^{\sharp}\) equipped with an automorphism to the derived category of constructible étale \(W_n\)-sheaves on \({Jac}\). The construction also uses the machine of \(\infty\)-categories. Moreover, this functor categorifies the geometric class field theory for \(p\)-torsion étale local systems on \(X\).
The main result is a criterion on full faithfulness of \(\mathbb{L}_n\), which is equivalent to the nilpotence of the Frobenius action on \(H^1(X_{\mathbb{F}_q},\mathscr{O}_{X_{\mathbb{F}_q}})\). The proof is based on the study of Verschiebung endomorphism action on the cohomologies of a Koszul complex. Jacobian; geometric class field theory; mod-\(p\) étale sheaves \(p\)-adic cohomology, crystalline cohomology, Differentials and other special sheaves; D-modules; Bernstein-Sato ideals and polynomials, Congruences for modular and \(p\)-adic modular forms, Drinfel'd modules; higher-dimensional motives, etc., Algebraic moduli problems, moduli of vector bundles, Geometric Langlands program (algebro-geometric aspects), Arithmetic ground fields for curves \(p\)-torsion étale sheaves on the Jacobian of a curve | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians Let \(M\) denote the universal plane curve of degree \(d\geq3\) defined over an algebraically closed field \(k\) of characteristic \(0\). The author shows that \(M\) can be viewed as a closed subvariety of codimension \(\frac{d(d-3)}2\) in the Simpson moduli space of semistable sheaves on \({\mathbb P}^2\) with Hilbert polynomial \(dm+\frac{d(3-d)}2+1\); moreover \(M\) is contained in the stable locus and coincides with the locus of sheaves with non-zero global sections (Proposition 3.2). Key steps in the proof are showing that the points of \(M\) are in bijective correspondence with isomorphism classes of sheaves possessing a particular type of 2-step resolution (Proposition 2.7) and then showing that these sheaves are in bijective correspondence with isomorphism classes of non-trivial extensions \(0\to{\mathcal O}_C\to{\mathcal F}\to k_p\to0\) with \((C,p)\in M\) (Lemma 2.8). The next step, in section 3, is to show that all these sheaves are stable. This gives a morphism from \(M\) to the appropriate Simpson moduli space, which is clearly injective with closed image. The proof that it is a closed embedding is completed using the Beilinson spectral sequence. Finally, to see that the image of this embedding coincides with the locus of sheaves with global sections, one uses again the results of section 2.
In section 4, the author shows that a point \((C,p)\in M\) corresponds to a sheaf which is not locally free on its support if and only if \(p\) is a singular point of \(C\) (Lemma 4.1). Let \(M'\) denote the subvariety of \(M\) consisting of such \((C,p)\). Then the blow-up \(\text{Bl}_{M'}(M)\) is a compactification of \(M\setminus M'\) whose points correspond to sheaves which are vector bundles on their supports (Theorem 5.7).
The paper is based on a section of the author's PhD thesis. coherent sheaves; Simpson moduli spaces; vector bundles on curves Iena O., Universal plane curve and moduli spaces of 1-dimensional coherent sheaves, Comm. Algebra, 2015, 43(2), 812-828 Algebraic moduli problems, moduli of vector bundles, Divisors, linear systems, invertible sheaves, Vector bundles on curves and their moduli Universal plane curve and moduli spaces of 1-dimensional coherent sheaves | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians The goal of this important paper under review is to develop the relative Gromov-Witten theory from a logarithmic approach. This approach was proposed by \textit{B. Siebert} ``Gromov-Witten invariants in relative and singular cases'', Lecture given in the workshop on Algebraic Aspects of Mirror Symmetry, Universität Kaiserslautern, Germany, Jun 26 (2001)] in 2001, and the original idea was to use tropical geometry and probing the stack by standard log point. In this paper the author applies somewhat different methods. An upshot is the notion of marked graphs associated to log maps, which allows the author to define the right base log structure (the minimal log structure).
The main results of the paper are the following. Let \(X\) be a projective variety, and let \(\mathcal M_X\) be a rank-one Deligne-Faltings log structure. Denote by \(X^{\text{log}} = (X, \mathcal M_X)\) the corresponding log scheme. Denote by \(\mathcal K_{\Gamma}(X^{\text{log}})\) the category fibered over the category of schemes, which for any scheme \(T\) associates the groupoid of minimal stable log maps over \(T\) with numerical data \(\Gamma\). The author proves that \(\mathcal K_{\Gamma}(X^{\text{log}})\) is a proper Deligne-Mumford stack, and that the natural map by removing the log structures from minimal stable log maps is representable and finite (Theorem 1.2.1). Moreover, denote by \(\mathcal M_{K_{\Gamma}(X^{\text{log}})}\) the universal minimal log structure. The author further proves that the pair \((K_{\Gamma}(X^{\text{log}}), \mathcal M_{K_{\Gamma}(X^{\text{log}})})\) defines a category fibered over the category of fine and saturated log schemes, which for any fine and saturated log scheme associates the category of stable log maps over it (Theorem 1.2.3).
Around the same time \textit{M. Gross} and \textit{B. Siebert} [J. Am. Math. Soc. 26, No. 2, 451--510 (2013; Zbl 1281.14044)] worked out Siebert's original approach, building on insights from tropical geometry. Several other approaches to the algebricity and boundedness of moduli of stable log maps have also been explored in recent years, including Kim's logarithmic stable maps [\textit{B. Kim}, Adv. Stud. Pure Math. 59, 167--200 (2010; Zbl 1216.14023)] and Parker's theory of exploded manifolds [\textit{B. Parker}, Adv. Math. 229, No. 6, 3256--3319 (2012; Zbl 1276.53092), Abh. Math. Semin. Univ. Hamb. 82, No. 1, 43--81 (2012; Zbl 1312.14130)]. relative Gromov-Witten theory; stable logarithmic maps Qile Chen. Stable logarithmic maps to Deligne-Faltings pairs I. \(Ann. of Math. (2)\), 180(2):455-521, 2014. Families, moduli of curves (algebraic), Stacks and moduli problems, Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects) Stable logarithmic maps to Deligne-Faltings pairs I | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians Let \(X\) be a scheme, locally of finite type over a locally noetherian scheme \(S\), and let \({ F}:={Hilb}_{X/S}\) be Grothendieck's Hilbert functor, i.e. the set \({\mathcal F}(T)\) consists of closed schemes of \( Z \subseteq X \times_S T\) which are flat and of proper support over \(T\). If \(X\) is quasiprojective (resp. separable) over \(S\), then the Hilbert funcor \(\text{Hilb}_{X/S}\) is representable by a quasiprojective scheme (resp. an algebraic space) due to fundamental theorems of \textit{A. Grothendieck} [Sem. Bourbaki 13(1960/61), No. 221 (1961; Zbl 0236.14003)] and \textit{A. Artin} [in: Global Analysis, Papers in Honor of K. Kodaira, 21--71 (1969; Zbl 0205.50402)]. One of Artin's necessary conditions for e.g. \({\mathcal F}\) to be representable, is that formal deformations are effective, i.e. that the map \({\mathcal F}(A) \to {\displaystyle\lim_{\longleftarrow}}{\mathcal F}(A/m^n)\) is surjective for any complete local ring \((A,m)\). The authors of this paper show that if \(X\) is not separated over \(S\), then the Hilbert functor \(\text{Hilb}_{X/S}^1\) of one point has non-effective formal deformations. Thus \(\text{Hilb}_{X/S}^1\) is not representable by a scheme or an algebraic space. Indeed \(\text{Hilb}_{X/S}^1 \simeq {\Hom}_S(-,X)\) if and only if \(X\to S\) is separated. Hilbert functor; representability; separated scheme; algebraic space Lundkvist C. and Skjelnes R., Non-effective deformations of Grothendieck's Hilbert functor, Math. Z. 258 (2008), no. 3, 513-519. Parametrization (Chow and Hilbert schemes), Local deformation theory, Artin approximation, etc., Generalizations (algebraic spaces, stacks) Non-effective deformations of Grothendieck's Hilbert 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 A former conjecture by \textit{S. Lang} [cf. Publ. Math., Inst. Hautes Étud. Sci. 6, 27-43 (1960; Zbl 0112.134)] about rational points on arithmetic abelian varieties has recently been confirmed by \textit{G. Faltings} [in: Memorial Meeting in honour of I. Barsotti, Abano Terme 1991, 175-182 (1994; see the preceding review)]. It states that, for any closed subvariety \(X\) of an abelian variety \(A\) over a number field \(K\), the set \(X(K)\) of K-rational points of \(X\) is contained in a finite union of translates of K-rational abelian subvarieties of \(A\). This theorem of G. Faltings can be generalized to semi-abelian schemes, according to a (yet unpublished) work of \textit{P. Vojta} [cf. ``Integral points on subvarieties of semi-abelian varieties'' (Preprint 1994)]. In the present note, the author uses the methods of Faltings and Vojta for investigating the more general case of families of subvarieties of an abelian variety over a number field. More precisely, let \(A\) be an abelian variety over a number field \(K\) and let \(X\) be a closed subvariety of \(A \times_ K S\), where \(S\) is a quasi-projective scheme over \(K\).
Under these assumptions, and with respect to chosen ample line bundles \(L\) (on a compactification \(\overline {S}\) of \(S\)) and \({\mathfrak L}\) (on \(A\)), a relative version of Faltings' theorem for the K-rational fibers \(X_ S (K)\), \(s\in S(K)\), is proved. The author's result is related to a conjecture of P. Vojta, called the ``effective Mordell conjecture'', and also to a conjecture due to A. Parshin, L. Szpiro, and L. Moret-Bailly [cf. \textit{L. Moret-Bailly} in: Les pinceaux de courbes elliptiques, Sémin., Paris 1988, Astérisque 183, 37-58 (1990; Zbl 0727.14015)] concerning the boundedness of height functions on the fiber of families of smooth curves of genus \(g\geq 2\).
With a view to his main theorem, the author proposes a generalization of these conjectures. rational points; subvarieties of an abelian variety; boundedness of height functions Rational points, Algebraic theory of abelian varieties, Arithmetic varieties and schemes; Arakelov theory; heights Complements on a theorem of Faltings | 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 Segal's \(\Gamma\)-rings provide a natural framework for absolute algebraic geometry. We use \textit{G. Almkvist}'s [J. Algebra 28, 375--388 (1974; Zbl 0281.18012)] global Witt construction to explore the relation with \textit{J. Borger} [``Lambda-rings and the field with one element''', Preprint, \url{arXiv:0906.3146}] \(\mathbb{F}_1\)-geometry and compute the Witt functor-ring \(\mathbb{W}_0 (\mathbb{S} )\) of the simplest \(\Gamma\)-ring \(\mathbb{S}\). We prove that it is isomorphic to the Galois invariant part of the BC-system, and exhibit the close relation between \(\lambda\)-rings and the Arithmetic Site. Then, we concentrate on the Arakelov compactification \(\overline{\mathrm{Spec}\, \mathbb{Z}}\) which acquires a structure sheaf of \(\mathbb{S}\)-algebras. After supplying a probabilistic interpretation of the classical theta invariant of a divisor \(D\) on \(\overline{\mathrm{Spec}\, \mathbb{Z}}\), we show how to associate to \(D\) a \(\Gamma\)-space that encodes, in homotopical terms, the Riemann-Roch problem for \(D\). Geometry over the field with one element, Riemann-Roch theorems, Chern characters Segal's gamma rings and universal arithmetic | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians Let \(J\) be the Jacobian of a curve \(C\) of genus 2, defined over \(\mathbb{Q}\). Let \(p\) be a prime number. Assume that the reduction of the Néron model of \(J\) over \(\mathbb{Q}_p\) is an extension of an elliptic curve by a torus. We denote by \(\overline\mathbb{Q}\) an algebraic closure of \(\mathbb{Q}\); the Galois group \(\text{Gal}(\overline\mathbb{Q}/\mathbb{Q})\) acts on the \(\ell\)-division points of \(J\). We denote by \(\rho_\ell\) the associated representation. Let \(q\) be a prime number where \(J\) has good reduction such that the Galois group over \(\mathbb{Q}\) of the characteristic polynomial of the Frobenius endomorphism associated to \(q\) is the dihedral group with 8 elements (this implies that \(J\) is absolutely simple). Then an infinite set of prime numbers can be found such that the image of \(\rho_\ell\) is \(\text{GSp}(4,\mathbb{F}_\ell)\). Two examples will be given at the end of this article. finite ground field; Jacobian of a curve; Néron model; Galois group; Frobenius endomorphism [15]P. Le Duff, Repr'esentations galoisiennes associ'ees aux points d'ordre l des jacobiennes de certaines courbes de genre 2, Bull. Soc. Math. France 126 (1998), 507--524. Jacobians, Prym varieties, Finite ground fields in algebraic geometry, Rational points, Universal profinite groups (relationship to moduli spaces, projective and moduli towers, Galois theory) Galois representations associated to the points of order \(\ell\) of the Jacobians of special curves of genus 2 | 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 Splice diagrams of germs of plane curve singularities were introduced by \textit{D. Eisenbud} and \textit{W. Neumann} [``Three-dimensional link theory and invariance of plane curve singularities'', Ann. Math. Stud. 110 (1985; Zbl 0628.57002)] and contain a significant amount of information on invariants of these singularities. In this article, the author uses splice diagrams, whose construction is briefly recalled at the beginning of the article, to characterize the set of Jacobian quotients of a map germ \((f,g)\).
Jacobian quotients can be considered as a generalization of the well-known polar quotients and correspond to the quotients appearing as lowest exponents in the Puiseux expansion of branches of the discriminant curve of \((f,g)\). A characterization thereof using the dual graph of the resolution of \(fg\) was already proved by \textit{H. Maugendre} [C. R. Acad. Sci., Paris, Sér. I 322, No. 10, 945--948 (1996; Zbl 0922.32022)] by topological methods. Here, the author obtains an analogous characterization by algebraic methods using splice diagrams, which she explains in great detail. Jacobian quotients; splice diagram Singularities in algebraic geometry, Local complex singularities, Modifications; resolution of singularities (complex-analytic aspects) Jacobian quotients: an algebraic approach | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians This article is an expository one based on lectures at the Summer Institute in Algebraic Geometry, Seattle, 2005. It gives a review of the main achievements in the subject.
The first lecture includes a brief review of \(p\)-adic integration, a result of Denef-Loeser giving a formula for the Euler characteristic of a smooth complex variety \(X\) in terms of its log resolution and the independence of the zeta function of a pair from the log resolution. There is also a result of Batyrev that the Betti numbers of two birationally equivalent Calabi-Yau varieties are equal, which led to his conjecture about the equality of their Hodge numbers, and so gave the main motivation for the invention of the motivic integration by Kontsevich.
The second lecture gives the basics of motivic integration. It discusses briefly arc spaces, additive invariants and Grothendieck rings. Then the change of variables formula and some of its major applications are given. Among them is a formula for the class of \(X\) in terms of its log resolution, the proof of the Batyrev conjecture by Kontsevich, the independence of the stringy invariant \(E_{st}(X)\) of a normal terminal \(\mathbb{Q}\)-Gorenstein variety on the log resolution, and the theorem of Mustată giving a formula for the log canonical threshold in terms of codimensions of some jet spaces.
In lecture 3, after introducing briefly the Milnor fiber and the nearby cycle functor, a relation between Euler characteristic and Lefschetz numbers is given. Then the motivic analogue of Igusa's local zeta function is defined, and the monodromy conjecture is formulated. After introducing briefly the Hodge spectrum and convolution product, two versions of Thom-Sebastani theorem are formulated, one for the Hodge spectrum, and another as the motivic version of the theorem. Also, the motivic version of a result of Saito proving a Steenbrink conjecture is given.
In the last lecture a general setting for motivic integration is developed, based on a series of works of the author with R. Cluckers. After an introduction to semialgebraic geometry and some notions from model theory, the Denef-Pas cell decomposition theorem is stated. Using the language of constructible motivic functions, the general motivic measure satisfying some axioms is defined, the general change of variables formula is stated, and the motivic analogues of exponential functions are introduced. In the last part is obtained a general transfer principle, allowing to transfer relations between integrals from \(\mathbb{Q}_p\) to \(\mathbb{F}_p((t))\), and vice versa. As a special case one has the Ax-Kochen-Eršov theorem. motivic integration; arc space; Grothendieck ring Loeser, François, Seattle lectures on motivic integration.Algebraic geometry---Seattle 2005. Part 2, Proc. Sympos. Pure Math. 80, 745-784, (2009), Amer. Math. Soc., Providence, RI Arcs and motivic integration Seattle lectures on motivic integration | 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 0626.00011.]
A theorem of \textit{A. Rojtman} [Ann. Math., II. Ser. 111, 553-569 (1980; Zbl 0504.14006)] says that for a smooth projective variety X over an algebraically closed field, the torsion subgroup of the group of zero- cycles modulo rational equivalence, \(CH_ 0(X)_{tors}\), is naturally isomorphic to the torsion subgroup of the Albanese variety of X, \(Alb(X)_{tors}\). This result was generalized by the author [Am. J. Math. 107, 737-757 (1985; Zbl 0579.14007)] for the case of a projective variety X, smooth in codimension one (modulo p-torsion in characteristic \(p,\) \(p>0)\), where \(CH_ 0(X)_{tors}\) had to be replaced by the torsion part of \(CH_ 0(X,X_{\sin g})\), the quotient of the free abelian group on the smooth points of X and the subgroup generated by cycles of the form \(i_{C\quad *}((f))\) with C a closed reduced, irreducible curve on X, not meeting \(X_{\sin g}\), and f a rational function on C.
In the general case one is led to define a relative Chow group \(CH_ 0(X,Y)\), where Y is closed in X and contains \(X_{\sin g}\). The definition was given in another article and is not repeated here. Now the main result of the paper under review is that for an affine variety X over an algebraically closed field \(CH_ 0(X,Y)\) is torsion free (modulo p-torsion in characteristic \(p>0).\)
As a corollary one obtains an injective map \(CH_ 0(X,Y)\to K_ 0(X)\), whose image is isomorphic to the subgroup of \(K_ 0\)(X) generated by the residue fields of the smooth points of X. torsion free relative Chow group; group of zero-cycles modulo rational equivalence; \(K_ 0\) M. Levine, Zero-cycles and \(K\)-theory on singular varieties , Algebraic geometry, Bowdoin, 1985 (Brunswick, Maine, 1985), Proc. Sympos. Pure Math., vol. 46, Amer. Math. Soc., Providence, RI, 1987, pp. 451-462. (Equivariant) Chow groups and rings; motives, Singularities in algebraic geometry, Applications of methods of algebraic \(K\)-theory in algebraic geometry, Algebraic cycles Zero-cycles and K-theory on singular varieties | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians Let \(C\) be a smooth complete connected curve of genus \(g>2\) embedded in its Jacobian variety \(JC\). The Ceresa cycle of \(C\) is the cycle \(Z=C-i(C)\subset CH^{g-1}(JC)\), where \(i=-1\) on \(JC\). It is known that \(Z\) is homologically but not algebraically equivalent to zero for generic \(C\).
If \(f:\mathcal C\to S\) is a family of curves of genus \(g>2\) as above and \(\mathcal J\to S\) is its relative Jacobian, then the relative Ceresa cycle \(\mathcal{Z}\in CH^{g-1}(\mathcal J)_{\hom}\) can be considered. The normal function associated to \(\mathcal Z\) is a section \(\nu\) of the relative intermediate Jacobian \(\mathcal J^{g-1}\to S\) defined by \(\nu(s)=AJ_s(Z_s)\), where \(AJ_{s} \) is the higher Abel-Jacobi map. A formula for computing the Griffiths' infinitesimal invariant of \(\nu\) has been given by \textit{A. Collino} and \textit{G. Pirola} [Duke Math. J. 78, 59--88 (1995; Zbl 0846.14016)]. In this paper the author, by means of Collino and Pirola's formula, expresses the Griffiths' infinitesimal invariant of the family in terms of a triple Massey product on a graded differential algebra \(\mathcal A\), defined similarly to the Kodaira-Spencer algebra. As a consequence of this result, it is proved that the differential graded algebra \(\mathcal A\) is not formal. Griffiths' infinitesimal invariant; Massey product Rizzi, C, Infinitesimal invariant and Massey products, Man. Math., 127, 235-248, (2008) Algebraic cycles, Massey products, Jacobians, Prym varieties, Differential graded algebras and applications (associative algebraic aspects), Secondary and higher cohomology operations in algebraic topology Infinitesimal invariant and Massey products | 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 An arithmetic method of proving the irrationality of smooth projective 3-folds is described, using reduction modulo \(p\). It is illustrated by an application to a cubic threefold, for which the hypothesis that its intermediate Jacobian is isomorphic to the Jacobian of a curve is contradicted by reducing modulo 3 and counting points over appropriate extensions of \(\mathbb F_3\). As a spin-off, it is shown that the 5-dimensional Prym varieties arising as intermediate Jacobians of certain cubic 3-folds have the maximal number of points over \(\mathbb F_q\) which attains Perret's and Weil's upper bounds. cubic hypersurfaces; Weil's conjectures; finite fields; Fano varieties; intermediate Jacobian; irrational varieties \(3\)-folds, Rationality questions in algebraic geometry, Finite ground fields in algebraic geometry, Fano varieties Irrationality of generic cubic threefold via Weil's conjectures | 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 Elliptic modules (nowadays called Drinfeld modules) introduced by \textit{V. G. Drinfeld} [Math. USSR, Sb. 23, 561--592 (1974); translation from Mat. Sb., Nov. Ser. 94(136), 594--627 (1974; Zbl 0321.14014)] are the analogues to elliptic curves within the the analogy between function fields and number fields: Many aspects of elliptic curves turned out to have a correspondent in the theory of Drinfeld modules. The present paper deals in particular with relative compactification of certain Drinfeld modular surfaces.
Let \(A\) be the coordinate ring of a fixed smooth projective curve minus one closed point. Drinfeld modules of rank \(d\) are defined over base schemes \(S\) lying over \(\text{Spec}(A)\), and give rise to moduli problems which are representable by a smooth algebraic stack. In order to have a representing scheme, one has to impose a certain level-\(I\)-structure for a suitable ideal \(I\subset A\). Let now \(M_I^d\to \text{Spec}(A)\) be the corresponding moduli scheme, which is regular of dimension \(d\) according to Drinfeld. In his pioneering paper [loc. cit.] Drinfeld sketched a relative compactification of the modular surfaces \(M_I^2\), and the aim of the present work is to fill in and work out all the details of Drinfeld's construction.
The book is organized as follows: The first two introductory chapters aim at a self-contained introduction to Drinfeld modules. Chapter 1 discusses line bundles considered as algebraic groups and their endomorphisms, which are roughly polynomials of the Frobenius endomorphism. Chapter 2 introduces Drinfeld modules, isogenies and level structures with an emphasis on the case of characteristic within the level, which is not treated in Drinfeld's paper. The following chapter is devoted to the infinitesimal deformation theory of Drinfeld modules. The cohomological approach of [\textit{G. Laumon}, Cohomology of Drinfeld modular varieties. Part 1: Geometry, counting of points and local harmonic analysis. Cambridge Studies in Advanced Mathematics. 41. Cambridge: Cambridge University Press (1996; Zbl 0837.14018) and Part II. Cambridge Studies in Advanced Mathematics. 56. Cambridge: Cambridge University Press (1997; Zbl 0870.14016)] for the case of characteristic away from the level is followed and via the technique of deformations of isogenies extended to the general case. This enables the author to show Drinfeld's theorem on the smoothness of the moduli space avoiding the original way of deforming formal groups. Then, chapter 3 describes the action of \(\text{GL}_d({\mathbb A}_f)\) on \(M^d:=\varprojlim_IM^d_I\), and proves that \(M^d_I\) is the quotient of \(M^d\) by a certain congruence subgroup. The fourth chapter discusses the boundary of \(M^d_I\) and constructs the so-called formal boundary, a formal scheme which can be thought of as the completion of a compactification yet to be constructed along the boundary. This construction suggests that the boundary itself should have a stratification with strata isomorphic to \(M^{d_1}_I, d_1<d\), as was confirmed in some special cases by Kapranov.
The last chapter deals with the case \(d=2\): The author constructs the formal boundary of \(M^2_I\), whose special fibre is a disjoint union of spaces isomorphic to \(M^1_I\). In the end, it is shown that this formal boundary indeed comes form an actual scheme.
This monograph is a revised version of the author's Habilitationsschrift, which was eventually reread by U. Stuhler and S. Wiedmann after the author passed away before he was able to incorporate the referee's suggestions. Drinfeld modular surface; Drinfeld modules; compactifications; infinitesimal deformations T. Lehmkuhl, Compactification of Drinfeld Modular Surfaces, Memoirs AMS 197(2009), no. 921. Drinfel'd modules; higher-dimensional motives, etc., Modular and Shimura varieties, Formal neighborhoods in algebraic geometry, Deformations and infinitesimal methods in commutative ring theory Compactification of the Drinfeld modular surfaces | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians We deal with the class of smooth projective curves \(C\) whose derived category of coherent sheaves is equivalent to the derived category of finite-dimensional modules over a finite-dimensional algebra. For an arbitrary base field \(k\) this happens if and only if \(C\) has genus zero. We collect a number of characterizations for \(C\) to be of genus zero. Some of the characterizing properties appear in print for the first time; others -- often in an implicit form -- are scattered throughout the literature, where usually they are stated only for an algebraically closed base field.
We approach the question from the point of view of hereditary Noetherian categories with Serre duality. The investigation of such categories has recently attracted much attention. For \(k\) algebraically closed, we refer in this context to the characterization of hereditary Noetherian categories with Serre duality by \textit{I. Reiten} and \textit{M. van den Bergh} [J. Am. Math. Soc. 15, 295--366 (2002; Zbl 0991.18009)], and to the characterization of hereditary categories with a tilting object by \textit{D. Happel} [Invent. Math. 144, 381--398 (2001; Zbl 1015.18006)]. derived category of coherent sheaves; hereditary Noetherian categories with Serre duality Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20], Special algebraic curves and curves of low genus, Representations of associative Artinian rings, Derived categories, triangulated categories Twenty-one characterizations of genus zero. | 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 More than 50 years ago \textit{D. Mumford} gave an example of a generically non-reduced irreducible component of the Hilbert scheme of smooth irreducible space curves [Am. J. Math. 84, 642--648 (1962; Zbl 0114.13106)]. Expanding on Mumford's example, \textit{J. O. Kleppe} used Hilbert flag schemes to show that if a general curve \(C\) in a Hilbert scheme component \(V\) lies on a smooth cubic surface \(X\) and \(H^1({\mathcal O}_X (-C)(3)) \neq 0\), then \(V\) is generically non-reduced [Lect. Notes Math. 1266, 181--207 (1985; Zbl 0631.14022)]. Recently \textit{J. O. Kleppe} and \textit{J. C. Ottem} extended these ideas to produce examples in which the general curve \(C\) lies on a quartic surface [Int. J. Math. 26, Article ID 1550017, 30 p. (2015; Zbl 1323.14005)], but their method fails for curves lying on smooth surfaces \(X\) of degree \(d \geq 5\) or having Picard number \(\rho (X) > 2\).
Here the author combines Hilbert flag scheme methods and the theory of Hodge loci to construct more examples of non-reduced Hilbert schemes whose general curve \(C\) lies on a smooth surface \(X\) of degree \(d \geq 5\) (and on no quartic) satisfying \(\rho (X) > 2\). He starts with a smooth surface \(X\) of degree \(d \geq 5\) containing two coplanar lines \(L_1\) and \(L_2\), noting that \(\rho (X) > 2\). The Hilbert scheme of extremal curves of the form \(D=2L_1 + L_2 \subset X\) is generically non-reduced by work of \textit{M. Martin-Deschamps} and \textit{D. Perrin} [Ann. Scient. École Norm. Sup. 29, 757--785 (1996; Zbl 0892.14005)]. Letting \(\gamma \in H^{1,1}(X, \mathbb Z)\) denote the cohomology class of of \(D\), the Zariski closure of the associated Hodge locus \(\text{NL}(\gamma)\) in the open set \(U \subset |{\mathcal O}_{\mathbb P^3} (d)|\) of smooth surfaces is irreducible and generically non-reduced. With arguments involving exact sequences and Mumford regularity, he shows that the general member \(C \in |{\mathcal O}_X (D) (m)|\) is a smooth connected curve for \(m \geq 2d-2\). Letting \(X\) and \(C\) vary and taking the closure in the Hilbert scheme gives an irreducible component which is generically non-reduced. The arguments are clearly written and easy to follow. Hilbert scheme of space curves; Hilbert flag scheme; Hodge locus; non-reduced components Parametrization (Chow and Hilbert schemes), Transcendental methods of algebraic geometry (complex-analytic aspects), Variation of Hodge structures (algebro-geometric aspects) On generically non-reduced components of Hilbert schemes of smooth 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 \(f:X\to Y\) be a proper smooth morphism of complex manifolds and let \(\omega^\bullet_{X/Y}\) be the relative de Rham complex. Then Poincaré's lemma asserts that the complex of sheaves \(\omega^\bullet_{X/Y}\) is a resolution of \(f^{-1}{\mathcal O}_Y\). From this it is easy to construct an isomorphism of \({\mathcal O}_Y\)-modules: \(\mathbb{R}^m f_*\mathbb{Q} \otimes{\mathcal O}_Y \simeq \mathbb{R}^m f_* \omega^c_{X/Y}\).
In a paper by \textit{F. Kato} [Duke Math. J. 93, No. 1, 179-206 (1998; Zbl 0947.32003)] a generalization of this result was proved for log analytic spaces as defined by \textit{L. Illusie} [Perspect. Math. 15, 183-203 (1994; Zbl 0832.14015)]. The aim of this paper is to prove a similar result for log Hodge structures.
More precisely, let \((Y,{\mathcal M}_Y)\) be a log analytic space and let \((Y^{\log}, {\mathcal O}^{\log}_Y\) be the corresponding ringed space endowed with a continuous surjective map \(\tau: Y^{\log} \to Y\). Let \(f:(X, {\mathcal M}_X)\to (Y,{\mathcal M}_X) \to(Y,{\mathcal M}_Y)\) be a morphism of log analytic space (satisfying some extra conditions). Then the author shows, using a log version of the relative Poincaré lemma, that there is an isomorphism of \({\mathcal O}^{\log}_Y\)-modules:
\[
\lambda: \mathbb{R}^m f^{\log}_* \mathbb{Q} \otimes {\mathcal O}^{\log}_Y \simeq \tau^*\mathbb{R}^m f_*\omega^\bullet _{X/Y}.
\]
Using this, he proves the following result for log Hodge structures: Let \({\mathcal H}_\mathbb{Q} =\mathbb{R}^mf^{\log}_* \mathbb{Q}\) and \({\mathcal H}_{\mathcal O}= \mathbb{R}^m f_* \omega^\bullet_{X/Y}\) endowed with a filtration \(\{\mathbb{R}^m_*\omega^\bullet\geq i_{X/Y}\}\) and let \(\lambda\) be the above isomorphism. Then the triplet \(({\mathcal H}_\mathbb{Q}, {\mathcal H}_{\mathcal O}, \lambda)\) is a log Hodge structure on \(Y\). log Hodge structures; log analytic space; Poincaré lemma T. MATSUBARA, On log Hodge structures of higher direct images, Kodai Math. J., 21 (1998), pp. 81-101. Zbl1017.32017 MR1645599 Transcendental methods of algebraic geometry (complex-analytic aspects), Transcendental methods, Hodge theory (algebro-geometric aspects), de Rham cohomology and algebraic geometry On log Hodge structures of higher direct images | 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 Following previous ideas of \textit{G. Frey} [ECC '98, Waterloo (1998)] and \textit{S. D. Galbraith} and \textit{N. P. Smart} [7th IMA Conference, Lect. Notes Comput. Sci. 1746, 191-200 (1999; Zbl 0981.94025)] the authors use the Weil descent technique to translate the discrete logarithm problem on an elliptic curve \(E\) over a finite field \(F_{q^n}\) to the discrete logarithm problem on the Jacobian of a hyperelliptic curve \(C\), built by intersecting \(n-1\) hyperplanes associated to the Weil restriction of \(E\) with an \(n\)-dimensional variety over \(F_q\).
The paper studies the case \(q=2^r\), \(r>1\) (the characteristic 2 assumption is crucial in the proof). Nevertheless, at the end of the paper, the authors discuss the situation in the cases \(q=2\), \(n\) prime (the common case in cryptography) and \(q\) odd.
As the title suggests, the paper studies two antagonistic applications of this technique: design of hyperelliptic cryptosystems and cryptanalytic attacks on the original elliptic cryptosystem. The cryptographic implications of the second possibility are obvious, however the authors stress that their method does ``not appear to be a threat to standards compliant elliptic curve systems in the real world''.
The GHS attack has been further analysed in other papers. For instance \textit{M. Jacobson}, \textit{A. Menezes} and \textit{A. Stein} [J. Ramanujan Math. Soc. 16, 231-260 (2001; Zbl 1017.11030)] show, for the particular case \(q= 2^5\), \(n=31\), that the method could be successful only with \(2^{33}\) out of the \(2^{156}\) total isomorphism classes of elliptic curves. However \textit{S. D. Galbraith}, \textit{F. Hess} and \textit{N. P. Smart} [EUROCRYPT 2002, Lect. Notes Comput. Sci. 2332, 29-44 (2002)] extend the GHS attack to a much larger class of elliptic curves (in the example of Jacobson, Menezes and Stein to around \(2^{104}\) curves). This seems to strengthen the idea of the cryptographic weakness of the elliptic curves over composite extension fields. Weil descent; discrete logarithm problem; elliptic curve; Jacobian; hyperelliptic curve; hyperelliptic cryptosystems; cryptanalytic attacks P. Gaudry, F. Hess, and N. P. Smart, \textit{Constructive and Destructive Facets of Weil Descent on Elliptic Curves}, Hewlett Packard Lab. Tech. Rep. (2000), http://www.lix.polytechnique.fr/Labo/Pierrick.Gaudry/papers.html. Cryptography, Applications to coding theory and cryptography of arithmetic geometry, Elliptic curves over local fields, Algebraic coding theory; cryptography (number-theoretic aspects) Constructive and destructive facets of Weil descent on elliptic 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 Inspired by Kummer theory on abelian varieties, we give similar looking descriptions of the Galois groups occurring in the differential Galois theories of Picard-Vessiot, Kolchin and Pillay, and mention some arithmetic applications.
The topic I had been assigned by the organizers of the Luminy September 09 School
was ``Algebraic \(D\)-groups and nonlinear differential Galois theories''. The present account is written in an applied maths spirit: how to compute the Galois groups, and what for?
Thus, we start with a motivating question which, in accordance with the theme of the
School, comes from diophantine geometry. We then describe the Galois groups of the
various theories under study, in terms that bear a strong similarity. Finally, we apply this description to the study of exponentials and logarithms on abelian schemes.
A general argument of Galois descent occurs along the text, hence the title of these
notes; its number theoretic prototype, given by Kummer theory, is recalled in an Appendix to the paper.
Although the presentation is sometimes novel, the results described here are not new. For original sources, we refer the reader to [\textit{M. van der Put} and \textit{M. F. Singer}, Galois theory of linear differential equations. Berlin: Springer (2003; Zbl 1036.12008)] for the Picard-Vessiot theory, [\textit{A. Pillay}, Pac. J. Math. 216, No. 2, 343--360 (2004; Zbl 1093.12004)] for Kolchin's and Pillay's theories, and to [\textit{Y. André}, Compos. Math. 82, No. 1, 1--24 (1992; Zbl 0770.14003)] and [the author and \textit{A. Pillay}, J. Am. Math. Soc. 23, No. 2, 491--533 (2010; Zbl 1276.12003)] for the applications to algebraic independence.
Actually, this text may serve as an introduction to the author's survey [Ann. Inst. Fourier 59, No. 7, 2773--2803 (2009; Zbl 1226.12002)], which is itself an introduction to the latter papers (and to the descent argument in the nonlinear case). linear and nonlinear differential Galois theory; Abelian varieties; Galois cohomology; Kummer theory Bertrand, D, Galois descente in Galois theories, Sem. Congr., 23, 1-24, (2011) Differential algebra, Arithmetic ground fields for abelian varieties, Abelian varieties of dimension \(> 1\), Galois cohomology Galois descent in Galois theories | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians Let \(\mathbb{C}\) be the base field. Let \(G\) be a reductive linear algebraic group acting properly on a smooth projective variety \(X\). By Geometric Invariant Theory, one has open subsets \(X^s \subset X^{ss} \subset X\) of stable and semi-stable points. The quotient \(X^{ss}/G\) is projective and generally singular; and the quotient \(X^{s}/G\) has at most finite quotient singularities. \textit{F. C. Kirwan} [Ann. Math. (2) 122, 41--85 (1985; Zbl 0592.14011)] described a procedure of blowing \(X\) up along a sequence of smooth \(G\)-invariant subvarieties to obtain a variety \(X'\) with a \(G\)-action, such that every semistable point of \(X'\) is stable. Hence, the quotient variety \((X')^{ss}/G\) is projective with only finite quotient singularities, and there is an induced projective birational morphism \((X')^{ss}/G \to X^{ss}/G\) which is an isomorphism over the open set \(X^s/G\). One can therefore consider \((X')^{ss}/G\) as a partial desingularization of \(X^{ss}/G\).
In this paper, the authors study a similar construction when \(\mathcal{X}\) is an Artin toric stack. To do this, they use \textit{Reichstein transformations}, which are certain birational transformations of Artin stacks with good moduli spaces. Let \(\mathcal{C} \subset \mathcal{X}\) be a closed substack. Then the Reichstein transformation of \(\mathcal{X}\) relative to \(\mathcal{C}\) is defined to be the complement of the strict transform of the saturation of \(\mathcal{C}\) relative to the quotient map \(q: \mathcal{X} \to M\) in the blow-up of \(\mathcal{X}\) along \(\mathcal{C}\). Theorem 4.7 states that the Reichstein transformation of a toric stack along a toric substack is another toric stack, which can moreover be described combinatorially in terms of the original toric stack. Applying this result repeatedly gives the result. Edidin, D.; More, Y., Partial desingularizations of good moduli spaces of Artin toric stacks, Michigan Math. J., 61, 451-474, (2012) Stacks and moduli problems, Toric varieties, Newton polyhedra, Okounkov bodies Partial desingularizations of good moduli spaces of Artin toric stacks | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians We prove a result describing the structure of a specific subgroup of the \(m\)-torsion of the Jacobian of a general superelliptic curve \(y^m=F(x)\), generalizing the structure theorem for the \(2\)-torsion of a hyperelliptic curve. We study existence of torsion on curves of the form \(y^q=x^p-x+a\) over finite fields of characteristic \(p\). We apply those results to bound from below the Mordell-Weil ranks of Jacobians of certain superelliptic curves over \(\mathbb{Q}\). Jacobian variety; superelliptic curves; Mordell-Weil group Jacobians, Prym varieties, Zeta functions and related questions in algebraic geometry (e.g., Birch-Swinnerton-Dyer conjecture), Special algebraic curves and curves of low genus On torsion of superelliptic Jacobians | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians The author develops some aspects of derived analytic geometry as introduced by \textit{J. Lurie} in [``Derived Algebraic Geometry V: Structured Spaces'', preprint, \url{arXiv:0905.0459}]. Therefore, a basic knowledge of Lurie's work and the language of \(\infty\)-categories is required for reading the paper.
The paper starts by reviewing the notion of derived analytic space in the sense of Lurie's work. Then, it is shown that there is a functor of points interpretation of the derived spaces obtained on a suitable site of derived Stein spaces. Next, the derived version of the analytication functor is introduced. This is a generalization of the functor studied by Serre extended to the category of derived stacks (more precisely derived Deligne-Mumford stack locally almost of finite presentation) over \(\mathbb{C}\), sending them to derived analytic stacks over \(\mathbb{C}\).
This derived analytification functor is used to prove the main results of the paper. These are a series of GAGA statements that compare algebraic and analytic notions on derived stacks. In particular, it is proved that for a proper morphism of analytic derived Artin stacks \(f: X \to Y\) the push-forward functor induces a functor between the \(\infty\)-categories of coherent sheaves \(f_*: \mathrm{Coh}^-(X) \to \mathrm{Coh}^-(Y)\), that for a morphism of algebraic derived Artin stacks \(f: X \to Y\) the identity \((-)^{\mathrm{an}} \circ f_* = f_* \circ (-)^{\mathrm{an}}\) holds, where \((-)^{\mathrm{an}}\) is the analytification functor, and that for an algebraic proper derived Artin stack \(X\) one has that \(\mathrm{Coh}(X) \cong\mathrm{Coh}(X^{\mathrm{an}})\).
These results are then applied for studying the essential image of the analytification functor and the deformation theory of derived analytic spaces. In particular, it is proved that a proper derived analytic space is in the image of \((-)^{\mathrm{an}}\) if and only if its classical underlying analytic space is, i.e. if and only if its \(0\)-th truncation is the analytification of an algebraic variety over \(\mathbb{C}\). Then, it proved that for any derived analytic space \(X\) and any \(x \in X\) the shifted tangent complex at \(x\), usually denoted \(\mathbb{T}_x X[-1]\), admits a differential graded Lie algebra structure. derived stack; Deligne-Mumford stack; spectral stack; HAG II; DAG V; analytic space; dereived analytic space Generalizations (algebraic spaces, stacks), Presheaves and sheaves, stacks, descent conditions (category-theoretic aspects), Topoi GAGA theorems in derived complex 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 Grothendieck gave two forms of his ``main conjecture of anabelian geometry'', namely the section conjecture and the hom conjecture. He stated that these two forms are equivalent and that if they hold for hyperbolic curves, then they hold for elementary anabelian varieties too. We state a stronger form of Grothendieck's conjecture (equivalent in the case of curves) and prove that Grothendieck's statements hold for our form of the conjecture. We work with DM stacks, rather than schemes. If \(X\) is a DM stack over \(k\subseteq\mathbb{C}\), we prove that whether \(X\) satisfies the conjecture or not depends only on \(X_{\mathbb{C}}\). We prove that the section conjecture for hyperbolic orbicurves stated by \textit{N. Borne} and \textit{M. Emsalem} [Bull. Soc. Math. Fr. 142, No. 3, 465--487 (2014; Zbl 1327.14103)] follows from the conjecture for hyperbolic curves. section conjecture; anabelian geometry Homotopy theory and fundamental groups in algebraic geometry, \(p\)-adic cohomology, crystalline cohomology, Varieties over global fields, Generalizations (algebraic spaces, stacks) Some implications between Grothendieck's anabelian conjectures | 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 central concept discussed in this research monograph is that of ``local moduli suite'' of an algebro-geometric object X. Roughly speaking this is a collection \(\{M_{\tau}\}\) of algebraic spaces each \(M_{\tau}\) prorepresenting the \(\tau\)-constant deformations of X occurring in the family \(\pi_{\tau}\) obtained by restricting an algebraisation \(\pi:\quad \tilde X\to H\) of the formal versal family of X to the ``\(\tau\)-constant stratum''.
An abstract theorem is proved first which guarantees the existence of the local moduli suite in the presence of certain axioms on the formal versal family. Results along this line were independently obtained by Palamodov and Saito. A conjecture of \textit{J. Wahl} [cf. Topology 20, 219-246 (1981; Zbl 0484.14012)] on the dimension of a smoothing component is then proved [this was independently proved by \textit{G.-M. Greuel} and \textit{E. Looijenga} in Duke Math. J. 52, 263-272 (1985; Zbl 0587.32038)].
Next the case of hypersurface singularities is investigated in detail (some modifications of the general setting of local moduli suites are here necessary). Here one is interested in the dimensions of the loci \(M_{\mu \tau}\) of all points in \(M_{\tau}\) corresponding to singularities with a given Milnor number \(\mu\).
Quite precise results are obtained in the case of weighted homogeneous plane curve singularities; in particular a coarse moduli space is proved to exist for all plane curve singularities with given semigroup \(\Gamma =<a_ 1,a_ 2>\), \((a_ 1,a_ 2)=1\) and minimal Tjurina number \(\tau\) and a computation is given for its dimension.
Part of these results are joint work of the authors with \textit{B. Martin}; they are a remarkable contribution to Zariski's program on the moduli problem for curve singularities. local moduli suite; formal versal family; hypersurface singularities; homogeneous plane curve singularities; coarse moduli space; Tjurina number O. A. Laudal and G. Pfister, ''Local moduli and singularities,'' In: Lecture Notes in Math., Vol. 1310, Springer, Berlin (1988). Structure of families (Picard-Lefschetz, monodromy, etc.), Singularities in algebraic geometry, Singularities of curves, local rings, Research exposition (monographs, survey articles) pertaining to algebraic geometry, Fine and coarse moduli spaces, Singularities of surfaces or higher-dimensional varieties, Families, moduli of curves (algebraic) Local moduli and singularities. Appendix (by B. Martin and G. Pfister): An algorithm to compute the kernel of the Kodaira-Spencer map for an irreducible plane curve singularity | 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 nicely written paper the authors develop the machinery of patching techniques in formal geometry which has been used to prove results on fundamental groups of algebraic curves. Among the results obtained we have the Abhyankar conjecture [\textit{M. Raynaud}, Invent. Math. 116, No. 1-3, 425-462 (1994; Zbl 0798.14013) and \textit{D. Harbater}, Invent. Math. 117, No. 1, 1-25 (1994; Zbl 0805.14014)], the Shafarevich conjecture [\textit{F. Pop}, Invent. Math. 120, No. 3, 555-578 (1995; Zbl 0842.14017) and \textit{D. Harbater} [in: Recent developments in the inverse Galois problem, Summer Res. Conf., Univ. Washington 1993, Contemp. Math. 186, 353-370 (1995; Zbl 0858.14013)], and realization of finite groups as Galois groups of projective curves [\textit{M. Saidi}, Compos. Math. 107, No. 3, 319-338 (1997; Zbl 0929.14016) and \textit{K. Stevenson}, J. Algebra 182, No. 3, 770-804 (1996; Zbl 0869.14011)].
One of the main goals of the paper is to develop a framework in which similar types of constructions can be facilitated. For instance, the authors show that singular curves over a field \(k\) can be thickened to curves over \(k[[t]]\) with prescribe behavior in a formal neighborhood of the singular locus.
The first result concerns patching problems for projective curves \(X^*\) over the formal power series ring \(R=k[[t_1,\cdots, t_r]]\). More precisely the authors show that ``giving a coherent projective module over \(X^*\) is equivalent to giving such module compatibly on the formal neighborhood of each singular point of the closed fiber of \(X\), and on the formal thickening along the component of the singular locus \(S\) of \(X\)''. This generalizes a previous result of \textit{D. Harbater} [Am. J. Math. 115, No. 3, 487-508 (1993; Zbl 0790.14027); theorem 1].
In the second section the authors apply this result to thickening problems. Building-up on these results applied to thickening and deforming covers, the authors obtain in the fourth section information on the fundamental groups of curves. In the affine case, they show that a result of Raynaud used in the proof of the Abhyankar conjecture for the affine line, which relied on Runge pairs, can be proved with the techniques of the paper. This result as well as further ones concerning finite quotients of fundamental groups of curves with prescribed ramification are stated for large fields rather than for algebraically closed fields. thickening; patching; formal geometry; Shafarevich conjecture; patching problems; fundamental groups of algebraic curves; Abhyankar conjecture D. Harbater and K. F. Stevenson, ''Patching and thickening problems,'' J. Algebra, vol. 212, iss. 1, pp. 272-304, 1999. Coverings of curves, fundamental group, Homotopy theory and fundamental groups in algebraic geometry Patching and thickening problems | 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 an algebraic function field in one variable over an algebraically closed field of positive characteristic \(p\), and let \(\overline K\) denote the algebraic closure of \(K\). An algebraic curve \(C/K\) is said to be non-conservative or genus-changing if its relative genus \(g_K\) is different from the absolute genus \(\overline g:=g_{\overline K}\). \textit{P. Samuel} [``Lectures on old and new results on algebraic curves'', Tata Inst. Fund. Res. (Bombay 1966; Zbl 0165.24102)] proved that a genus-changing algebraic curve \(C\) of absolute genus \(\overline g\geq 2\) has only finitely many \(K\)-rational points. \textit{J. P. Voloch} [Bull. Soc. Math. Fr. 119, No. 1, 121-126 (1991; Zbl 0735.14018)] established the same finiteness result for algebraic curves of absolute genus \(0\) or \(1\), when the constant field of the function field \(K\) is finite.
In this paper the author considers genus-changing algebraic curves of absolute genus \(0\) for general \(K\), and asks if they have only finitely many \(K\)-rational points. The main result of this paper is formulated as follows:
Theorem: Let \(K\) be a function field in one variable over an algebraically closed field of positive characteristic \(p\). Then every non-conservative algebraic curve \(C\) over \(K\) has finitely many \(K\)-rational points, that is, the set \(C(K)\) of \(K\)-rational points is finite.
The result is first established for an algebraic curve \(C\) of the form \(y^p=r(x)\) that admits genus change. From this result, the author deduces the finiteness for \(C(K)\) for every genus changing algebraic curve \(C\) over \(K\). genus-changing algebraic curves; finite number of rational points; characteristic \(p\); function field; non-conservative algebraic curve Jeong, S.: Rational points on algebraic curves that change genus. J. number theory 67, 170-181 (1998) Rational points, Algebraic functions and function fields in algebraic geometry, Special algebraic curves and curves of low genus Rational points on algebraic curves that change genus | 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 proper curve over a base scheme \(B\), and let \(G\) be a reductive algebraic group. The classification theory of rank-\(r\) vector bundles (or, more generally, of principal \(G\)-bundles) over \(C\) has recently gained crucial significance in the relationship between algebraic geometry and conformal quantum field theory. Especially the construction and the geometric investigation of the relevant moduli spaces of bundles (or, more generally, of the algebraic moduli stacks of \(G\)-torsors) are of fundamental importance.
In the present paper, the author exhibits two general constructions for such moduli spaces and moduli stacks, respectively. As to that, the underlying strategy is to show that various invariants are independent of the curve \(C\), and then to let \(C\) degenerate to a rational nodal curve, hoping that in this case the relevant invariants can be explicitly computed. This amounts to study the behavior of moduli spaces and moduli stacks under degenerations. The first construction is rather geometric, but applies only to some particular linear groups \(G\). Nonetheless, this approach leads to global moduli stacks for torsion-free \(G\)-sheaves of given rank \(r\) on \(C\) and, under suitable semi-stability conditions, to classifying algebraic spaces (moduli schemes). The second approach uses loop groups and works for arbitrary reductive structure groups. It is based on an alternative treatment of the local structure of \(G\)-torsors on semi-stable curves and, henceforth, brings about the drawback of being non-canonical and non-global. The resulting object of this construction is a stack which, in characteristic zero, satisfies the valuative criterion of properness. Although, in general, this stack does not classify particular torsion-free sheaves over \(C\), it sometimes maps naturally to one of the previously constructed stacks and, for this reason, might be useful for the purpose of comparison.
As the author points out, this second method of construction was inspired by his recent general proof of the Verlinde formula [cf. J. Algebr. Geom. 3, No. 2, 347-374 (1994; Zbl 0809.14009)]. semistable curves; torsors; conformal quantum field theory; moduli spaces; moduli stacks; loop groups; Verlinde formula Hobson, N.: Quantum Kostka and the rank one problem for \(\mathfrak{s}l_{2m}\) (2015). arXiv:1508.06952 [math.AG] Vector bundles on curves and their moduli, Algebraic moduli problems, moduli of vector bundles, Homogeneous spaces and generalizations, Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20], Other algebraic groups (geometric aspects) Moduli-stacks for bundles on semistable curves | 0 |
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