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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 We prove that the moduli spaces \(\mathcal A_3(D)\) of polarized abelian threefolds with polarizations of types \(D=(1,1,2)\), \((1,2,2)\), \((1,1,3)\) or \((1,3,3)\) are unirational. The result is based on the study of families of simple coverings of elliptic curves of degree \(2\) or \(3\) and on the study of the corresponding period mappings associated with holomorphic differentials with trace \(0\). In particular, we prove the unirationality of the Hurwitz space \(\mathcal H_{3A}(Y)\) which parametrizes simply branched triple coverings of an elliptic curve Y with determinants of the Tschirnhausen modules isomorphic to \(A^{-1}\). Hurwitz spaces; abelian threefolds; Prym varieties; moduli; unirationality Kanev, V.: Irreducibility of Hurwitz spaces. Preprint N. 241, Dipartimento di Matematica ed Applicazioni, Università degli Studi di Palermo (2004); arXiv: math. AG/0509154 Algebraic moduli of abelian varieties, classification, Jacobians, Prym varieties, Rational and unirational varieties, Families, moduli of curves (algebraic), Coverings of curves, fundamental group, Variation of Hodge structures (algebro-geometric aspects) Hurwitz spaces of triple coverings of elliptic curves and moduli spaces of abelian threefolds | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians This paper consists of a direct proof that a recursive formula proved by \textit{S. Keel} in [Trans. Am. Math. Soc. 330, No. 2, 545--574 (1992; Zbl 0768.14002)] for the Betti numbers of the moduli space of stable complex rational curves with marked points also holds for the number of points of the space of stable rational curves over any finite field. moduli space; stable rational curves; rational points Families, moduli of curves (algebraic), Curves over finite and local fields, Finite ground fields in algebraic geometry Poincaré polynomial of the space \(\overline{\mathcal M_{0,n}}(\mathbb{C})\) and the number of points of the space \(\overline{\mathcal M_{0,n}}(F_q)\) | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is 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 preface: The objective of this work is to shed some light on \(\overline{{\mathcal M}_g}\), the compactification of the moduli space \({\mathcal M}_g\) of Riemann surfaces of genus \(g\), by considering only the Riemann surfaces with nodes \(X\) in \(\overline{{\mathcal M}_g}\) which have a given homomorphic image \(G\): Let
\[
{\mathcal M}_{g,G}: =\bigl\{X \in\overline {{\mathcal M}_g} :\text{ it exists a surjective group homomorphism }\pi_1 (X)\to G\bigr\}.
\]
In chapter 1 we will exhibit which groups \(G\) need to be considered and prove some combinatorial group theoretic results about them. To give some understanding of \({\mathcal M}_{g,G}\), we construct an \(X\) in the interior of \({\mathcal M}_{g,G}\). The main result of this thesis:
If \(X\in{\mathcal M}_{g,G}\) and \(\varepsilon: \pi_1(X) \to G\) is given, the regular covering with \(\text{Deck} (Y/X) @>\sim> \varepsilon>G\) is a tree of planar Riemann surfaces.
This is proved in chapter 2. There is some hope that the universal family \({\mathcal X}_{g,G} @>p>>{\mathcal M}_{g, G}\) (with \(p^{-1} (\{X\}) \cong X\) for every \(X\in {\mathcal M}_{g,G})\) is related to the deformation space of \(G\) as a Kleinian group. In chapter 3, we show that this deformation space is a finite-dimensional complex analytic manifold and exhibit global holomorphic coordinates for it. compactification of the moduli space of Riemann surfaces; deformation space; Kleinian group Coverings of curves, fundamental group, Kleinian groups (aspects of compact Riemann surfaces and uniformization), Riemann surfaces; Weierstrass points; gap sequences, Families, moduli of curves (algebraic) Schottky-like coverings | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians The authors extend the results of the paper reviewed above on the structure of Picard groups of ``generic'' Zariski surfaces to the ``general'' ones by using specialization methods. factorial ring Picard groups, Special surfaces Appendix to ``Picard groups of Zariski 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 \(C\) be a general smooth curve of genus \(g\) and \(L\) a general line bundle on \(C\) with Clifford dimension \(\mathrm{Cliff}(L)\) equal to the Clifford index \(\mathrm{Cliff} (C)\) of \(C\) (it implies \(h^1(L)\leq 2)\). If \(h^1(L)=0\), then \(L\) embeds \(C\) as a projectively normal curve [\textit{E. Ballico} and \textit{Ph. Ellia}, Math. Z. 188, 215--223 (1985; Zbl 0631.14029)]. Here the author proves the cases \(h^1(L) =1, 2\). The result seems to be covered by \textit{E. Ballico} and \textit{C. Fontanari} [J. Pure Appl. Algebra 214, No. 6, 837--840 (2010; Zbl 1184.14039); J. Pure Appl. Algebra 214, No. 8, 1450--1455 (2010; Zbl 1185.14028)], but the proof is new and interesting. It uses deformation theory (at all orders, not just a tangent space computation) taking as \(C\) a smoothing of a general curve of compact type \(X\cup Y\) with two components. The line bundle on \(X\cup Y\) has bad postulation, but its badness is killed at a finite order of the deformation. projectively normal curve; deformation theory; reducible curve; postulation; Clifford index; Clifford dimension Special divisors on curves (gonality, Brill-Noether theory), Families, moduli of curves (algebraic), Special algebraic curves and curves of low genus On the projective normality of line bundles of extremal degree | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is 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 discuss geometric invariant theory (GIT) for canonically embedded genus 4 curves and the connection to the Hassett-Keel program. A canonical genus 4 curve is a complete intersection of a quadric and a cubic, and, in contrast to the genus 3 case, there is a family of GIT quotients that depend on a choice of linearization. We discuss the corresponding variation of GIT (VGIT) problem and show that the resulting spaces give the final steps in the Hassett-Keel program for genus 4 curves. geometric invariant theory; Hassett-Keel program; log canonical model Casalaina-Martin, S.; Jensen, D.; Laza, R., Log canonical models and variation of GIT for genus 4 canonical curves, J. Algebraic Geom., 23, 727-764, (2014) Geometric invariant theory, Families, moduli of curves (algebraic), Minimal model program (Mori theory, extremal rays), Divisors, linear systems, invertible sheaves, Parametrization (Chow and Hilbert schemes), Special algebraic curves and curves of low genus Log canonical models and variation of GIT for genus 4 canonical curves | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians In this paper, a birational invariant for varieties in characteristic \(p\), as a quotient of the étal cohomology, is defined. The non-vanishing of this invariant guarantees the non-uniruledness of the variety. For \(n\geq p\geq 11\), the theory of moduli forms is used to show that this invariant for the moduli \(\overline{M}_{1,n,\mathbb{F}_p}\) of elliptic curves with \(n\) marked points, is non-zero, and hence \(\overline{M}_{1,n,\mathbb{F}_p}\) is not uniruled. unirational; uniruled; characteristic \(p\); moduli of curves; elliptic curves; modular forms; étale cohomology Rational and unirational varieties, Families, moduli of curves (algebraic) \(\overline M_{1,n}\) is usually not uniruled in characteristic \(p\) | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is 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 is now an algebraic version of Chen and Ruan's orbifold cohomology defined for smooth Deligne-Mumford stacks by \textit{D. Abramovich}, \textit{T. Graber} and \textit{A. Vistoli} [in: Orbifolds in mathematics and physics, Contemp. Math. 310, 1--24 (2002; Zbl 1067.14055)]. In that same paper, the authors determine this ring with integral coefficients for the stacks \({\mathcal M}_{1,1}\) and \(\overline{{\mathcal M}}_{1,1}\) of one-pointed nonsingular (respectively stable) genus-one curves. The purpose of this note is to give an in-depth treatment of the exercise of computing the stringy Chow ring with rational coefficients for the stack \({\mathcal M}_2\) of nonsingular genus-two curves. I have also computed, by an approach similar to the one used in this paper, the ring with integral coefficients for \({\mathcal M}_2\) and \(\overline{{\mathcal M}}_2\). Those results will appear in subsequent publications. Spencer, J.: The orbifold cohomology of the moduli of genus-two curves. Gromov--Witten theory of spin curves and orbifolds. Contemp. Math. 403, 167--184 (2006) Families, moduli of curves (algebraic), Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects), Gromov-Witten invariants, quantum cohomology, Frobenius manifolds The orbifold cohomology of the moduli of genus-two curves | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians The main result of this paper is that the nef and ample cones of the Kontsevich space \(\overline{\mathcal{M}}_{0,n}(\mathbb{P}^r,d)\) are determined by the corresponding cones of \(\overline{\mathcal{M}}_{0,n+d}\). Describing these cones for these and other moduli spaces is an active area of research combining tools from moduli theory and the minimal model program. The reduction in this paper from the space of maps to the space of curves is quite elegant, though one should keep in mind that the nef/ample cones for \(\overline{\mathcal{M}}_{0,n}\) are at this point only conjectural, the most prevalent description being known as the F-conjecture [see \textit{A. Gibney, S. Keel} and \textit{I. Morrison}, J. Am. Math. Soc. 15, No. 2, 273--294 (2002; Zbl 0993.14009)].
A more precise statement of the main result of this paper is as follows: the nef cone of \(\overline{\mathcal{M}}_{0,n}(\mathbb{P}^r,d)\) equals the nef cone in the product \(\text{Pic}(\overline{\mathcal{M}}_{0,n+d})^{S_d}\times\text{Pic}(\mathbb{P}^{r-1})\times\text{Pic}(\mathbb{P}^1)^n\times\text{Pic}(\mathbb{P}^1)\) and consequently the ample cone of this Kontsevich space corresponds to the product of the ample cones of these factors. The various projective spaces in this product correspond to natural divisors on the Kontsevich moduli space.
A consequence the authors derive from this result is a certain contraction of the Kontsevich space analogous to one for \(\overline{\mathcal{M}}_{0,n}\) first constructed by Keel and McKernan. Kontsevich space; stable maps; moduli of curves; NEF cone; ample cone; contraction Coskun, Izzet; Harris, Joe; Starr, Jason, The ample cone of the Kontsevich moduli space, Canad. J. Math., 61, 1, 109-123, (2009) Families, moduli of curves (algebraic), Birational geometry The ample cone of the Kontsevich moduli 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 Let \(X\) be a homogeneous, projective variety, and let \(\overline M_{0,n}(X,\beta)\) be the space of genus zero, degree \(\beta\) stable curves to \(X\). \textit{B.~Kim} and \textit{R. Pandharipande} [in: Symplectic geometry and mirror symmetry, Proc. 4th KIAS Conf., Seoul 2000, 187--201 (World Scientific, Singapore) (2001; Zbl 1076.14517)] proved that \(\overline M_{0,n}(X,\beta)\) is a normal and irreducible, projective variety. Moreover they described its Bialynicki-Birula stratification corresponding to the maximal torus action. The virtual Poincaré polynomial of \(\overline M_{0,n}(X,\beta)\) has been computed by \textit{Yu.~Manin} [Topol. Methods Nonlinear Anal. 11, 207--217 (1998; Zbl 0987.14033)].
The tautological ring of \(\overline M_{0,n}(X,\beta)\) is the smallest subring of \(H^*(\overline M_{0,n}(X,\beta))\) having the following properties: it contains the pull-back by the evaluation morphisms of the cohomology ring of \(X\), and is closed under pull-backs and push-forwards by the forgetful morphisms, corresponding to various values of \(n\).
The goal of the article under review is to find generators, and to compute the dimensions of the cohomology (resp. Chow) groups of \(\overline M_{0,n}(X,\beta)\). The author succeeds in doing this for the Chow groups of codimension one and two.
At a first stage, the author describes the generators of the codimension one Chow group of \(\overline M_{0,n}(X,\beta)\); it turns out that all these are tautological classes. Assuming \(X\) is homogeneous for the special linear group, that is \(X\) is a flag variety, he computes moreover the dimension of \(A^1(\overline M_{0,n}(X,\beta))\).
Secondly, using a localization argument, the author describes a basis for the codimension two Chow group of \(\overline M_{0,n}(\mathbb P^r,d)\), for \(n=0,1,2\). One should compare his computations with those of \textit{K.~Behrend} and \textit{A.~O'Halloran} [Invent. Math. 154, 385--450 (2003; Zbl 1092.14019)].
Thirdly, a basis of \(A^2(\overline M_{0,n}(\text{Grass}(r,N),d))\), for \(n=0,1\), is determined. The computation is used for proving a reconstruction theorem, similar to that of Kontsevich-Manin, for genus zero Gromov-Witten invariants of \(\text{Grass}(r,N)\). stable maps; flag spaces; tautological ring; cohomology groups; Chow groups Oprea, D, Divisors on the moduli spaces of stable maps to flag varieties and reconstruction, J. Reine Angew. Math., 586, 169-205, (2005) Families, moduli of curves (algebraic), Families, moduli, classification: algebraic theory, (Equivariant) Chow groups and rings; motives, Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects), Grassmannians, Schubert varieties, flag manifolds, Other algebro-geometric (co)homologies (e.g., intersection, equivariant, Lawson, Deligne (co)homologies) Divisors on the moduli spaces of stable maps to flag varieties and reconstruction | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians Let \(C\) be a smooth projective complex curve with a cyclic group of automorphisms of prime order. By using the well-known isomorphism between NS(J) (the Neron-Severi group of the jacobian of \(C)\) and the invariant part (under the Rosati involution) of its ring of endomorphisms, the author gives a geometric description of certain elements of NS(J) which allows him to estimate rank NS(J) from below. Then the author proves the main result which asserts that under certain hypotheses, about the eigenspaces of the space of holomorphic differentials, those found elements span NS(J). cyclic group automorphisms; Neron-Severi group; Jacobian; ring of endomorphisms Picard groups, Jacobians, Prym varieties, Relationships between algebraic curves and integrable systems, Special algebraic curves and curves of low genus On the Neron-Severi group of Jacobians of curves with automorphisms | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is 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 0561.00011.]
The author considers a moduli problem, namely, the classification of nilpotent endomorphisms of indecomposable semi-stable vector bundles of rank 2 over a compact connected Riemann surface. Defining first a global (universal) family of endomorphisms of vector bundles parametrized by a variety, and using the universal property of the Picard variety and the universal family of extensions given by \textit{M. S. Narasimhan} and \textit{S. Ramanan} [Ann. Math., II. Ser. 89, 14-51 (1969; Zbl 0186.549)], the author constructs a universal family of the above endomorphisms for P(2,d), where P(n,d) denotes the set of isomorphism classes of the pairs [E, ], E being vector bundles of rank n and d being the slope of E. endomorphisms of indecomposable semi-stable vector bundles; compact connected Riemann surface; Picard variety; universal family L. Brambila, Moduli of endomorphisms of vector bundles over a compact Riemann surface, preprint. Sheaves and cohomology of sections of holomorphic vector bundles, general results, Algebraic moduli problems, moduli of vector bundles, Families, moduli of curves (algebraic), Compact Riemann surfaces and uniformization Endomorphisms of vector bundles over a compact Riemann surface | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians Let \(S\) be a smooth projective \(K3\) surface over a number field \(k\), \(k\subset \mathbb C\), \((k:\mathbb Q)<\infty\), and let \(\ell\) be a prime number. Let
\[
\rho_{\ell}: \text{Gal}(\bar k/k)\to \mathrm{GL}(H^ 2_{\ell}(S\otimes_ k\bar k))
\]
be the natural \(\ell\)-adic representation and let \(\text{MT}(S_{\mathbb C})\) (resp. \(\text{NS}(S_{\mathbb C}))\) be the Mumford-Tate group (resp. the Néron-Severi group) of the surface \(S_{\mathbb C}=S\otimes_ k\mathbb C\). Let \(V_{\mathbb Q}(1)\) be the orthogonal complement of the subspace \(\text{NS}(S_{\mathbb C})\otimes_{\mathbb Z}\mathbb Q\subset H^ 2(S_{\mathbb C},\mathbb Q)(1)\) with respect to the intersection multiplicity. Yu. G. Zarkhin proved that the canonical representation of \(\text{MT}(S_{\mathbb C})\) in \(V_{\mathbb Q}(1)\) is irreducible and \(K=\text{End}_{\text{MT}(S_{\mathbb C})}V_{\mathbb Q}(1)\) is a field. Set \(e=(K:\mathbb Q)\). The main result of the paper is contained in the following theorem:
(1) For every smooth projective \(K3\) surface over the number field \(k\) the canonical map
\[
\text{NS}(S\otimes_ k\bar k)\otimes_{\mathbb Z}\mathbb Q_{\ell}\rightarrow [H^ 2_{\ell}(S\otimes_ k\bar k)(1)]^{\text{Lie}\,(\text{Im}(\rho_{\ell}))}
\]
is an isomorphism.
(2) If \(e=1\), then the canonical map
\[
\text{Lie}\,\text{Im}(\rho_{\ell})\rightarrow \text{Lie}\, \text{MT}(S_{\mathbb C})(\mathbb Q_{\ell})
\]
is an isomorphism.
Thus, for every \(K3\) surface over a number field the Tate conjecture for algebraic cycles is true. If \(e=1\) (that is, the canonical representation of \(\text{MT}(S_{\mathbb C})\) in \(V_{\mathbb Q}(1)\) is absolutely irreducible), then the Mumford-Tate hypothesis for \(S\) is true. projective K3 surface; Mumford-Tate group; Néron-Severi group; intersection multiplicity; Tate conjecture for algebraic cycles Tankeev, S. G., Surfaces of type \(K3\) over number fields, and \(l\)-adic representations, Math. USSR-Izv., 0373-2436, 52 33, 3, 575-595, (1989) \(K3\) surfaces and Enriques surfaces, Global ground fields in algebraic geometry, Cycles and subschemes, Local ground fields in algebraic geometry, Picard groups K3 surfaces over number fields and \(\ell\)-adic representations | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians In this paper, it is proved that the generating series of the Hodge integrals over the moduli space of stable curves is a solution of a certain deformation of the KdV hierarchy. This hierarchy is constructed in the framework of the Dubrovin-Zhang theory of hierarchies of topological type. It occurs that this deformation of the KdV hierarchy is closely related to the hierarchy of the intermediate long wave equation. deformed KdV hierarchy; cohomological field theories; intermediate long wave equation Buryak, A., Dubrovin--{Z}hang hierarchy for the {H}odge integrals, Communications in Number Theory and Physics, 9, 2, 239-272, (2015) Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.), Families, moduli of curves (algebraic), , Relations of infinite-dimensional Hamiltonian and Lagrangian dynamical systems with algebraic geometry, complex analysis, and special functions Dubrovin-Zhang hierarchy for the Hodge 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 Let R be an associative ring with unit element and let R-mod be the category of left (unitary) R-modules. An (idempotent) kernel functor \(\kappa\) is a left exact subfunctor of the identity on R-mod such that \(\kappa (M/\kappa (M))=0\) for all M in R-mod. Such kernel functors define a ''localization'' on R-mod: The quotient module of M in R-mod is \(Q_{\kappa}(M)\cong \lim_{L\in \vec L(\kappa)}Hom_ R(L,M/\kappa (M))\), where \({\mathcal L}(\kappa)\) is the filter associated to \(\kappa\), i.e. the set of left ideals L of R such that \(\kappa (R/L)=R/L\). Since invariants as the Picard group, Brauer group and cohomology groups, classically associated to commutative rings are defined categorically it is possible to define them relative to some localization with respect to a kernel functor \(\kappa\).
As an example we define the \(\kappa\)-Picard group. A module M in R-mod is \(\kappa\)-closed if it is faithfully \(\kappa\)-injective, i.e. every homomorphism f fatorizes over a monomorphism in M with \(\kappa\)-invariant cokernel. - An R-module P is said to be \(\kappa\)-invertible if it is \(\kappa\)-closed and there exists a \(\kappa\)-closed R-module Q such that \(Q_{\kappa}(P\otimes Q)\cong Q_{\kappa}(R)\). The set of isomorphism classes [P] of \(\kappa\)-invertible R-modules P is denoted Pic(R,\(\kappa)\) and is called the Picard group of R relative to \(\kappa\). The group structure is given by the product \([P]\cdot [Q]=[Q_{\kappa}(P\otimes Q)].\)
In the book under review the theory of relative invariants (Picard groups, Brauer groups, etc.) is built up. Special attention goes to the relative Picard groups and to the relative Brauer groups. The first chapter deals with generalities on localizations and invariants. - In chapter 2 the relative Picard group is introduced, in chapter 3 the relative Brauer group. - Chapter 4 and 5 deal with applications of the introduced notions. In the first the Brauer groups and Picard groups of ringed spaces and quasi-affine schemes are studied using the relative techniques. In the latter Noetherian integrally closed domains are considered. - In the last chapter the Brauer group of a projective scheme is studied. In chapter 4 it was shown that the Brauer group of a quasi- affine open subscheme X(I) of Spec(R) is nothing but the relative Brauer group \(Br(R,\kappa_ I)\). One would expect that if one can take into account graded structures of rings an analogous result should hold for projective schemes, too. However one has to be careful here, e.g. the Brauer group of Proj(R), with R a graded ring is not the graded Brauer group \(Br^ g(R)\), but it is a relative (graded) Brauer group. - This last chapter uses of course also techniques and facts from graded ring theory. These are fully provided by chapter 6.
Finally a few remarks on the presentation. The typing is indeed incredibly good but the correction of the text afterwards is not that good. References are not filled in at several places and several other errors occur. - At the end of the book some exercises to each chapter are brought together. As one can expect several of them concern calculating explicitly relative invariants. It wouldn't have been a bad idea to put some explicit examples in the text, too. kernel functors; localization; invariants; relative Picard groups; relative Brauer groups; ringed spaces; quasi-affine schemes; projective scheme; graded ring theory Van Oystaeyen, F.; Verschoren, A.: Relative invariants of rings, the commutative theory. (1983) Brauer groups of schemes, Picard groups, Research exposition (monographs, survey articles) pertaining to commutative algebra, Grothendieck groups, \(K\)-theory and commutative rings, Schemes and morphisms Relative invariants of rings. The commutative theory | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians Let \(M\) be the moduli space of rank 2 stable vector bundles \(E\) on a complex curve \(C\) of genus \(g\), with \(\det(E)\cong \xi\), where \(\xi\) is a fixed line bundle on \(C\). By a result of Drezet and Narasimhan, \(\text{Pic}(M) \cong {\mathbb Z}\). If \(\Theta \) is the ample generator of \(\text{Pic}(M)\), by ``rational curve of degree \(k\)'' is meant a non-constant morphism \(f: {\mathbb P}^1 \to M\), such that \(f^\ast \Theta \cong {\mathcal O}(k)\). The main achievements of this paper are:
1. determining the irreducible components of the Hilbert scheme of rational curves of degree \(k\geq 1\) on \(M\), \(\Hom_k({\mathbb P}^1,M)\). Previously, only the cases \(k=1\), \(k=2\) where known in the literature [cf. \textit{S. Kilaru}, Proc. Indian Acad. Sci., Math. Sci. 108, No. 3, 217--226 (1998; Zbl 0947.14008)];
2. determining the maximal rationally connected fibrations of these components. rational curve; rationally connected; rank 2 vector bundle; intermediate Jacobian Castravet A.-M., Rational families of vector bundles on curves, Internat. J. Math., 2004, 15(1), 13--45 Vector bundles on curves and their moduli, Fano varieties, Rational and unirational varieties, Families, moduli of curves (algebraic), Algebraic moduli problems, moduli of vector bundles Rational families of vector bundles on curves. | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians [For the entire collection see Zbl 0583.00007.]
The author extends the notion of Hecke-action on Picard groups, studied by Roggenkamp and Scott to Hecke actions on class groups by representing the class group as a relative Picard group. Some results concerning schemes are derived from this. Hecke actions on class groups Verschoren, A.: Hecke actions on relative Picard groups. Lecture notes in mathematics 1197, 207-224 (1986) Theory of modules and ideals in commutative rings, Brauer groups of schemes, Picard groups Hecke actions on relative Picard groups | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians Let \(J \to \Delta^*\) be a family of intermediate Jacobians arising from a variation of polarized Hodge structures of weight \(-1\) with a unipotent monodromy of a punctured disk. The intermediate Jacobians are isomorphic to the extension groups of the Hodge structures in the category of mixed Hodge structures, and a section of \(J \to \Delta^*\) is a variation of mixed Hodge structures. For a variation of Hodge structure over a punctured disk, \textit{M. Green, P. Griffiths} and \textit{M. Kerr} [Compos. Math. 146, No. 2, 288--366 (2010; Zbl 1195.14006)] introduced a Néron model which is a Hausdorff space that includes values of admissible normal functions. On the other hand, another Néron model was introduced by \textit{K. Kato} and \textit{S. Usui} [Classifying spaces of degenerating polarized Hodge structures. Annals of Mathematics Studies 169. Princeton, NJ: Princeton University Press (2009; Zbl 1172.14002)] as a logarithmic manifold using the log mixed Hodge theory. In this paper the author constructs a homeomorphism between these two models. Néron models; mixed Hodge structures; intermediate Jacobians Tatsuki Hayama, Néron models of Green-Griffiths-Kerr and log Néron models, Publ. Res. Inst. Math. Sci. 47 (2011), no. 3, 803 -- 824. Variation of Hodge structures (algebro-geometric aspects), Families, moduli of curves (algebraic), Transcendental methods, Hodge theory (algebro-geometric aspects) Néron models of Green-Griffiths-Kerr and log Néron models | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians Let \(C\) be a stable curve of genus \(g\). The gonality of \(C\) is defined to be the minimal integer \(k\) such that \(C\) is a flat limit of smooth \(k\)-gonal curves. The gonality of \(C\) is the minimal integer \(k\) such that \(C\) fits as an admissible cover of \(\mathbb {P}^1\) in the sense of J. Harris and D. Mumford. In this paper, first the authors compare the gonality of \(C\) and the gonality of the normalization of its components, giving both upper bounds and lower bounds. Then they give \(4\) constructions of admissible covering to get generically injective maps between certain Hurwitz schemes and pointed Hurwitz schemes and describe the closures of the images of these maps. There is a very different definition of gonality in terms of balanced line bundles on a stable curve due to L. Caporaso and the interested reader may find results to both definitions in [\textit{G. Bini} et al., Geometric invariant theory for polarized curves. Cham: Springer (2014; Zbl 1328.14072)]. gonality; stable curves; admissible covers; Hurwitz schemes Families, moduli of curves (algebraic), Special divisors on curves (gonality, Brill-Noether theory) On the gonality of stable curves | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians The author formulates, interrelates and in special cases proves conjectures pertaining to the three topics of the title.
Let \(G_{(n,s)}\) denote the maximum genus of a smooth curve of degree \(n\) in \(\mathbb{P}^ 3\) (complex projective 3-space) not contained in a surface of degree less than \(s\), and let \(H_{n,G_{(n,s)}}\) denote the Hilbert scheme of smooth curves of degree \(n\) and genus \(G_{(n,s)}\) in \(\mathbb{P}^ 3\).
Conjecture (H). If \({(s^ 2 + 4s + 6) \over 6} \leq n < {(s^ 2 + 4s + 6) \over 3}\) then \(H_{n,G_{(n,s)}}\) is pure dimensional of dimension \(4n\).
This conjecture has been proven for certain special cases, e.g. \(s=2\).
Next recall that an instanton bundle on \(\mathbb{P}^ 3\) is a rank 2 bundle \(F\) with \(c_ 1(F) = 0\), \(H^ 0(F) = H^ 1(F(-2)) = 0\). There is a well defined moduli space \(MI(0,c_ 2)\) of instantons with \(c_ 2(F) = c_ 2\).
Conjecture \((\text{I}')\). Every component of \(MI (0,c_ 2)\) has dimension \(8c_ 2 -3.\)
(Conjecture I is the conjecture that \(MI (0,c_ 2)\) is irreducible, and is known to imply \(\text{I}'\), where the notation. It has been proven for \(c_ 2 \leq 4.)\)
A further technical conjecture, attributed to \textit{R. Hartshorne} [in: Vector bundles and differential equations, Proc., Nice 1979, Prog. Math. 7, 83-112 (1980; Zbl 0452.14005)] is:
Conjecture (T). Let \(W\) be a component of \(M(0, c_ 2)\) and let \(F\) be a general element of \(W\). Then \(H^ 0 (F(t)) = 0\) for \(t \leq \sqrt {3c_ 2 + 1} - 2\).
Let \(S(d)\) be the space of smooth surfaces of degree \(d\) in \(\mathbb{P}^ 3\) and let \(NL(d)\) be the Noether-Lefschetz locus, i.e., the subset of \(S(d)\) consisting of smooth surfaces whose Picard group is larger than \(\mathbb{Z}\). It is known that \(NL(d)\) is the union of countably many closed proper subsets of \(S(d)\), called components. A component of \(NL(d)\) is general if its codimension in \(S(d)\) is the expected value \(\left( \begin{smallmatrix} d - 1 \\ 3 \end{smallmatrix} \right)\), otherwise special. Green and Ciliberto conjectured that for a generic element \(S\) of a special component of \(NL(d)\), some canonical divisor on \(S\) has a component whose class is not a multiple of the hyperplane class. This would imply that there are only finitely many special components of \(NL(d)\), but this is now known to be false [\textit{C. Voisin}, C. R. Acad. Sci., Paris, Sér. I 313, 685-687 (1991; Zbl 0751.14020)]. Nevertheless, there is some hope that restricted versions of the Green-Ciliberto conjecture may be true. In the present paper the author introduces the notion of regular component \(W(d)\) of \(NL(d)\); the definition involves the choice of a component \(W\) of the Hilbert scheme of smooth curves of \(\mathbb{P}^ 3\). If \(C\) is a curve representing the generic point of \(W\), and \(S\) a general surface of degree \(d\) containing \(C\) then \(C^ 2\) means the self-intersection of \(C\) in \(S\).
Conjecture (GCR). The Green-Ciliberto conjecture holds for special regular components of \(NL(d)\) with \(C^ 2 \leq 0\).
The principal results of the paper are the following:
Theorem. \((d \geq 7)\). Conjecture (H) is equivalent to conjecture (GCR) for components of \(NL(d)\) with \(C^ 2 \leq - 2\).
Theorem. \((d \geq 7)\). Conjecture (GCR) is true for components of \(NL(d)\) with \(C^ 2 = 0\) if \(d \neq 10, d\equiv 1\pmod 3\).
Let \({\mathcal J}_ C\) be the ideal of \(C\) in \({\mathcal O}_{\mathbb{P}^ 3}\).
Theorem. Suppose \(d \equiv 0,2 \pmod 6\), \(d \geq 8\), and let \(c_ 2 = {d^ 2 - 8d + 12 \over 12}\). Then
Conjecture \((\text{I}')\) implies conjecture (GCR) for components of \(NL(d)\) with \(H^ 1({\mathcal J}_ C({d \over 2} - 2)) = 0\) and \(C^ 2 = 0\).
Conversely, conjecture (GCR) for components of \(NL(d)\) with \(C^ 2 = 0\) and \(H^ 1({\mathcal J}_ C ({d \over 2} - 2)) = 0\), together with conjecture (T), implies conjecture \((\text{I}')\). dimension of Hilbert scheme; dimension of moduli space of instantons; space curves; instanton bundle; Noether-Lefschetz locus Algebraic moduli problems, moduli of vector bundles, Families, moduli of curves (algebraic), Hilbert-Samuel and Hilbert-Kunz functions; Poincaré series, Families, moduli, classification: algebraic theory Noether-Lefschetz, space curves and mathematical instantons | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians The author studies the divisibility of the images of the specialization maps for an elliptic surface \(E\to {\mathbb{P}}^ 1\) defined over \({\mathbb{Q}}\). To be more precise, let \(E\to {\mathbb{P}}^ 1\) be an elliptic surface over \({\mathbb{Q}}\) with non-constant j-invariant. Let \(t\in {\mathbb{P}}^ 1({\mathbb{Q}})\) be a point such that the fibre \(E_ t\) is a non-singular elliptic curve, and let \(\sigma_ t: E({\mathbb{P}}^ 1)\to E_ t({\mathbb{Q}})\) be the specialization homomorphism (which is known to be injective for all but finitely many t). Using the theory of canonical heights the author proves that the set of positive integers t, for which the quotient group \(E_ t({\mathbb{Q}})/\sigma_ t(E({\mathbb{P}}^ 1))\) has no points of finite order, is of density 1. divisibility of the images of the specialization maps; elliptic surface; canonical heights J. Silverman, Divisibility of the specialization map for families of elliptic curves , Amer. J. Math. 107 (1985), no. 3, 555-565. JSTOR: Arithmetic ground fields for surfaces or higher-dimensional varieties, Special surfaces, Families, moduli of curves (algebraic), Arithmetic ground fields (finite, local, global) and families or fibrations Divisibility of the specialization map for 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 author computes the descendant Gromov-Witten invariants -- gravitational correlators of \(\mathbb{P}^1\) in genera zero, one and two by using the topological recursion relations of Witten, Eguchi-Hori-Xiong and Getzler. By the approach of this paper it is shown that the known topological recursion relations allow one to compute all the Gromov-Witten invariants up to genus-2 and completely determine all the Hodge integrals for \(\mathbb{P}^1\) up to genus-2.
Next, the author checks the Virasoro conjecture in gravitational quantum cohomology by explicitly showing that the gravitational correlators satisfy the Virasoro constraints. There are also derived the recursion relations of simple Hurwitz numbers, which count the number of inequivalent ramified coverings of a sphere by Riemann surfaces with specified branching conditions over an infinity.
The author uses a computer program which computes up to 7-point invariants of arbitrary degree and presents some examples of computer calculations, as well as some open questions regarding the relation between the topological recursion relations and the Virasoro constraints. Hurwitz number; Virasoro conjecture; topological recursion relation; Gromov-Witten invariants Song, J.S., Descendant Gromov-Witten invariants, simple Hurwitz numbers, and the Virasoro conjecture for \(P\)\^{}\{1\}, Adv. Theor. Math. Phys., 3, 1721-1768, (1999) Gromov-Witten invariants, quantum cohomology, Frobenius manifolds, Families, moduli of curves (algebraic), Coverings of curves, fundamental group, Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects) Descendant Gromov-Witten invariants, simple Hurwitz numbers, and the Virasoro conjecture for \({\mathbb P}^1\) | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians Let \(M_{g,n}\) be the moduli stack of complex nonsingular curves of genus \(g\) with \(n\) distinct marked points, and denote by \(\overline M_{g,n}\) its Deligne-Mumford compactification via stable curves. It is well-known that \(\overline M_{g,n}\) is a nonsingular orbifold of dimension \(3g- 3+n\), and that the Hodge bundle \(\mathbb{E}\) on \(M_{g,n}\) extends to the compactification \(\overline M_{g,n}\). Denote the \(\lambda_g\) the determinant bundle of \(\mathbb{E}\), which is called the Hodge line bundle on \(\overline M_{g,n}\).
The main result of the paper under review is a formula for integrating tautological classes on \(\overline M_{g,n}\) against the Hogde line bundle \(\lambda_g\). More precisely, the authors derive a closed formula for integrals of the cotangent line classes against \(\lambda_g\) in the form
\[
\int_{\overline M_{g,n}} \psi^{\alpha_1}_1\cdots \psi^{\alpha_n}_n \lambda_g= {2g+ n-3\choose \alpha_1\cdots\alpha_n}\, \int_{\overline M_{g,1}} \psi^{2g- 2}\lambda_g,
\]
where the integrals on the righthand side had been calculated by the same authors in a previous paper [\textit{C. Faber} and \textit{R. Pandharipande}, Hodge integrals and Gromov-Witten theory, Invent. Math. 139, No. 1, 173--199 (2000; Zbl 0960.14031)]. The study of integration against the Hodge line bundle is motivated by predictions in Gromov-Witten theory, on the one hand, and by the enumerative geometry (tautological ring) of the moduli space \(M^c_g\subset \overline{M}_g\) of stable curves of compact type. The above integrals are computed via relations obtained from virtual localization in Gromov-Witten theory. moduli spaces of curves; stable curves; Hodge bundles; enumerative geometry of moduli spaces; Hodge integrals; Gromov-Witten theory; partition matrices Faber, C.; Pandharipande, R., Hodge integrals, partition matrices, and the \(\lambda _g\) conjecture, Ann. Math. (2), 157, 97-124, (2003) Families, moduli of curves (algebraic), Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects), Transcendental methods, Hodge theory (algebro-geometric aspects) Hodge integrals, partition matrices, and the \(\lambda_g\) 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 [For the entire collection see Zbl 0667.00008.]
Let H be the Hilbert scheme of rational curves of degree d in \({\mathbb{P}}^ n\quad (n\geq 2).\) The paper shows that H is rational. This follows from Katsylo's theorem on the rationality of the scheme of invariants of binary forms [\textit{P. I. Katsylo}, Math. USSR, Izw. 25, 45-50 (1985); translation from Izv. Akad. Nauk SSSR, Ser. Mat. 48, No.4, 705-710 (1984; Zbl 0593.14017)]. The proof depends on the parity of d. For \(d\quad even\) it is invariant theoretic and for \(d\quad odd\) it depends on the local triviality of certain conic bundles. Hilbert scheme of rational curves; rationality Parametrization (Chow and Hilbert schemes), Families, moduli of curves (algebraic), Rational and unirational varieties La rationalité des schémas de Hilbert de courbes gauches rationnelles suivant Katsylo. (The rationality of the Hilbert schemes of rational skew curves according to Katsylo) | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is 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 that the moduli space of tetragonal curves of genus \(g>6\) is rational when \(g\) is congruent to \(1, 2, 5, 6, 9, 10\) modulo 12 and not equal to \(9, 45\). Families, moduli of curves (algebraic), Special algebraic curves and curves of low genus, Rationality questions in algebraic geometry Rationality of some tetragonal loci | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians Let \(C\) be a complex projective smooth curve of genus \(g \geq 3\). Let \(E\) be a vector bundle of rank \(r\) over \(C\) which is semistable and of integral slope \(m := \mathrm{deg}(E)/r\). For \(h := g - 1 - m\), consider the locus
\[
\{ N \in \mathrm{Pic}^h (C) : h^0 (\mathrm{gr}(E) \otimes N) \geq 1 \} .
\]
If not the whole of \(\mathrm{Pic}^h(C)\), this locus is the support of a divisor \(\Theta_E \in |r \Theta_M |\), where \(\Theta_M\) is a translate of the Riemann theta divisor on \(\mathrm{Pic}^{g-1}(C)\).
The author shows that if \(C\) is a Petri curve of genus \(g \geq 4\), and \(E\) is a general stable bundle, then \(\Theta_E\) is an integral divisor with singular locus of dimension \(g-4\). Moreover, in this case a line bundle \(N\) is a singular point of \(\Theta_E\) if and only if \(h^0 ( E \otimes N ) \geq 2\).
Furthermore, if \(C\) is nonhyperelliptic of genus \(3\), then for general stable \(E\) the divisor \(\Theta_E\) is smooth and integral.
The proof relies on the Petri condition and uses various facts on generalised Brill-Noether loci from the references \textit{S. B. Bradlow} et al. [Int. J. Math. 14, No. 7, 683--733 (2003; Zbl 1057.14041)], \textit{S. Casalaina-Martin} and \textit{M. Teixidor i Bigas} [Math. Nachr. 284, No. 14--15, 1846--1871 (2011; Zbl 1233.14025)], \textit{Y. Laszlo} [Duke Math. J. 64, No. 2, 333--347 (1991; Zbl 0753.14023)], \textit{V. Mercat} [Manuscr. Math. 98, No. 1, 75--85 (1999; Zbl 0976.14021)] and \textit{M. Teixidor i Bigas} [Math. Nachr. 196, 251--257 (1998; Zbl 0954.14024)] and \textit{M. Teixidor i Bigas} [Duke Math. J. 62, No. 2, 385--400 (1991; Zbl 0739.14006)]. vector bundle; curve; generalised theta divisor Brivio, S, A note on theta divisors of stable bundles, Rev. Mat. Iberoam., 31, 601-608, (2015) Vector bundles on curves and their moduli, Families, moduli of curves (algebraic) A note on theta divisors of stable 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 defines the modular isogeny complex and proves a vanishing result for the cohomology of this complex. Let \(E/S\) be an elliptic scheme over a scheme \(S,\) let \([N\text{-Isog}](E/S)\) be the set of locally free finite commutative \(S\) subgroup schemes \(G\subset E\) of rank \(N\) over \(S\) (cf. [\textit{N. M. Katz} and \textit{B. Mazur}, Arithmetic moduli of elliptic curves. Princeton, New Jersey: Princeton University Press (1985; Zbl 0576.14026)]). For \(S=\mathrm{Spec}A\) let \({\mathcal S}_{N}(E/S)\) be the function ring of \([N\text{-Isog}](E/S).\) Similarly for integers \(N_{1},\dots , N_{q}\), let \([N_{1},\dots , N_{q} \text{-Isog}](E/S)\) denote the set of sequences \({\underline G}= G_{1} \subsetneq G_{2} \dots \subsetneq G_{q}, \) where \(G_{i}\) is a locally free commutative \(S\)-subgroup scheme of \(E\) of rank \(N_{1},\dots, N_{i}.\) For \(S=\mathrm{Spec} A\) denote by \({\mathcal S}_{N_{1},\dots , N_{q}}(E/S)\) the function ring of \([N_{1},\dots , N_{q} \text{-Isog}](E/S).\) The modular \(N\)-isogeny complex \({\mathcal K}_{N}^{\bullet}(E/S)\) of \(A\)-modules is defined in the following way:
\[
{\mathcal K}_{N}^{q}(E/S)={\prod}_{N_{1},\dots , N_{q}}{\mathcal S}_{N_{1},\dots , N_{q}}(E/S),
\]
where the product runs over all \(q\)-tuples \((N_{1},\dots , N_{q})\) of integers such that \(N_{1}\cdots N_{q}=N\) and \(N_{i}>1.\) The coboundary map \({\delta}: {\mathcal K}_{N}^{q-1} \rightarrow {\mathcal K}_{N}^{q}\) is defined by the formula:
\[
{{\delta} f}_{N_{1},\dots , N_{q}} = {\sum}_{i=1}^{q-1} (-1)^i u_{i}(f_{N_{1},\dots, N_iN_{i+1},\dots , N_{q}}),
\]
where \(u_{i}\) is induced by forgetting the \(i\)-the group in the sequence \({\underline G}= G_{1} \subsetneq G_{2} \dots \subsetneq G_{q} .\) The results have applications to study the power operations in Morava \(E\)-theory at height \(2.\) elliptic curves; elliptic schemes; cohomology; Morava E-theory Families, moduli of curves (algebraic), Elliptic cohomology, \(K\)-theory operations and generalized cohomology operations in algebraic topology Modular isogeny complexes | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians By results of \textit{D. Mumford} [Enseign. Math., II. Ser. 23, 39--100 (1977; Zbl 0363.14003)] and \textit{D. Gieseker} [Lectures on moduli of curves. Tata Inst. Fund. Res. (1982; Zbl 0534.14012)] the compactified moduli space \(\bar{M}_g\) of curves of genus \(g\) is isomorphic to the GIT quotients \(\text{Hilb}_{g,n}/ /{SL}_{(2n-2)(g-1)}\) and \(\text{Chow}_{g,n}/ /{SL}_{(2n-2)(g-1)}\) of the Hilbert scheme and the Chow variety of \(n\)-canonical images, for any \(n \geq 5\). There are several ways for constructing compact moduli spaces of hyperelliptic curves \(H_g\), two of which are concerned in this paper. The first is to take the closure \(\bar{H}_g\) in \(M_g\); it yields that for any \(n \geq 5\),\ \(\bar{H}_g\) is isomorphic to the GIT quotients \(\text{Hilb}^{he}_{g,n}/ /{SL}_{(2n-2)(g-1)}\) and \(\text{Chow}^{he}_{g,n}/ /{SL}_{(2n-2)(g-1)}\) of the hyperelliptic loci of \(\text{Hilb}_{g,n}\) and \(\text{Chow}_{g,n}\).
This paper studies the above GIT quotients for hyperelliptic curves in some of the problematic cases corresponding to the values \(n < 5\). The general hyperelliptic curve \(C\) of genus \(g\) is defined by the \((2g+2)\) branched points of the double covering \(C \rightarrow {\mathbb P}^1\), equivalently -- by the binary forms of degree \(2g+2\). By a result of Avritzer and Lange, there is a natural morphism \(f_g: \bar{H}_g \rightarrow \bar{B}_{2g+2}\) to the GIT compactified moduli space of such forms, contracting \(g-1\) special divisors \(\tilde{B}_3,\dots,\tilde{B}_{g+1}\), [see \textit{D. Avritzer, H. Lange}, Math. Z. 242, No. 4, 615--632 (2002; Zbl 1080.14031)]. This paper studies the maps \(f_g\) from the viewpoint of log Mori theory. For this purpose, for any \(\alpha \in {\mathbb Q}\), the authors introduce \({\mathbb Q}\)-Cartier divisors \(L_{\alpha}\), and define the varieties \(\bar{H}_g(\alpha)\) = \(\text{Proj}\;\bigoplus_{n \geq 0} \Gamma(\bar{H}_g,n(4g+2)L_{\alpha})\). Under some additional conditions (the \(F\)-conjecture -- see section 3) they verify first Theorem 1, which in particular states that for \(g \geq 3\) there is a birational contraction \(\bar{H}_g \rightarrow \bar{H}_g(9/11)\) with exceptional locus \(\tilde{B}_3\); and also Theorem 2, which in particular shows that \(\bar{H}_g(7/10)\) is isomorphic to \(\text{Chow}^{he}_{g,2}//{SL}_{3g-3}\). Especially for genus \(g =3\), Theorem 3 states that \(\text{Chow}^{he}_{3,2}\) is isomorphic to the compactified space of binary forms \(\bar{B}_8\). Together with Theorem 2, this yields an isomoprphism \(\bar{H}_3(7/10) \cong \bar{B}_8\). Theorem 4 shows that this isomorphism extends the natural isomorphism \(H_3 \cong B_8\). Hyeon, A new look at the moduli space of stable hyperelliptic curves, Math. Z. 264 pp 317-- (2010) Families, moduli of curves (algebraic), Minimal model program (Mori theory, extremal rays) A new look at the moduli space of stable 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 aim of this paper is to generalize a lemma by \textit{L. Caporaso} [Trans. Am. Math. Soc. 355, No. 9, 3475--3484 (2003; Zbl 1030.14022)] to families of \(n\)-gonal curves, namely curves that have a \(n\) to \(1\) map to \(\mathbb{P}^1\) (or equivalently a base-point free \(g_n^1\)). Families, moduli of curves (algebraic), Special divisors on curves (gonality, Brill-Noether theory) Families of \(N\)-gonal curves with maximal variation of moduli | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians For a Deligne-Lusztig variety \(\overline X (w)\) arising from one of the classical (possibly twisted) groups, we show that the Picard group of \(\overline X(w)\) is generated by the finitely many Deligne-Lusztig subvarieties of \(\overline X (w)\). It is conjectured that this more generally should hold in any codimension, and a good deal of supporting evidence for this claim is presented.
This article is based on Chapter 3 of the author's PhD thesis ``The geometry of Deligne-Lusztig varieties. Higher-dimensional AG codes`` [University of Aarhus (1999)]. Hansen S.H.: Picard groups of Deligne-Lusztig varieties--with a view toward higher codimensions. Beiträge Algebra Geom. 43(1), 9--26 (2002) Grassmannians, Schubert varieties, flag manifolds, Algebraic cycles, (Equivariant) Chow groups and rings; motives, Picard groups Picard groups of Deligne-Lusztig varieties -- with a view toward higher codimensions | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is 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 \(R\) be the coordinate ring of a union of \(n\) hyperplanes in an affine \(m\)-space over an algebraically closed field \(k\). The author calculates the Picard group in the case that the planes all contain the same linear subvariety of codimension one, and \(K_0\) in the case that the planes are all through a line in affine 3-space and \(k\) is of characteristic zero. These results extend the work of \textit{B. H. Dayton} [J. Algebra 88, 534--569 (1984; Zbl 0537.13010)] about planes through the origin, \textit{B. H. Dayton} and \textit{C. A. Weibel} [Trans. Am. Math. Soc. 257, 119--141 (1980; Zbl 0424.18011)] about planes satisfying a general position condition and \textit{F. Orecchia} [J. Pure Appl. Algebra 142, No. 1, 49--61 (1999; Zbl 0942.14025)] about lines through a point. Picard group; affine hyperplane; coordinate ring; \(K_0\); lines through a point Picard groups, Applications of methods of algebraic \(K\)-theory in algebraic geometry, Low codimension problems in algebraic geometry, Algebraic \(K\)-theory and \(L\)-theory (category-theoretic aspects), \(K_0\) of other rings On the \(K\)-theory of affine hyperplanes through a linear subvariety of codimension one | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians The main result of the paper is an explicit construction of an étale local resolution of the rational moduli map \(B\dashrightarrow \overline {M}_g\) for a family of curves \(X\to B\) with ADE-singularities of genus \(g\geq 2\). The approach is similar to \textit{E. Brieskorn}'s simultaneous resolution of singularities for families of surfaces with ADE-singularities [Actes Congr. internat. Math. 1970, 2, 279--284 (1971; Zbl 0223.22012)]. ADE-singularities; moduli spaces of curves; simultaneous resolution Casalaina-Martin S. and Laza R., Simultaneous semi-stable reduction for curves with ADE singularities, Trans. Amer. Math. Soc. 365 (2013), no. 5, 2271-2295. Families, moduli of curves (algebraic), Geometric invariant theory, Minimal model program (Mori theory, extremal rays) Simultaneous semi-stable reduction for curves with ADE singularities | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians Let \(g\geq 2, d\geq 1, s\geq 0\) be fixed integers. Let \(V\) be a smooth, irreducible, projective variety of degree \(d\) defined over \( {\mathbb C}\). Let \(T \subset V\) be a closed subscheme of degree \(s\). The author introduces the set \(F_g(V,T)\) of equivalence classes of non-isotrivial families of smooth curves of genus \(g\) over \(V \setminus T\) and proves (theorem 1) that there exists a number \(H(g,d,s)\) such that \(|F_g(V,T)|\leq H(g,d,s)\). Moreover, the bound \(H\) does not depend on \(s\), if \(T\) has codimension at least 2 in \(V\). With the above notation, let \(X\) be a non-isotrivial curve of genus \(g\) defined over \(L= {\mathbb C}(V)\) and having good reduction outside \(T\). It is proven (theorem 2) that \(|X(L)|\leq N(g,d,s)\) for some integer \(N(g,d,s)\). Let \(C_g^2(L)\) be the set of \(L\)-isomorphism classes of non-isotrivial curves of genus \(g\) over \(L\) having good reduction in codimension 1. It is shown that \(C_g^2(L)\) is a finite set, and that there exists a number \(N_g^2(L)\) such that \(|X(L)|\leq N_g^2(L)\) for any curve \(X\in C_g^2(L)\) (theorem 3).
The proofs of the first two theorems are based on the results from an earlier paper [\textit{L. Caporaso}, Compos. Math. 130, 1-19 (2002; Zbl 1067.14022)]. The proof of theorem 3 uses moduli maps. Note that if \( \dim V=1\) then theorem 3 implies a theorem of \textit{A. N. Parshin} [Izv. Akad. Nauk SSSR, Ser. Mat. 32, 1191-1219 (1968; Zbl 0181.23902)].
The author discusses an example suggested by J. de Jong, which shows that the cardinality of the set of fibrations with fixed degeneracy locus is not bounded as the cardinality of the degeneracy locus grows. The paper contains also some results on uniform boundedness of rational points which are independent of the degeneracy locus. It is concluded by introducing the notion of modular degree of a curve \(X\) and by proving some properties which relate this notion to \(|X(L)|\) and to the geometric Lang conjecture on the distribution of rational points on varieties of general type. arithmetic of rational points; varieties over function fields; cardinaltiy of the set of fibrations; uniform boundedness of rational points; distribution of rational points Enumerative problems (combinatorial problems) in algebraic geometry, Algebraic functions and function fields in algebraic geometry, Families, moduli of curves (algebraic), Rational points Remarks about uniform boundedness of rational points over 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 We present a topological recursion formula for calculating the intersection numbers defined on the moduli space of open Riemann surfaces. The spectral curve is \(x = \frac{1}{2} y^2\), the same as the spectral curve used to calculate intersection numbers for closed Riemann surfaces, but the formula itself is a variation of the usual Eynard-Orantin recursion. It looks like the recursion formula used for spectral curves of degree \(3\), and also includes features present in \(\beta\)-deformed models. The recursion formula suggests a conjectural refinement to the generating function that allows for distinguishing intersection numbers on moduli spaces with different numbers of boundary components. Safnuk, B., Topological recursion for open intersection numbers, Commun. Number Theory Phys., 10, 4, 833-857, (2016) Families, moduli of curves (algebraic), Relationships between algebraic curves and physics, Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects) Topological recursion for open intersection numbers | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians Here is essentially a very good mathematical-computational paper strongly emphasizing several important problems related to local mirror symmetry with special attention on the cases where a certain parameter \(k\) can be greater or equal to one. A straight forward calculational scheme is provided using the basic tools of equivariant \(I\)-functions and their Birkhoff factorization.
This article has 6 sections: Introduction, Overview (Background, Equivalent local mirror symmetry of curves, Non-nef toric varieties), Equivalent mirror symmetry (Equivalent Picard-Fuchs equation, Non-vanishing invariants, \(k\geq 1\), A model computation) Mirror symmetry (Motivation, Verification using \(I\)-function for \(k= 1, 2\), Alternative proof of connection matrices) Mirror symmetry for \(F_n\) and \(KF_n\), \(n\geq 3< KF_3\), connection matrices for \(F_3\), \(F_4\), Quantum differential equations) and Conclusion. This valuable contribution is of great current significance. local mirror symmetry; topological string; quantum cosmology Forbes, B.; Jinzenji, M., \(J\) functions, non-nef toric varieties and equivariant local mirror symmetry of curves. int, J. Mod. Phys. A, 22, 2327-2360, (2007) Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects), Families, moduli of curves (algebraic), Calabi-Yau manifolds (algebro-geometric aspects), Toric varieties, Newton polyhedra, Okounkov bodies \(J\) functions, non-nef toric varieties and equivariant local mirror symmetry 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 Generically minimally rigid graphs in the plane (Laman graphs) admit only finitely many realizations in the plane for a generic assignment of edge-lengths. Recently, substantial progress has been made in the computation of the number of non-isometric realizations in the complex extension of the Euclidean plane [\textit{J. Capco} et al., SIAM J. Appl. Algebra Geom. 2, No. 1, 94--125 (2018; Zbl 1439.14182)].
This paper presents a recursive algorithm for computing the number of non-isometric realizations of Laman graphs in the complex sphere (where they are also known to be generically minimally rigid). It differs fundamentally from the algorithm for computing the number of planar realizations presented in the aforementioned article. The basic idea is to view a realization with prescribed edge lengths, up to sphere isometries, as a tuple of points on the complex projective line with prescribed cross-ratios. The compactification of this space is the moduli space of stable rational curves with marked points. Its Chow rings, as described information [\textit{S. Keel}, Trans. Am. Math. Soc. 330, No. 2, 545--574 (1992; Zbl 0768.14002)], play a crucial role in the algorithm for computing the desired number of realizations.
Even if detailed timing information is missing, the devised algorithm seems to be substantially faster than naive approaches via Gröbner bases. It allows a systematic exploration for Laman graphs of low vertex numbers. An interesting observation is that the graphs with a maximal number of realizations need not be the same for the planar and the spherical version of the problem. generic minimal rigidity; moduli space; Chow ring; intersection theory Graph algorithms (graph-theoretic aspects), Enumeration in graph theory, Families, moduli of curves (algebraic), Metric geometry, Rigidity and flexibility of structures (aspects of discrete geometry) Counting realizations of Laman graphs on the sphere | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians Raynaud, M.: Compactification du module des courbes. Sém. Bourbaki23 (1970/1971), No. 385. Families, moduli of curves (algebraic) Compactification du module des courbes | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians Erratum concerned with ibid. 76, No. 1, 203-272 (1994; see the review above). Ginzburg V., Kapranov M.: Erratum to: ''Koszul duality for operads''. Duke Math. J. 80(1), 293 (1995) Monoidal categories (= multiplicative categories) [See also 19D23], Other \(n\)-ary compositions \((n \ge 3)\), Infinite loop spaces, Families, moduli of curves (algebraic), Associative rings and algebras arising under various constructions, Chain complexes (category-theoretic aspects), dg categories Erratum to ``Koszul duality for operads'' | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is 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 note can be considered as a sequel of an earlier paper of the authors [in: Algebraic transformation groups and algebraic varieties, Invariant theory and algebraic transformation groups 3, 1--7 (2004; Zbl 1068.14010)] where an approach of \textit{F. Severi} [Rom. Acc. L. Rend. (5) \(15_2\), 691--696 (1906; JFM 37.0131.01)] was developed. Combining their previous results with some considerations from Hodge theory they prove the following.
Let \(V\subset {\mathbb P}^5\) be a reduced and irreducible threefold of degree \(s,\) complete intersection on a smooth hypersurface of degree \(t,\) with \(s>t^2-t.\) Let also assume that the singular locus of \(V\) consists of \(\delta< 3s/8t\) ordinary double points. Then any projective surface contained in \(V\) is a complete intersection on \(V.\) In particular, \(V\) is \(\mathbb Q\)-factorial. Hence one obtains some new examples of \(\mathbb Q\)-factorial projective varieties [cf. \textit{Y. Miyaoka} and \textit{T. Peternell}, ``Geometry of higher dimensional algebraic varieties'' (1997; Zbl 0865.14018)]. The authors also underline that their method allows to conclude \(\mathbb Q\)-factoriality of threefolds complete intersection \(V\) in \({\mathbb P}^5\) with few isolated singularities ``under suitable hypothesis''. \(\mathbb{Q}\)-factoriality; Castelnuovo-Halphen theory; Hodge theory; Lefschetz theorem Ciliberto, C.; Gennaro, V., Factoriality of certain threefolds complete intersection in P\^{}\{5\} with ordinary double points, Commun. Algebra, 32, 2705-2710, (2004) Complete intersections, Picard groups, Singularities of surfaces or higher-dimensional varieties, Varieties defined by ring conditions (factorial, Cohen-Macaulay, seminormal) Factoriality of certain threefolds complete intersection in \(\mathbb{P}^5\) with ordinary double 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 \(N\) be a numerical semigroup of genus \(g\geq 4\) whose set of gaps \(\mathbb N_0\setminus N=\{\ell_1<\ldots<\ell_g\}\) satisfy \(\ell_2=2\) and \(\ell_g=2g-3\). It is important to notice the existence of two gaps of \(N\), say \(\lambda, \mu\), which are uniquely determined by \(\mu+\lambda=\ell_g\) and \(\mu<\lambda\). In the paper under review, the authors study the moduli space \(\mathcal M\) of genus \(g\) (non-singular, projective, irreducible over an algebraically closed field \(k\)) algebraic pointed curves \((X,P)\) being \(N\) the Weierstrass semigroup at \(P\). [Let us point out that we can assume that \(X\) is contained in the \(g\)-dimensional projective space over \(k\) as a curve of degree \(2g-2\)]. They follow closely \textit{K.-O. Stöhr}'s approach [J. Reine Angew. Math. 441, 189--213 (1993; Zbl 0771.14009)] (case of a symmetric semigroup or \(\ell_g=2g-1\)); cf. [\textit{G. Oliveira} and \textit{K.-O. Stöhr}, Geom. Dedicata 67, No. 1, 65--82 (1997; Zbl 0904.14019)] (case of a pseudo-symmetric semigroup or \(\ell_g=2g-2\)). In particular, Gröbner basis techniques are considered.
The main result is the existence of a stratification \(\mathcal M=\mathcal M_0\cup\mathcal M_1\) by means of two disjoint locally closed varieties which are closely related to the aforementioned gaps \(\mu\) and \(\lambda\). To explain this one is led to consider the sequence of positive divisors of the curve \(X\), \(E_{\ell_2}\leq\ldots\leq E_{\ell_g}\), where \(E_{\ell_i}:=X\cdot T^{(i)}-(\ell_i-1)P\) with \(T^{(i)}\) being the \(i\)-th osculating space at \(P\). By results of \textit{F. L. R. Pimentel} [Geom. Dedicata 85, No. 1--3, 125--134 (2001; Zbl 0991.14014)] and \textit{N. Medeiros} [J. Pure Appl. Algebra 170, No. 2--3, 267--285 (2002; Zbl 1039.14015)], \(E_\mu=0\) and \(\deg(E_\lambda)\in\{0,1\}\). Thus \(\mathcal M_0\) (resp. \(\mathcal M_1\)) is made of those pointed curves where \(\deg(E_\lambda)=0\) (resp. \(\deg(E_\lambda)=1\)).
As a matter of fact in both sets \(\mathcal M_i\) Stöhr's method can be applied, and the following useful arithmetical criterion: \(\lambda-\mu\) is also a gap at \(P\) and \(2\mu\geq \lambda\) whenever \(E_\lambda)\neq 0\) can be used to decide the emptiness property of the strata. moduli spaces; Weierstrass semigroups; Weierstrass gap sequence Riemann surfaces; Weierstrass points; gap sequences, Algebraic moduli problems, moduli of vector bundles, Families, moduli of curves (algebraic) A stratification of the moduli space of pointed non-singular curves | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians The paper is devoted to the study of the possible dimension of a germ of a totally geodesic submanifold of the moduli space of polarised abelian varieties of a given dimension, which is contained in the Prym locus of a (possibly) ramified double cover. The main result improves estimations previously obtained in a couple of papers. The general approach consists in studying the second fundamental form of the Prym map and deduce an estimate by using the gonality. In this spirit they adapt some techniques previously developed with P. Pirola and A. Ghigi on the study of the second fundamental group of the Torelli map to the case of the Prym map, which allow to conclude their stronger result. Families, moduli of curves (algebraic), Families, moduli of curves (analytic), Jacobians, Prym varieties, Subvarieties of abelian varieties On the dimension of totally geodesic submanifolds in the Prym loci | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians The problem of counting ramified covers of a Riemann surface up to homeomorphism was proposed by Hurwitz in the late 1800's. This problem translates combinatorially into factoring a permutation of specified cycle type, with certain conditions on the cycle types of the factors, such as minimality and transitivity.
\textit{I. P. Goulden} and \textit{D. M. Jackson} [Proc. Am. Math. Soc. 125, 51--60 (1997; Zbl 0861.05006)] have given a proof for the number of minimal, transitive factorizations of a permutation into transpositions. This proof involves a partial differential equation for the generating series, called the join-cut equation. Recently, \textit{M. Bousquet-Mélou} and \textit{G. Schaeffer} [Adv. Appl. Math. 24, 337--368 (2000; Zbl 0955.05004)] have found the number of minimal, transitive factorizations of a permutation into arbitrary unspecified factors. This was proved by a purely combinatorial argument, based on a direct bijection between factorizations and certain objects called \(m\)-Eulerian trees.
In this paper, we give a simple partial differential equation for Bousquet-Mélou and Schaeffer's generating series, and for Goulden and Jackson's generating series, as well as a new proof of the result by Bousquet-Mélou and Schaeffer. We apply algebraic methods based on Lagrange's theorem, and combinatorial methods based on a new use of Bousquet-Mélou and Schaeffer's \(m\)-Eulerian trees. minimal; transitive; permutation factorizations; ramified covers; exact enumeration; generating functions Goulden, I. P.; Serrano, Luis G., A simple recurrence for covers of the sphere with branch points of arbitrary ramification, Ann. Comb., 10, 4, 431-441, (2006) Exact enumeration problems, generating functions, Families, moduli of curves (algebraic), Moduli problems for topological structures A simple recurrence for covers of the sphere with branch points of arbitrary ramification | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians [For the entire collection see Zbl 0741.00067.]
For a degenerating family of varieties, the degenerate fibre of the family of conormal varieties is described using refined polar classes. A formula for the degenerate fibre of the family of dual varieties follows immediately. In some cases (e.g. hypersurfaces with isolated singularities, plane curves) a more explicit description is given. degenerate fibre; family of conormal varieties Structure of families (Picard-Lefschetz, monodromy, etc.), Projective techniques in algebraic geometry, Families, moduli of curves (algebraic) Degenerations of conormal varieties | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians This paper studies the moduli space, \(P^8_1\), of semi-stable \(8\) ordered points on the projective line. It is known that \(P^8_1\) is isomorphic to the Satake-Baily-Borel compactification of an arithmetic quotient of \(5\)-dimensional complex ball, by \textit{P. Deligne} and \textit{G. D. Mostow} [Publ. Math., Inst. Hautes Étud. Sci. 63, 5--89 (1986; Zbl 0615.22008)]. Deligne and Mostow used the theory of periods of families of curves.
In this paper, the periods of \(K3\) surfaces are used as opposed to those of curves. For each stable point from \(P_1^8\), the author associates a \(K3\) surface with a non-symplectic automorphism of order \(4\). Then it is proved that the period domain of these \(K3\) surfaces is a \(5\)-dimensional complex ball \({\mathcal{B}}\), and that \(P^8_1\) is isomorphic to the Satake-Baily-Borel compactification \(\bar{\mathcal{B}}/\Gamma(1-i)\) of the arithmetic quotient \({\mathcal{B}}/\Gamma(1-i)\) where \(\Gamma(1-i)\) is an arithmetic subgroup of a unitary group of a hermitian form of signature \((1,5)\) defined over the Gaussian integers \(\mathbb Z[i]\). The symmetric group \(S_8\) acts on \(P^8_1\), and it is shown that the isomorphism \(P^8_1\simeq \bar{\mathcal{B}}/\Gamma(1-i)\) is \(S_8\)-equivariant. Next the theory of automorphic forms is used to give a projective model of the moduli space \(P^8_1\). It is shown that there is a \(14\)-dimensional space of automorphic forms on \({\mathcal{B}}\) which gives an \(S_8\)-equivariant map from the arithmetic quotient \(\bar{\mathcal{B}}/\Gamma(1-i)\) into \(\mathbb P^{13}\), and furthermore, this map coincides with the map defined by the cross ratios of \(8\) points on the projective line. \(K3\) surfaces; automorphic forms; complex ball uniformization Shigeyuki Kondō, The moduli space of 8 points of \Bbb P\textonesuperior and automorphic forms, Algebraic geometry, Contemp. Math., vol. 422, Amer. Math. Soc., Providence, RI, 2007, pp. 89 -- 106. Families, moduli of curves (algebraic), \(K3\) surfaces and Enriques surfaces, Relations with algebraic geometry and topology The moduli space of 8 points on \(\mathbb P^1\) and automorphic forms | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians We give a formula for the cohomological invariants of a root stack, which we apply to compute the cohomological invariants and the Brauer group of the compactification of the stacks of hyperelliptic curves given by admissible double coverings. cohomological invariants; root stacks; hyperelliptic curves; admissible coverings Brauer groups of schemes, Families, moduli of curves (algebraic), (Equivariant) Chow groups and rings; motives, Generalizations (algebraic spaces, stacks) Cohomological invariants of root stacks and admissible double coverings | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians The universal Abel map \(\mathcal M_{g,n} \rightarrow \mathcal J_{g,n}\) extends in general only to a rational map \(\overline{ \mathcal M}_{g,n} \dashrightarrow \overline {\mathcal J}_{g,n}\) from the Deligne-Mumford compactification of the moduli space of smooth curves to a compactification of the universal Jacobian given by some choice of universal stability condition. Roughly speaking, the issue is that the line bundles one obtains as images of the Abel map need not be stable, and a stable representative depends on the choice of a one-parameter smoothing of the curve. There are two natural approaches to this issue: first, one can modify \(\overline{ \mathcal M}_{g,n}\) to resolve the indeterminancy as for example in [\textit{D. Holmes}, J. Inst. Math. Jussieu 20, No. 1, 331--359 (2021; Zbl 1462.14031)] or in [\textit{S. Marcus} and \textit{J. Wise}, Proc. Lond. Math. Soc. (3) 121, No. 5, 1207--1250 (2020; Zbl 1455.14021)]; or second, one can tailor a stability condition to obtain a compactified Jacobian that avoids the issue for a given Abel map as in [\textit{J. L. Kass} and \textit{N. Pagani}, Trans. Am. Math. Soc. 372, No. 7, 4851--4887 (2019; Zbl 1423.14187)].
In this paper, the authors follow the first approach and describe a blow-up of \(\overline{ \mathcal M}_{g,n}\) that resolves the indeterminancy of the universal Abel map. This resolution is formulated in terms of tropical geometry. Namely, the tropical universal Abel map is not a morphism of generalized cone complexes, for the analogous reason as in the algebro-geometric setting: given a divisor on a tropical curve, there is a unique stable representative linearly equivalent to it, but this representative depends on the edge lengths of the underlying graph. Refining the cone structure of the moduli space of tropical curves turns the tropical universal Abel map into a morphism of cone complexes, which describes the desired blow-up of \(\overline{ \mathcal M}_{g,n}\) by a standard construction of toric geometry. Much of the tropical analysis is done in the authors' previous work [\textit{A. Abreu} and \textit{M. Pacini}, Proc. Lond. Math. Soc. (3) 120, No. 3, 328--369 (2020; Zbl 1453.14082)], to which the current paper serves as an algebro-geometric counterpart. As an application, the authors give descriptions of the algebro-geometric and tropical double ramification cycles. geometric Abel map; tropical Abel map; double ramification cycle Families, moduli of curves (algebraic), Jacobians, Prym varieties, Applications of tropical geometry The resolution of the universal Abel map via tropical geometry and applications | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians Soit C une courbe lisse, complète, de genre \(g>1\) sur un corps, K, algébriquement clos. Soit \(C_ n\) le produit symétrique n-ième de C. Un résultat de Kempf [\textit{G. Kempf}, Proc. 1978 Stony Brook Conf., Ann. Math. Stud. 97, 319-341 (1981; Zbl 0465.14013)] affirme que si C n'est pas hyperelliptique, alors toute petite déformation de \(C_ n\) est le produit symétrique d'un déformation de C. L'hypothèse C non hyperelliptique permet d'utiliser le théorème de Noether (i.e. \(H^ 0(\omega_ C)\otimes H^ 0(\omega_ C^{\otimes m})\to^{\alpha_ m}H^ 0(\omega_ C^{\otimes (n+1)})\) est surjective pour \(m\geq 1)\). Dans l'article recensé, l'A. étend le résultat de Kempf aux courbes hyperelliptiques avec \(g>2\). Pour ce faire, il utilise un résultat antérieur de l'A. [dans An. Inst. Mat., Univ. Nac. Auton. Méx. 19, 141-147 (1979; Zbl 0471.14006)] et parvient à compléter le théorème de Noether (si ch(K)\(\neq 2\) et C est hyperelliptique de genre \(g>1\), \(\alpha_ m\) est surjective sauf si \(m=1\) et \(m=2=g)\). L'article se termine par une analyse du cas \(g=2\). En particulier, l'A. montre que: \(h^ 1(C_ n,T_{C_ n})=h^ 1(C,T_ C)+1\) et que si \(K={\mathbb{C}}\), il existe des déformations de \(C_ 2\) qui ne sont même pas algébriques. deformation of symmetric product of curves; hyperelliptic curve Families, moduli of curves (algebraic), Algebraic moduli problems, moduli of vector bundles, Algebraic functions and function fields in algebraic geometry, Special algebraic curves and curves of low genus Symmetric products 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 construction of non-trivial factorial domains using methods of algebraic geometry has a long and rich history. Based on some results of \textit{M. Demazure} [Anneaux gradués normaux, séminaire ``Singularites des surfaces'', École Polytechnique, Paris 1979; (Zbl 0686.14005)], a very general construction was given by \textit{L. Robbiano} [Factorial and almost factorial schemes in weighted projective spaces, Lect. Notes Math. 1092, Springer Verlag, Heidelberg (1984; Zbl 0564.14021)]. The main result of the paper under review is to give a characterization of the factorial rings that can be obtained via Robbiano's construction. A similar, more general structure theorem was proved by \textit{M. Tomari} and \textit{K. Watanabe} [Cyclic covers of normal graded rings, Kodai Math. J. 24, 436--457 (2001; Zbl 1078.14512)] under slight additional assumptions about the characteristic of the base field. In the last part of the paper the author uses his results to construct an algorithm for computing the factorization of an element in an UFD \(K[x_1,\dots, x_n]/I\) if \(I\) is a homogeneous ideal. This algorithm uses only operations in \(K[x_1,\dots,x_n]\). If \(I\) is a principal ideal, an alternative method is given, too. factorial ring; Weil divisor; projectively normal scheme; factorication algorithm Commutative rings defined by factorization properties (e.g., atomic, factorial, half-factorial), Divisors, linear systems, invertible sheaves, Picard groups, Varieties defined by ring conditions (factorial, Cohen-Macaulay, seminormal), Polynomials, factorization in commutative rings Some remarks on factorial quotient rings | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians Let \(X\) be a \(K3\) surface defined over a number field \(k\). Suppose that for a finite place \(\mathfrak{p}\) of \(k\), \(X\) has a good reduction \(X_{\mathfrak{p}}\). There is a natural specialization homomorphism that induces an inequality of geometric Picard numbers of \(X\) and \(X_{\mathfrak{p}}\): \(\rho(\overline{X}) \leq \rho(\overline{X}_{\mathfrak{p}})\). In fact, if we define \(\eta(\overline{X})\) by
\[
\eta(\bar{X}) := \begin{cases} 0 & \text{if } E_X \text{ is a CM-field or } \dim_{E_X} T_X \text{ is even},\\ [E_X: \mathbb{Q}] & \text{if } E_X \text{ is a totally real field and } \dim_{E_X} T_X \text{ is odd}, \end{cases}
\]
where \(T_X\) is the transcendental lattice of \(X\) and \(E_X\) is the endomorphism algebra of the Hodge structure underlying \(T_X\), then, we have \(\rho(\overline{X}) + \eta(\overline{X})\leq \rho(\overline{X}_{\mathfrak{p}})\). Let
\[
\Pi_{\text{jump}} := \left\{ \mathfrak{p} \, | \, \rho(\overline{X}) +\eta(\overline{X}) < \rho(\overline{X}_{\mathfrak{p}}) \right\}
\]
and
\[
\gamma(X, B) := \frac{\# \{ p\leq B \text{ and } p\in \Pi_{\text{jump}}\}}{\# \{ p\leq B\}}.
\]
The aim of the paper under review is to study the amount \(\gamma(X, B)\).
The authors calculate \(\rho(\overline{X})\) and \(\rho(\overline{X}_p)\) for \(2<p<2^{16}\) and \(\gamma(X, B)\) for \(B<2^{16}\), and then plot the results \(\gamma(X, B)\) for several quartic \(K3\) surfaces \(X\) using Kedlaya's algorithm by studying Frobenius action on some cohomology group throughout the reduction process. With this computer-aided computation experiments, in the Main Result, the authors get evidences that \(\displaystyle\lim \sup _{B\to \infty} \gamma(X, B)\) is at least \(1/2\) if \(\rho(\overline{X}) = 2\), and that \(\rho(\overline{X}_p)\) jumps with probability proportional to \(1/\sqrt{p}\) if \(\rho(\overline{X}) = 1\) and \(E_X = \mathbb{Q}\). Kedlaya's algorithm; Frobenius action; good reduction; cohomology group; Hodge structure; \(K3\) surface E. Costa, Y. Tschinkel, Variation of Néron-Severi ranks of reductions of K3 surfaces, Exp. Math. 23(4), 475-481 (2014) \(K3\) surfaces and Enriques surfaces, Picard groups Variation of Néron-Severi ranks of reductions of \(K3\) 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 Il existe deux invariants algébriques importants associés à une surface de Riemann \(X\): son corps des modules et ses corps de définition. Un corps de définition est un sous-corps de \(\mathbb{C}\) sur lequel est défini une courbe algébrique isomorphe à \(X\). Le corps des modules de \(X\) est l'intersection de tous les corps de définition de \(X\). Une surface de Riemann (ou la courbe algébrique associée) est dite pseudo-réelle si son corps des modules est \(\mathbb{R}\), mais \(\mathbb{R}\) n'est pas un corps de définition. Le critère de descente de Weil donne une condition nécessaire et suffisante pour que le corps des modules d'une surface de Riemann soit un corps de définition. Toutefois, il est en général difficile de vérifier ce critère en pratique. Grâce aux travaux de Mestre, Huggins, Lemercier, Ritzenthaler et Sijsling, on comprend bien le cas hyperelliptique (voir en particuler [\textit{R. Lercier} et al., Math. Comput. 85, No. 300, 2011--2045 (2016; Zbl 1343.14047)]). Dans le cas non hyperelliptique, Huggins montre dans sa thèse [\textit{B. Huggins}, ``Fields of moduli and fields of definition of curves'', Preprint, \url{arXiv:math/0610247}] qu'il existe des surfaces pseudo-réelles pour tout \(k\geq2\).
Dans cet article, l'auteur considère le cas des surface de Riemann \(X\) qui sont données par une courbe algébrique planaire. Dans ce cas, la thèse d'Huggins implique que si \(X\) est pseudo-réelle, alors son groupe d'automorphismes est conjugué par \(\mathrm{PGL}_{3}\) à \(\mathrm{Hess}_{38}\) ou à un groupe diagonal. Il existe des exemples où le groupe est \(\mathrm{Hess}_{38}\) et l'auteur se pose donc la question de l'existence d'exemples où le groupe est diagonal.
Il est montré qu'il existe des courbes planaires pseudo-réelles de degré \(d=2(2k+1)\geq6\) dont le groupe d'automorphismes est \(\mathbb{Z}_{d}\) (Corollary 3.13) et de degré \(d=2pm\) avec \(p\) un nombre premier et \(m\) un nombre impair supérieur ou égal à \(3\) dont le groupe d'automorphismes est \(\mathbb{Z}_{d/p}\) (Theorem 3.15). Dans tous les autres cas il n'existe pas de surfaces de Riemann pseudo-réelles.
Les preuves reposent de manière essentielles sur la description explicite des courbes planaires dont le groupe d'automorphismes est diagonal par [\textit{E. Badr} and \textit{F. Bars}, Int. J. Algebra Comput. 26, No. 2, 399--433 (2016; Zbl 1357.14041)] et de nombreux résultats donnant des conditions pour qu'une surface de Riemann soit pseudo-réelle. L'article est globalement bien écrit même s'il est très dommage que ses résultats ne soient pas exposés dans l'introduction. pseudo-real Riemann surface; field of moduli; field of definition; plane curve; automorphism group Plane and space curves, Automorphisms of curves, Compact Riemann surfaces and uniformization, Families, moduli of curves (algebraic) A class of pseudo-real Riemann surfaces with diagonal automorphism group | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians A rope \(\tilde Y\) is a non-reduced scheme over a smooth projective curve \(Y\), whose local structure is the infinitesimal neighbourhood of \(Y\) in the total space of some vector bundle \(E\). Ropes may arise as flat limits of smooth curves, in the Hilbert scheme. Consequently, the study of ropes can spread some light in our knowledge of families of embedded curves.
The authors consider ropes, from the following point of view: they study which ropes are smoothable, i.e. lie in the boundary of the Hilbert scheme of smooth curves. To give an answer, they prove the following interesting
Lemma. Finite covers onto smooth irreducible curves can be flatly deformed to \(1:1\) mappings. As a corollary, it turns out that ropes of non-negative arithmetic genus are smoothable, provided that the bundle \(E\) is the trace \(0\) module of a smooth, irreducible, finite cover of \(Y\).
Ropes of multiplicity \(2\) have been considered in several papers, in the literature. The authors consider then ropes of multiplicity \(3\) over \(\mathbb P^1\). They are able to prove that such ropes are always smoothable, whenever the arithmetic genus is non-negative, with no additional assumptions on \(E\). curves Gallego, FJ; González, M; Purnaprajna, BP, Deformation of finite morphisms and smoothing of ropes, Compos. Math., 144, 673-688, (2008) Families, moduli of curves (algebraic) Deformation of finite morphisms and smoothing of ropes | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is 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 reduced graded algebra which is finitely generated by forms of various positive degrees over a field \(k\) is called a reduced G-algebra. It is well known that for a G-algebra \(A\) with seminormalization B there is a close relationship between \(Pic(A)\) and \(B/A\), for example if A is seminormal so \(B=A\) then \(Pic(A)=0\). This paper makes this relationship explicit by giving an actual calculation of \(Pic(A)\) in case either \(B/A\) is finite dimensional as a k-vector space or \(char(k)=0.\)
In the first case the calculation is accomplished by a straightforward and explicit identification of the elements of \(Pic(A)\) and the results is that Pic(A) is isomorphic to the group formed by taking the quotient of the muliplicative monoid
\[
1+B_+ = \{1+a_1 +a_2 +\cdots +a_ n| a_ i \text{ is homogeneous of degree }i\}
\]
by the relation \(f\equiv g\) if and only if there exists \(h\in 1+B_+\) so that both \(fh,gh\in A.\)
The second case (where \(char(k)=0)\) relies on the fact that \(Pic(A)\) is a module over the ring of Witt vectors \(W(k)\). Here the calculation is accomplished by viewing the monoid \(1+B_+\) described above as a subset of \(W(B)\) and applying the ghost map. The result is that \(Pic(A)=B/A\) and this isomorphism is a \(k\)-module isomorphism of the natural k-module structures on both \(Pic(A)\) and \(B/A\). Picard group; reduced G-algebra; seminormalization; Witt vectors Dayton Barry H., J. Pure Appl. Algebra Grothendieck groups, \(K\)-theory and commutative rings, General commutative ring theory, , Picard groups, Witt vectors and related rings The Picard group of a reduced G-algebra | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is 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 work with a smooth relative curve \(X_U/U\) with nodal reduction over an excellent and locally factorial scheme \(S\). We show that blowing up a nodal model of \(X_U\) in the ideal sheaf of a section yields a new nodal model and describe how these models relate to each other. We construct a Néron model for the Jacobian of \(X_U\) and describe it locally on \(S\) as a quotient of the Picard space of a well-chosen nodal model. We provide a combinatorial criterion for the Néron model to be separated. compactified Jacobians; Néron models; stable curves; nodal curves Families, moduli of curves (algebraic), Jacobians, Prym varieties Néron models of Jacobians over bases of arbitrary dimension | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is 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 \textit{pseudo-real} Riemann surface is a Riemann surface admitting anticonformal automorphisms but not anticonformal involutions. It is shown that the order of the group of conformal and anticonformal automoprhisms of a pseudo-real Riemann surface is always divisible by 4, and if the genus of the surface is \(g\), the maximal order is \(12(g-1)\). This bound is sharp. A point of comparison is Hurwitz's classical \(84(g-1)\) bound. Riemann surfaces; involutions; chiral maps Bujalance, E., Conder, M. D. E., Costa, A. F.: Pseudo-real Riemann surfaces and chiral regular maps. Trans. Amer. Math. Soc., 362(7), 3365--3376 (2010) Compact Riemann surfaces and uniformization, Families, moduli of curves (algebraic), Relations of low-dimensional topology with graph theory Pseudo-real Riemann surfaces and chiral regular maps | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians In [\textit{S. Kobayashi}, Hyperbolic manifolds and holomorphic mappings, Marcel Dekker, New York (1970; Zbl 0207.37902)], Kobayashi conjectured that a generic hypersurface of \(\mathbb{P}_{\mathbb{C}}^n\) of sufficiently high degree \(d\) (expected bound being \(d \geq 2n-1\)) is hyperbolic. It is true for \(n=2\), but for \(n>2\) only some examples are known. For \(n=3\), Brody and Green obtained the first example of a smooth hyperbolic surface in \(\mathbb{P}_{\mathbb{C}}^3\) of any degree \(d=2p \geq 50\) [\textit{R. Brody} and \textit{M. Green}, Duke Math. J. 44, 873-874 (1977; Zbl 0383.32009)], and Nadel obtained such examples of any degree \(d=6p+3\geq 21\) [\textit{A. M. Nadel}, Duke Math. J. 58, No. 3, 749-771 (1989; Zbl 0686.32015)].
In the present paper, the author improves Nadel's technique and produces examples of smooth hyperbolic surfaces in \(\mathbb{P}_{\mathbb{C}}^3\) of arbitrary degree \(d \geq 14\), which is closer to the expected bound \(5\). Furthermore, if \(H_{3,d}\) is the subset of all hyperbolic surfaces in the projective space \(P_{3,d}\) of all surfaces of degree \(d\) in \(\mathbb{P}_{\mathbb{C}}^3\), and if \(d>9+ \sum_{i=0}^3 k_i\) where at least two of the integers \(k_i\) are \(\geq 2\), then the author proves that \(H_{3,d}\) contains a Zariski open subset of \(P_{3,d}\) of dimension \(\sum_{i=0}^3 \binom{k_i+3}{k_i} -1\). hyperbolic surfaces; complex projective 3-space El Goul, J., Algebraic families of smooth hyperbolic surfaces of low degree in \(\mathbf{P}^{3}_{\mathbf{C}}\), Manuscr. Math., 90, 521-532, (1996) Hyperbolic and Kobayashi hyperbolic manifolds, Families, moduli of curves (algebraic), Singularities of curves, local rings, Enumerative problems (combinatorial problems) in algebraic geometry Algebraic families of smooth hyperbolic surfaces of low degree in \(\mathbb{P}_ \mathbb{C}^ 3\) | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians The moduli spaces \(\bar{M}_{g,n}\) of stable \(n\)-pointed complex curves of genus \(g\) carry natural rational cohomology classes \(\omega_{g,n}(a)\) of degree \(2a\), which were introduced by Mumford for \(n=0\) and subsequently by \textit{E. Arbarello} and \textit{M. Cornalba} [J. Algebr. Geom. 5, No. 4, 705-749 (1996; Zbl 0886.14007)] for all \(n\). Integrals of products of these classes over \(\bar{M}_{g,n}\) are called higher Weil-Petersson volumes; if only \(\omega_{g,n}(1)\) is involved they reduce to classical WP volumes.
\textit{P. Zograf} [in: Mapping class groups and moduli spaces of Riemann surfaces, Proc. Workshops Göttingen 1991, Seattle 1991, Contemp. Math. 150, 367-372 (1993; Zbl 0792.32016)] obtained recursive formulas for the classical WP volumes involving binomial coefficients. The authors generalise them in several ways: first they give both recursive formulas and closed formulas involving multinomial coefficients for higher WP volumes in genus 0, secondly they obtain a closed formula for higher WP volumes in arbitrary genus, where the multinomial coefficients get replaced by the less well known correlation numbers \(\langle \tau_{d_1} \cdots \tau_{d_n}\rangle\).
Finally the authors describe the 1-dimensional cohomological field theories occurring in an article by \textit{M. Kontsevich} and \textit{Yu. Manin} with an appendix by \textit{R. Kaufmann} [Invent. Math. 124, No. 1-3, 313-339 (1996; Zbl 0853.14021)] explicitly using the generating function they found for the higher WP volumes in genus 0. This last description has been generalised by \textit{A. Kabanov} and \textit{T. Kimura} [``Intersection numbers and rank one cohomological field theories in genus one'', preprint 97-61, MPI Bonn] to the genus one case. moduli spaces of stable pointed curves; Weil-Petersson volumes; 1-dimensional cohomological field theories; rational cohomology classes Kaufmann, R.; Manin, Yu.; Zagier, D., Higher {W}eil-{P}etersson volumes of moduli spaces of stable {\(n\)}-pointed curves, Comm. Math. Phys., 181, 3, 763-787, (1996) Families, moduli of curves (algebraic), Singularities of curves, local rings Higher Weil-Petersson volumes of moduli spaces of stable \(n\)-pointed 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 One of the fundamental discoveries of \textit{S. Mori} [cf. Ann. Math., II. Ser. 110, 593-606 (1979; Zbl 0423.14006) and 116, 133-176 (1982; Zbl 0557.14021)] is the cone theorem. In its simplest form it can be stated as follows:
Theorem. Let \(X\) be a smooth projective variety and \(NE(X) \subset H_ 2 (X, \mathbb{R})\) the convex cone generated by the homology classes of algebraic curves on \(X\). Then \(NE(X)\) is ``locally polyhedral'' in the open half-space \(\{z \in H_ 2 (X, \mathbb{R}) | z \cdot K_ X < 0\}\), where \(K_ X\) is the canonical class.
Subsequently a cohomological approach to the cone theorem was developed in a series of articles (Kawamata, Reid, Shokurov and the author). This approach relies heavily on vanishing results of Kodaira type, and thus is applicable only in characteristic zero.
The aim of this article is to develop Mori's geometric approach further. There are at least two reasons to attempt this. The first is the hope of doing Mori's program in positive characteristic. Second, the geometric approach yields information about the cone of curves that is not yet available via the cohomological approach even in characteristic zero. I was unable to prove the formulas about deformations of curves in singular varieties that would make Mori's original proof work. Therefore the current proofs proceeds along slightly different lines. The first step is the observation that, in order to establish the cone theorem, one needs the desired formulas only for certain curves. More precisely, instead of deforming \(C \subset X\) it is sufficient to deform a morphism \(D \to C \subset X\) where \(D\) is any finite cover of \(C\) (possibly ramified or even inseparable). To study deformations of \(D \to X\) the following construction is used:
Lemma. Let \(X\) be a projective variety over \(\mathbb{C}\) with at worst isolated quotient singularities. Then there is a unique algebraic space \(Y\) and a morphism \(p : Y \to X\) with the following properties:
(1) \(Y\) is of finite type and \(Y \to \text{Spec} \mathbb{C}\) is universally closed;
(2) \(Y\) is smooth;
(3) \(p\) is an isomorphism over \(X - \text{Sing} X\);
(4) \(p\) is one-to-one on closed points.
I call \(Y\) the bug-eyed cover of \(X\) [compare the picture in \textit{M. Artin}'s paper in J. Algebra 29, 330-348 (1974; Zbl 0292.14013), p. 331]. -- For suitable choices of \(D\) there is a factorisation \(D \to Y \to X\). Since \(Y\) is smooth, the deformation theory of \(D\to Y\) is as expected. This way one obtains deformations of \(D \to X\) as well. These questions are discussed in \S2.
The main theorem (5.1) proves the cone theorem for varieties whose singularities are quotients of complete intersection singularities by an equivalence relation that is étale in codimension one. In characteristic zero this is a reasonable restriction; however in positive characteristic one would like to allow quotients by nonreduced group schemes as well (and possibly certain other inseparable covers). As an application of this approach one obtains the following result:
Theorem. Let \(X\) be a normal projective threefold over a field of characteristic zero. Assume that \(c_ 1(K_ X) \in N(S(X) \otimes \mathbb{Q}\) exists (e.g. \(K_ X\) is \(\mathbb{Q}\)-Cartier). Then \(NE(X)\) is ``locally polyhedral'' in the open half-space \(\{z \in N_ 1 (X) | z\cdot c_ 1 (K_ X) < 0\}\).
The aim of the last section is to investigate curves on a threefold \(X\) that have no deformations at all. A curve \(C \subset X\) is called very rigid if for every subscheme \(\overline C \subset X\) whose support is \(C\), every flat deformation of \(\overline C\) is supported on \(C\) (except possibly at finitely many points). Very rigid curves arise in connection with flips and flops. cone theorem; deformations of curves; bug-eyed cover; threefold János Kollár, Cone theorems and bug-eyed covers, J. Algebraic Geom. 1 (1992), no. 2, 293 -- 323. Coverings in algebraic geometry, \(3\)-folds, Formal methods and deformations in algebraic geometry, Families, moduli of curves (algebraic) Cone theorems and bug-eyed 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 Let \(k\) be a field, \(X\) a smooth projective surface over \(k,Y\) a smooth projective curve over \(k\), and \(f:X\to Y\) a generically smooth semistable curve of genus \(g\geq 2\) over \(Y\). Let \(K\) be the function field of \(Y\), \(\overline K\) the algebraic closure of \(K\), and \(C\) the generic fiber of \(f\). For \(D\in\text{Pic}^1(C)(\overline K)\), let \(j_D:C_{\overline K}\to\text{Pic}^0 (C)_{\overline K}\) be an embeddlng defined by \(j_D(x)=x-D\). Then, we have the following
Conjecture 1 (Bogomolov conjecture over function fields). If \(f\) is non-isotrivial, then, for any embedding \(j_D\), the image \(j_D(C (\overline K))\) is discrete in terms of the semi-norm \(\|\;\|_{NT}\) given by the Néron-Tate height pairing on \(\text{Pic}^0(C)(\overline K)\), i.e., for any point \(P\in\text{Pic}^0 (C)(\overline K)\), there is a positive number \(\varepsilon\) such that the set \(\{x\in C(\overline K)| j_D(x)-P \|_{NT} \leq\varepsilon\}\) is finite.
In this paper, we will prove the above conjecture under the assumption that the stable model of \(f:X\to Y\) has only geometrically irreducible fibers.
Theorem 2. If the stable model of \(f:X \to Y\) has only geometrically irreducible fibers, then conjecture 1 holds. More strongly, there is a positive number \(A\) with the following properties.
(1) \(A\geq \sqrt{{g-1\over 12g(2g+1)} \delta}\), where \(\delta\) is the number of singularities in singular fibers of \(f_{\overline k}:X_{\overline k}\to Y_{\overline k}\).
(2) For any small positive number \(\varepsilon\), the set \(\{x\in C(\overline K)\mid\| j_D(x)- P\|_{NT}\leq (1-\varepsilon)A\}\) is finite for any embedding \(j_D\) and any point \(P\in\text{Pic}^0(C)(\overline K)\).
Our proof of theorem 2 is based on the admissible pairing on semistable curves due to \textit{S. Zhang} [Invent. Math. 112, No. 1, 171-193 (1993; Zbl 0795.14015)], Cornalba-Harris-Xiao's inequality [\textit{M. Cornalba} and \textit{J. Harris}, Ann. Sci. Éc. Norm. Supér., IV. Sér. 21, No. 3, 455-475 (1988; Zbl 0674.14006) and \textit{G. Xiao}, Math. Ann. 276, 449-466 (1987; Zbl 0596.14028)] over an arbitrary field and an excact calculation of a Green function on a certain metrized graph. The estimation of a Green function also gives the following result, which strengthen S. Zhang's theorem [loc. cit.].
Theorem 3. Let \(K\) be a number field, \(O_K\) the ring of integers, \(f:X\to\text{Spec}(O_K)\) a regular semistable arithmetic surface of genus \(g\geq 2\) over \(O_K\). If \(f\) is not smooth, then
\[
(\omega_{X/O_K}^{Ar} \cdot\omega_{X/O_K}^{Ar}) \geq{\log 2\over 6(g-1)}.
\]
Bogomolov conjecture over function fields; discrete embedding of curve; Néron-Tate height pairing; admissible pairing; Green function; semistable arithmetic surface A. Moriwaki, Bogomolov conjecture over function fields for stable curves with only irreducible fibers, Compos. Math. 105 (1997), 125-140. Algebraic functions and function fields in algebraic geometry, Arithmetic varieties and schemes; Arakelov theory; heights, Picard groups Bogomolov conjecture over function fields for stable curves with only irreducible fibers | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians The paper gives a description of the Picard groups \(\text{Pic}(\Gamma)\) of algebraic line bundles of the Siegel modular varieties \(M_ \Gamma\) of genus 2 in terms of the Saito-Kurokawa lift. Furthermore it is shown, that the absolute Galois group over \(\mathbb Q\) acts only via its factor commutator group on \(\text{Pic}(\Gamma)\otimes\mathbb Q\). Picard groups; Siegel modular varieties; Saito-Kurokawa lift R. Weissauer, The Picard group of Siegel modular threefolds, J. Reine Angew. Math. 430 (1992), 179 -- 211. With an erratum: ''Differential forms attached to subgroups of the Siegel modular group of degree two'' [J. Reine Angew. Math. 391 (1988), 100 -- 156; MR0961166 (89i:32074)] by the author. Siegel modular groups; Siegel and Hilbert-Siegel modular and automorphic forms, Arithmetic aspects of modular and Shimura varieties, Picard groups, Modular and Shimura varieties, \(3\)-folds The Picard group of Siegel modular threefolds | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians In [\textit{A. Zimmermann}, CMS Conf. Proc. 18, 721-749 (1996; Zbl 0855.16015)] and \textit{R. Rouquier} and \textit{A. Zimmermann} [``Picard groups for derived module categories'', Proc. Lond. Math. Soc. 87, 197-225 (2003; Zbl 1058.18007)] Rouquier and the reviewer and independently \textit{A. Yekutieli} [J. Lond. Math. Soc., II. Ser. 60, 723-746 (1999; Zbl 0954.16006)] defined the group of standard self-equivalences of the derived category of an algebra. One of the statements proved in these papers in varying generality was the fact that for a commutative indecomposable ring \(R\) any self-equivalence of the derived category of bounded complexes of \(R\)-modules is given by a composition of a shift in degree and a Morita self-equivalence. The author proves in the paper under review basically the same kind of statement replacing a commutative ring by a commutative unital ringed Grothendieck topos with enough points. The method of proof is similar to the one in [Rouquier and Zimmermann, loc. cit.], though, since the setting is much more general, many technical difficulties are considerably harder in the paper under review. self-equivalences for derived categories; derived category of commutative rings Fausk, H.: Picard groups of derived categories, J. Pure Appl. Algebra 180, 251--261 (2003) Monoidal categories (= multiplicative categories) [See also 19D23], Derived categories, triangulated categories, Picard groups, Étale and other Grothendieck topologies and (co)homologies, Module categories in associative algebras Picard groups of derived categories | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians We give a proof of Mirzakhani's recursion formula of Weil-Petersson volumes of moduli spaces of curves using the Witten-Kontsevich theorem. We also describe properties of intersections numbers involving higher degree \(\kappa\) classes. Weil-Petersson volume; Mirzakhani recursion formula; Witten-Kontsevich theorem 21. K. Liu and H. Xu, Mirzakhani's recursion formula is equivalent to the Witten-Kontsevich theorem, Astérisque328 (2009) 223-235. Families, moduli of curves (algebraic), Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects), Relationships between algebraic curves and physics Mirzakhani's recursion formula is equivalent to the Witten-Kontsevich 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 We establish the basic properties of the Mal'tsev completion of a discrete group relative to a Zariski dense representation of the group into an affine algebraic group. We then study the completion of a mapping class group with respect to its standard representation on the first homology of the corresponding Riemann surface. In particular, we prove that the pro-unipotent completion of the corresponding Torelli group is a central extension of the pro-unipotent radical of this completion of the mapping class group and that the kernel is one-dimensional, provided the genus is sufficiently large. The central extension is related to the normal function of the algebraic 1-cycle \(C-C^-\) in the Jacobian of an algebraic curve \(C\). Mal'tsev completion of a discrete group; affine algebraic group; mapping class group; pro-unipotent completion; Torelli group; algebraic 1-cycle; Jacobian of an algebraic curve Hain, R., Completions of mapping class groups and the cycle \(C - C^-\), Contemp. math., 150, 75-105, (1993) Topology of Euclidean 2-space, 2-manifolds, General low-dimensional topology, Families, moduli of curves (algebraic), Fundamental groups and their automorphisms (group-theoretic aspects), Algebraic cycles, Affine algebraic groups, hyperalgebra constructions, Infinite-dimensional Lie (super)algebras, Infinite-dimensional Lie groups and their Lie algebras: general properties, Rational homotopy theory, Differential topological aspects of diffeomorphisms Completions of mapping class groups and the cycle \(C-C^ -\) | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians The authors give a construction of the moduli space of stable maps to the classifying stack \(B\mu_r\) of a cyclic group by a sequence of \(r\)-th root constructions on \({\overline{\mathcal{M}}}_{0, n}\) and prove a closed formula for the total Chern class of \(\mu_r\)-eigenspaces of the Hodge bundle. From this, linear recursions for all genus-zero Gromov-Witten invariants of the stacks \([{\mathbb{C}}^N/\mu_r]\) are deduced.
More precisely, let \({\overline{\mathcal{M}}}_{0, n}(e_1, \dots, e_n; B\mu_r)\) be the moduli space of twisted stable maps to \(B\mu_r \cong [0/\mu_r] \subset [{\mathbb{C}}^N/\mu_r]\), where the branching behavior at the \(i\)-th section is prescribed by \(e_i \in \mu_r\). For every proper subset \(T \subset \{1, \dots, n-1\}\) having at least 2 elements, let \(D^T\) be the boundary divisor of \({\overline{\mathcal{M}}}_{0,n}\) consisting of curves having a node which separates the marking labels \(1,\ldots, n\) into \(T\) and \(\{1,\dots,n\}\setminus T\), and let \(r_T=\prod_{i \in T} e_i\). The authors show that \({\overline{\mathcal{M}}}_{0, n}(e_1, \dots, e_n; B\mu_r)\) is a \(\mu_r\)-gerbe over the stack constructed from \({\overline{\mathcal{M}}}_{0, n}\) by successively doing the \(r_T\)-th root construction at the boundary divisor \(D^T\) for all proper subsets \(T \subset \{1,\dots,n-1\}\) having at least 2 elements.
To determine a formula for the Chern class of the obstruction bundle, the authors introduce an ad-hoc definition of a ``moduli space of weighted stable maps to \(B\mu_r\)'', inspired by by the notion of weighted stable curves [\textit{B.~Hassett}, Adv. Math. 173, No. 2, 316--352 (2003; Zbl 1072.14014)] and weighted stable maps [\textit{V.~Alexeev, G.~M.~Guy}, J. Inst. Math. Jussieu 7, No. 3, 425--456 (2008; Zbl 1166.14034); \textit{A.~M.~Mustaţă, A.~Mustaţă}, J. Reine Angew. Math. 615, 93--119 (2008; Zbl 1139.14043); \textit{A.~Bayer, Yu I.~Manin}, Mosc. Math. J. 9, No. 1, 3--32 (2009; Zbl 1216.14051)]. When the weights of the linear action of \(\mu_r\) on \({\mathbb{C}}^N\) are chosen such that all fibers of the universal curve are irreducible, the obstruction bundle can easily be computed from general facts about the \(r\)-th root construction [\textit{C.~Cadman}, Am. J. Math. 129, No. 2, 405--427 (2007; Zbl 1127.14002)]. In particular, this is done for the weight data that give a moduli space isomorphic to \({\mathbb{P}}^{n-3}\). Next, by a careful analysis of the wall--crossing for changing weights, the authors are able to lift this to a closed formula for the equivariant top Chern class of the obstruction bundle for the standard (non-weighted) stable maps.
Finally, by a generalized inclusion-exclusion principle, the Chern class formula leads to linear recursions for all Gromov-Witten invariants of \([{\mathbb{C}}^N/\mu_r]\) by a sum over partitions, where every partition corresponds to a moduli space of comb curves, and an explicit formula for the non-equivariant invariants of \([{\mathbb{C}}^3/\mu_3]\) is deduced. stable maps; root construction; Gromov-Witten invariants; stacks; quantum orbifold cohomology A. Bayer and C. Cadman, Quantum cohomology of \([\mathbb{C}^N/\mu_r]\), Compos. Math. 146 (2010), no. 5, 1291-1322. MR2684301 (2012d:14095) Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects), Stacks and moduli problems, Families, moduli of curves (algebraic) Quantum cohomology of \([\mathbb C^N/ \mu _r]\) | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians From the introduction: ''Let \(X\) be a variety over a field \(k\). An alteration of \(X\) is a dominant proper morphism \(X'\to X\) of varieties over \(k\), with \(\dim X=\dim X'\). We prove that any variety has an alteration which is regular. This is weaker than resolution of singularities in that we allow finite extensions of the function field \(k(X)\). In fact, we can choose \(X'\) to be a complement of a divisor with strict normal crossings in some regular projective variety \(\overline X'\) (see theorem 4.1 and remark 4.2.). If the field \(k\) is local, we can find \(X' \subset \overline X'\) such that \(\overline X'\) is actually defined over a finite extension \(k\subset k'\) and has semi-stable reduction over \({\mathcal O}_{k'}\) in the strongest possible sense (see theorem 6.5).
As an application, we note that theorem 4.1 implies that for any variety \(X\) over a perfect field \(k\), there exist \((\alpha)\) a simplicial scheme \(X_\bullet\) projective and smooth over \(k\), \((\beta)\) a strict normal crossings divisor \(D_\bullet\) in \(X_\bullet\); we put \(U_\bullet=X_\bullet\setminus D_\bullet\), and \((\gamma)\) an augmentation \(a:U_\bullet\to X\) which is a proper hypercovering of \(X\). In case \(k\) is local, we may assume that the pairs \((X_n,D_n)\) are defined over finite extensions \(k_n\) of \(k\) and extend to strict semi-stable pairs over \({\mathcal O}_{k_n}\) (see 6.3). This should be interpreted as saying that in a suitable category \({\mathcal M}{\mathcal M}_k\) of mixed motives over \(k\), any variety \(X\) may be replaced by a complex of varieties which are complements of strict normal crossing divisors in smooth projective varieties.
We give a short sketch of the argument that proves our results in case \(X\) is a proper variety. The idea is to fibre \(X\) over a variety \(Y\) such that all fibres are curves and work by induction on the dimension of \(X\). After modifying \(X\), we may assume \(X\) is projective and normal and we can choose the fibration to be a kind of Lefschetz pencil, where the morphism is smooth generically along any component of any fibre. Next one chooses a sufficiently general and sufficiently ample relative divisor \(H\) on \(X\) over \(Y\). After altering \(Y\), i.e. we take a base change with an alteration \(Y'\to Y\), we may assume that \(H\) is a union of sections \(\sigma_i:Y\to X\). The choice of \(H\) above gives that for any component of any fibre of \(X\to Y\), there are at least three sections \(\sigma_i\) intersecting it in distinct points of the smooth locus of \(X\to Y\). The generic fibre of \(X\to Y\), together with the points determined by the \(\sigma_i\) is a stable pointed curve. By the existence of proper moduli spaces of stable pointed curves, we can replace \(Y\) by an alteration such that this extends to a family \({\mathcal C}\) with sections \(\tau_i\) of stable pointed curves over \(Y\). An important step is to show that the rational morphism \({\mathcal C}\cdots\to X\) extends to a morphism, possibly after replacing \(Y\) by a modification; this follows from the condition on sections hitting components of fibres above. Thus we see that we may replace \(X\) by \({\mathcal C}\). We apply the induction hypothesis to \(Y\) and we get \(Y\) regular. However, our induction hypothesis is actually stronger and we may assume that the locus of degeneracy of \({\mathcal C}\to Y\) is a divisor with strict normal crossings. At this point it is clear that the only singularities of \({\mathcal C}\) are given by equations of the type \(xy=t_1^{n_1} \cdot\dots \cdot t_d^{n_d}\). These we resolve explicitly.
Section 2 contains definitions and results, which we assume known in the rest of the paper. In section 3 we resolve singularities for a family of semi-stable curves over a regular scheme, which is degenerate over a divisor, with normal crossings. This we use in section 4, where we prove the theorem on varieties. Section 5 deals with the problem of altering a family of curves into a family of semi-stable curves. This we use in section 6, where we do the relative case, i.e. the case of schemes over a complete discrete valuation ring.
In the final two sections we indicate how to refine the method of proof of theorem 4.1 and theorem 6.5 to get results where one has additional restraints or works over other base schemes. In section 7 we prove that our method works (over algebraically closed fields) to get resolution of singularities up to quotient singularities and purely inseparable function field extensions. In fact we deal with the situation where there is a finite group acting. In section 8 we do the arithmetic case. In particular, we show that any integral scheme \(X\), flat and projective over \(\text{Spec} \mathbb{Z}\) can be altered into a scheme \(Y\) which is semi-stable over the ring of integers in a number field (theorem 8.2).
In a follow-up of this article the author proves that one can alter any family of curves into a semi-stable family of curves [see \textit{A. J. de Jong}, Ann. Inst. Fourier 47, No. 2, 599-621 (1977; Zbl 0868.14012)]. This is stronger than the result of section 5. In this cited paper the author deals with group actions as well. Thus the reader can find therein a number of results that extend the results of this article to (slightly) more general situations. For example it is shown that regular alterations exist of schemes of finite type over two-dimensional excellent base schemes.
A.Nobile:
This very important article describes a solution to a weak version of the desingularization problem for algebraic varieties over a field, as well as generalizations. The key concept to express the main results is that of alteration. An alteration of an integral scheme \(S\) is an integral scheme \(S'\) together with a morphism \(\varphi:S'\to S\) which is surjective, proper and such that, for a suitable open dense set \(U\subseteq X\), the induced morphism \(\varphi_U:\varphi^{-1}(U)\to U\) is finite.
The basic main result of this paper says: Given a variety \(X\) over an arbitrary field \(k\) and a proper closed set \(Z\subset X\), then there is an alteration \(\varphi:X_1\to X\) such that \(X_1\) is an open set of a regular scheme \(X_1'\), projective over \(k\), such that \(\varphi^{-1}(Z) \cup(X_1'-X_1)\) is a strict normal crossings divisor \(D\) of \(X_1'\) (i.e., the irreducible components of \(D\) are regular and meet transversally). There are also: (a) a \(G\)-equivariant version, where \(G\) is a finite group acting on \(X\), (b) a ``relative'' version, where \(X\) is an irreducible, separated scheme, flat and of finite type over \(S=\text{Spec}\,R\), with \(R\) a complete discrete valuation ring; (c) an arithmetic version, where the basic situation is as in part (b), but now \(R\) is a Dedekind domain whose field of fractions is a global field. -- The result of (b) may be interpreted as ``weak'' semistable reduction theorem (weak because certain alterations are allowed), without restrictions on the characteristics involved. As a tool to obtain (b) or (c), some interesting results about improving a family of curves via alterations are discussed.
This is the main technique to show the relevant theorems, sketched in the case of varieties over a field.
Moreover, to simplify \(X\) is assumed projective. One tries to find an alteration \(f:X'\to X\) such that there is flat morphism \(g:X'\to T\), with \(T\) regular, \(\dim(T)= \dim (X')-1\), such that the general fiber of \(g\) is regular and any fiber is a curve with, at worst, ordinary double points as singularities, moreover the set of points of \(T\) where \(g^{-1}(t)\) is singular is contained in a strict normal crossings divisor. Then, to desingularize such a \(X'\) by means of monoidal transformations is easy. An alteration as above is obtained by using some classical projective techniques, the theory of moduli for pointed semi-stable curves and an induction hypothesis (applied to \(T\), whose dimension is one less than that of \(X)\).
These results, although in a sense weak (because they involve alterations and not birational morphisms) are strong enough to solve a number of cohomological problems that ``require'' resolutin) of singularities. Some are described in the introduction. See also: \textit{P. Berthelot}, ``Altérations de varietés algébriques, Séminaire Bourbaki, Volume 1995/96, Exposé No. 815, Astérisque 24l, 273--311 (1997; Zbl 0924.14087)]. alteration; resolution of singularities; semi-stable family of curves de Jong, A. J., Smoothness, semi-stability and alterations, Publ. Math. Inst. Hautes Études Sci., 83, 51-93, (1996) Global theory and resolution of singularities (algebro-geometric aspects), Families, moduli of curves (algebraic), Singularities in algebraic geometry Smoothness, semi-stability and alterations | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians Let \(G\) be a finite group. For a \(G\)-ring \(A\), let \(\operatorname{ Pic}^G(A)\) denote the equivariant Picard group of \(A\). We show that if \(A\) is a finite type algebra over a field \(k\) then \(\operatorname{Pic}^G(A)\) is contracted in the sense of Bass with contraction \(H_{\operatorname{et}}^1(G; \operatorname{Spec}(A), \mathbb{Z})\). This gives a natural decomposition of the group \(\operatorname{Pic}^G (A[t, t^{-1}])\). equivariant Picard groups; contracted functor; \(G\)-sheaves Applications of methods of algebraic \(K\)-theory in algebraic geometry, \(K\)-theory of schemes, Abelian categories, Grothendieck categories, Picard groups Equivariant Picard groups and Laurent polynomials | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is 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 give a complete description of the integral Chow ring of the stack \(\mathscr{H}_{g,1}\) of 1-pointed hyperelliptic curves, lifting relations and generators from the Chow ring of \(\mathscr{H}_g\). We also give a geometric interpretation for the generators. Families, moduli of curves (algebraic), (Equivariant) Chow groups and rings; motives, Stacks and moduli problems The integral Chow ring of the stack of 1-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 This article is the author's doctoral dissertation, and is a detailed study of the moduli space \(M_ k\) of hyperelliptic curves of arbitrary genus g over an algebraically closed field k of characteristic 0. In the first part, the author constructs moduli spaces for hyperelliptic curves with group action: this includes a complete description of the groups which can act, and the possible specialisations of such actions. The second part is concerned with the stratification of \(M_ k\) given by the images of these moduli spaces in \(M_ k\). moduli space of hyperelliptic curves of arbitrary genus; moduli spaces for hyperelliptic curves with group action Families, moduli of curves (algebraic), Algebraic moduli problems, moduli of vector bundles, Group actions on varieties or schemes (quotients), Geometric invariant theory Automorphismengruppen und Moduln hyperelliptischer Kurven. (Automorphism groups and moduli 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 studies some properties of the moduli space of parabolic connections over a Riemann surface. To list the main results of this paper, we first fix some notations according to the paper.
Let \(X\) be a compact Riemann surface of genus \(g\geq 3\) with a finite set of punctures \(S=\{x_1,\dots,x_m\}\). Fix integers \(r\) and \(d\) with \(r>0\), let \(\alpha=\{\alpha_1^x,\cdots,\alpha_m^x\}_{x\in S}\) with each \(\alpha_i^x\in\mathbb{Q}\) and \(0\leq\alpha_1^x<\dots<\alpha_r^x<1\) be a generic system of weights so that \(\sum\limits_{x\in S}\sum\limits_{i=1}^r\alpha_i^x=-d\), and let \(\xi\) be a fixed line bundle of degree \(d\) over \(X\). Denote by \(\xi_*\) the induced trivial parabolic bundle with weight system \(\beta=\{\beta^x=\sum\limits_{i=1}^r\alpha_i^x\}_{x\in S}\). A parabolic bundle over \((X,S)\) of type \((r,d,\alpha)\) (resp. \((r,\alpha,\xi)\)) is a holomorphic vector bundle \(E\) of rank \(r\), degree \(d\) over \(X\) (resp. with determinant \(\xi\)), so that each fiber \(E_x\) at \(x\in S\) admits a full flag of decreasing linear subspaces and an attached weight system \(\alpha_x:=\{\alpha_1^x,\dots,\alpha_r^x\}\) to this flag. Let \(\mathcal{M}(r,d,\alpha)\) (resp. \(\mathcal{M}(r,\alpha,\xi)\)) be the moduli space of stable parabolic bundles over \((X,S)\) of type \((r,d,\alpha)\) (resp. of type \((r,\alpha,\xi)\)), and let \(\mathcal{M}_{pc}(r,d,\alpha)\) (resp. \(\mathcal{M}_{pc}(r,\alpha,\xi)\)) be the moduli space of stable parabolic connections of the corresponding type, which contains the moduli space \(\mathcal{M}'_{pc}(r,d,\alpha)\) (resp. \(\mathcal{M}_{pc}'(r,\alpha,\xi)\)) of parabolic connections so that the underlying parabolic bundles are stable as an open dense subset.
The first result of this paper is the property on the Picard groups of the moduli spaces mentioned above, namely \(\mathrm{Pic}(\mathcal{M}_{pc}(r,d,\alpha))\cong\mathrm{Pic}(\mathcal{M}(r,d,\alpha))\). And there is a natural compactification of \(\mathcal{M}'_{pc}(r,d,\alpha)\) and \(\mathcal{M}_{pc}'(r,\alpha,\xi)\) by smooth divisors, the compatification comes from a construction of an algebraic vector bundle over these moduli spaces.
Then the author studies the space of holomorphic connections \(\mathcal{C}(L)\) on an ample line bundle \(L\) over \(\mathcal{M}(r,d,\alpha)\), and showed \(\mathrm{Pic}(\mathcal{C}(L))\cong\mathrm{Pic}(\mathcal{M}(r,d,\alpha))\) and \(H^0(\mathcal{C}(L),\mathcal{O}_{\mathcal{C}(L)})=\mathbb{C}\), namely there is no non-constant algebraic function on \(\mathcal{C}(L)\). parabolic connection; moduli space; compactification; Picard group Algebraic moduli problems, moduli of vector bundles, Picard groups, Algebraic functions and function fields in algebraic geometry Line bundles on the moduli space of parabolic connections over a compact Riemann surface | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is 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 \(R\) be a Noetherian integral domain with integral closure \(\overline R\) and let \(C\) be the conductor \((R:_R \overline R)\). Assume that \(C\neq 0\) and that \(\overline R\) is a principal ideal domain. The main result, as modified in the Corrigendum, is that \(\text{Pic} (R) \cong U (\overline R/C)/ (U (R/C) \cdot W)\) where \(U(\;)\) denotes the group of units and \(W\) is the image of \(U(\overline {R})\) in \(U(\overline {R}/ C)\). Picard group; Krull dimension one; Noetherian integral domain; integral closure DOI: 10.1080/00927879508825512 Integral domains, Integral closure of commutative rings and ideals, Commutative Noetherian rings and modules, Picard groups The Picard group of Noetherian integral domains whose integral closures are principal ideal 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 the entire collection see Zbl 0651.00012.]
In an exposition apparently aimed at mathematical physicists, the author provides a gentle survey of the mysteries of the space of moduli of closed Riemann surfaces - its construction, its basic properties, its compactification, and some recent work on its divisors.
The mathematician who has difficulty locating the publication will probably be better served by the author's paper from the algebraic geometry conference in Bowdoin, 1985 [see Algebraic geometry, Proc. Summer Res. Inst., Brunswick/Maine 1985, Part 1, Proc. Symp. Pure Math. 46, 99-144 (1987; Zbl 0646.14019)] or his recent joint survey with \textit{D. Eisenbud} [Bull. Am. Math. Soc., New Ser. 21, 205-232 (1989)]. string theory; space of moduli of closed Riemann surfaces Harris, R. (1987).The language machine. London: Duckworth. Families, moduli of curves (algebraic), String and superstring theories; other extended objects (e.g., branes) in quantum field theory An introduction to the moduli space of curves | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians Let \(\overline{M}_3\) be the moduli space of stable curves of genus \(3\) and \(\delta\) be the total boundary. In this paper, the authors compute the log canonical models for the pairs \((\overline{\mathcal M}_3, \alpha \delta )\) given by
\[
\overline{M}_3(\alpha ):=\mathbf{Proj}\bigoplus _{m\geq 0}\Gamma (\overline{M}_3, m(K_{\overline{\mathcal M}_3}+\alpha \delta))
\]
for all \(\alpha \in [0,1]\). They give a modular interpretation of each model \(\overline{M}_3\) and of the birational maps between them. Hyeon D., An outline of the log minimal model program for the moduli space of curves, preprint 2010, . Families, moduli of curves (algebraic), Minimal model program (Mori theory, extremal rays) Log minimal model program for the moduli space of stable curves 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 \textit{M. M. Kapranov} [J. Algebr. Geom. 2, No. 2, 239--262 (1993; Zbl 0790.14020)] gave a compactification \(\bar X(r,n)\) of the moduli space \(X(r,n)\) of ordered \(r\)-tuples of hyperplanes in \({\mathbb P}^{r-1}\) in linear general position. The construction works by looking at the Chow points of orbit closures of an action of an algebraic group, in this case \(\text{ PGL}_r\), on a Grassmannian. If \(r=2\) one recovers the usual compactification \(\bar M_{0,n}\) of \(M_{0,n}\), the space of \(n\) ordered points on \({\mathbb P}^1\). Like \(\bar M_{0,n}\), \(\bar X(r,n)\) carries a family of pairs and the fibres have good (actually toroidal) singularities. All these facts were proved by Kapranov [loc. cit.] and \textit{L. Lafforgue} [Invent. Math. 136, No. 1, 233--271 (1999; Zbl 0965.14024)].
It was observed by Hacking, and independently by the authors (the proof is not given here), that \(\bar X(r,n)\) admits a moduli interpretation as a moduli space of certain semi log canonical pairs, which is what one would hope for as an analogue of the moduli properties of \(\bar M_{0,n}\). One would also hope to be able to complete a 1-parameter family (say over a punctured disc, over the complex numbers) in a natural way. One of the main results of this paper is that this is indeed possible: the necessary description of the central fibre is similar to the one given by Kapranov for \(\bar M_{0,n}\) and involves combinatorics controlled by the Bruhat-Tits building of~\(\text{ PGL}_r\)
By analogy with the case of \(\bar M_{0,n}\) one could also hope that the normalisation of \(\bar X(r,n)\) would be the log canonical model. The authors show, however, that this is not the case: on the contrary, already for \(r=3\) the singularities that can occur are essentially arbitrary. This is slightly disappointing but perhaps not wholly surprising, and leads the authors naturally to the question of what the log canonical compactification is. They conjecture that it is \(\bar X(r,n)\) in the few cases where they cannot show that it is not (there is an exception in characteristic~2).
The proofs form part of a rather detailed study of of Chow quotients in general and \(\bar X(r,n)\) in particular, which is really the content of the paper. This is not readily summarised, especially as in order to keep the length of the paper within reasonable bounds the authors have not repeated material from Kapranov's and Lafforgue's papers. Chow quotient; compactification; log canonical model; families Keel S., Tevelev J.: Geometry of Chow quotients of Grassmannians. Duke Math. J. 134, 259--311 (2006) Families, moduli, classification: algebraic theory, Families, moduli of curves (algebraic), Rational and birational maps, Group actions on varieties or schemes (quotients), Arrangements of points, flats, hyperplanes (aspects of discrete geometry) Geometry of Chow quotients of Grassmannians | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is 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 derive effective recursion formulae of top intersections in the tautological ring \({R^*(\mathcal{M}_g)}\) of the moduli space of curves of genus \(g \geq 2\). As an application, we prove a convolution-type tautological relation in \({R^{g-2}(\mathcal{M}_g)}\) and some interesting Bernoulli number identities. moduli spaces of curves; intersection numbers; tautological rings Kefeng Liu and Hao Xu, Computing top intersections in the tautological ring of \Cal M_{\?}, Math. Z. 270 (2012), no. 3-4, 819 -- 837. Families, moduli of curves (algebraic), Intersection theory, characteristic classes, intersection multiplicities in algebraic geometry Computing top intersections in the tautological ring of \({\mathcal{M}_g}\) | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians Magnus proved in 1930 that, given two elements \( x\) and \( y\) of a finitely generated free group \( F\) with equal normal closures \( \langle x\rangle ^F=\langle y\rangle ^F\), \( x\) is conjugated either to \( y\) or \( y^{-1}\). More recently, this property, called the Magnus property, has been generalized to oriented surface groups. In this paper, we consider an analogue property for profinite surface groups. While the Magnus property, in general, does not hold in the profinite setting, it does hold in some restricted form. In particular, for \( \mathscr {S}\) a class of finite groups, we prove that if \( x\) and \( y\) are \textit{algebraically simple} elements of the pro-\( \mathscr {S}\) completion \( \widehat {\Pi }^{\mathscr {S}}\) of an orientable surface group \( \Pi \) such that, for all \( n\in \mathbb{N}\), there holds \( \langle x^n\rangle ^{\widehat {\Pi }^{\mathscr {S}}}=\langle y^n\rangle ^{\widehat {\Pi }^{\mathscr {S}}}\), then \( x\) is conjugated to \( y^s\) for some \( s\in (\widehat {\mathbb{Z}}^{\mathscr {S}})^\ast \). As a matter of fact, a much more general property is proved and further extended to a wider class of profinite completions.
The most important application of the theory above is a generalization of the description of centralizers of profinite Dehn twists given in [\textit{M. Boggi}, Trans. Am. Math. Soc. 366, No. 10, 5185--5221 (2014; Zbl 1298.14030)] to profinite Dehn multitwists. Limits, profinite groups, Geometric group theory, Fuchsian groups and their generalizations (group-theoretic aspects), Teichmüller theory for Riemann surfaces, Families, moduli of curves (algebraic), Coverings of curves, fundamental group A restricted Magnus property for profinite surface 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 To determine the degree, for example, of the space of plane curves of degree \(d\) and \(\delta\) ordinary nodes, the author generalizes the method of his previous paper [Ann. Math., II. Ser. 130, 121-157 (1989; Zbl 0704.14018)], where he showed that the space of such curves is irreducible by degenerating a blow-up of the ambient \({\mathbb{P}}^ 2\) to a union of ruled surfaces. The point of the degeneration, here as in the previous paper, is to reduce to curves of lower degree, while keeping the singularities manageable. Here, the new feature is to keep careful track of certain limit multiplicities, which is accomplished via local deformation theory. degree of the space of plane curves; enumerative geometry; degeneration Z. Ran, Enumerative geometry of singular plane curves, Invent. Math. 97 (1989), no. 3, 447-465. Enumerative problems (combinatorial problems) in algebraic geometry, Singularities of curves, local rings, Families, moduli of curves (algebraic) Enumerative geometry of singular plane 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 book focuses on the theory of Riemann surfaces from the modern viewpoint of complex algebraic curves and their moduli spaces. Chapters 1--3 introduce preliminaries of curves, including their algebraic structure, complex structure, topology, branched coverings, realization in the plane, and rational parametrization. Chapters 4--6 study curves in projective spaces, Plücker Formulas, and mappings of curves with a focus on elliptic curves. Chapters 7--12 highlight classical objects associated with curves, including differential forms, line bundles and divisors, Riemann-Roch Formula, Weierstrass points, and Jacobian. Chapters 13--18 focus on various moduli spaces, including Hilbert schemes, moduli spaces of curves with marked points, and moduli spaces of stable maps. Finally in Chapter 19 many exam questions are given to test the reader's understanding of the subject. algebraic curve; Riemann surface; moduli space Families, moduli of curves (algebraic), Riemann surfaces; Weierstrass points; gap sequences, Plane and space curves Algebraic curves. Towards moduli spaces. Translated from the Russian by Natalia Tsilevich | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is 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 Harnack-Thom inequality states that the total mod \(2\) Betti number of the real part of a real algebraic variety does not exceed the total Betti number of the complexification. The present paper contains upper bounds to the first and the second mod \(2\) Betti numbers of the real part of a real algebraic surface given via the Picard and Brauer groups of the complexification and the number of real connected components of the Albanese variety. Necessary and sufficient conditions for the equalities in these relations are found. The new upper bounds are close to the other analogues of the Harnack-Thom inequality found earlier by \textit{V. A. Krasnov} [Math. USSR, Izv. 22, 247-275 (1984); translation from Izv. Akad. Nauk SSSR, Ser. Mat. 47, No.~2, 268-297 (1983; Zbl 0537.14035)]. real algebraic surface; Galois cohomology; Picard group; Brauer group; real cycle map; Harnack-Thom inequality; Betti number; complexification V. A. Krasnov, ''Analogs of the Harnack-Thom inequality for a real algebraic surface,''Izv. Ross. Akad. Nauk Ser. Mat. [Russian Acad. Sci. Izv. Math.] (to appear). Topology of real algebraic varieties, Picard groups, Brauer groups of schemes Analogues of the Harnack-Thom inequality for a real algebraic surface | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is 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 present an algorithm for computing the Berkovich skeleton of a superelliptic curve \(y_n = f(x)\) over a valued field. After defining superelliptic weighted metric graphs, we show that each one is realizable by an algebraic superelliptic curve when \(n\) is prime. Lastly, we study the locus of superelliptic weighted metric graphs inside the moduli space of tropical curves of genus \(g\). tropical curves; superelliptic curves; semistable models; Berkovich spaces; trees; moduli spaces; stacky polyhedral fans Geometric aspects of tropical varieties, Families, moduli of curves (algebraic), Rigid analytic geometry Tropical superelliptic 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 \(\Sigma\) (d,k) be a complete irreducible family of plane curves of degree d in the complex projective plane whose general member has k nodes and no other singularity. The author reports that J. Harris has recently proved Severi's conjecture that \(\Sigma\) (d,k) is unique when \(0\leq k\leq (d-1)(d-2).\) Here it is shown that the sequence \(\Sigma (d,1)\supset \Sigma (d,2)...\supset \Sigma (d,(d-1)(d-2)/2)\) is unique; cf. \textit{C. Giacinti-Diebolt} [Math. Ann. 266, 321-350 (1984; Zbl 0504.14017)]. node; Severi conjecture; complete irreducible family of plane curves Families, moduli of curves (algebraic), Singularities of curves, local rings On the Galois property of some plane nodal curves | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians The sublocus of the moduli space of curves of genus g consisting of those curves which possess a Weierstrass point p whose non-gap sequence starts with k is an irreducible subvariety of dimension \(2g+k-3\). For a generic point of this sublocus, the space of functions \(L((kp)^{-1})\) has dimension exactly two. Here, using first order deformation theory, the author proves that for a generic point the non-gaps are simply multiples of k until after the largest gap. Thus in particular, Weierstrass points with gap sequence \(1,2,...,g-2,g,g+2\) (g\(\geq 4)\) give exceptional points and such are known to exist [\textit{D. Eisenbud} and \textit{J. Harris}, Bull. Am. Math. Soc., New. Ser. 10, 277-280 (1984; Zbl 0533.14013)]. The author shows that these give rise to non-transversal intersection beween the sublocus above and the sublocus defined by the condition that \(L(((g+1)p)^{-1})\) has dimension at least three. moduli space of curves; Weierstrass point; non-gaps Diaz, S.: Deformations of exceptional Weierstrass points. Proc. A.M.S., 96, 7--10 (1986) Riemann surfaces; Weierstrass points; gap sequences, Families, moduli of curves (algebraic), Compact Riemann surfaces and uniformization Deformations of exceptional Weierstrass 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 We compare the compactified Torelli morphism (as defined by \textit{V. Alexeev} [Publ. Res. Inst. Math. Sci. 40, No. 4, 1241--1265 (2004; Zbl 1079.14019)]) and the tropical Torelli map (as defined by the author in a joint work with \textit{S. Brannetti} and \textit{M. Melo} [Adv. Math. 226, No. 3, 2546--2586 (2011; Zbl 1218.14056)], and furthered studied by M. Chan). Our aim is twofold: on one hand, we will review the construction and main properties of the above mentioned two Torelli maps, focusing in particular on the description of their fibers achieved by the author in joint works with L. Caporaso; on the other hand, we will clarify the relationship between the two Torelli maps via the introduction of the reduction maps and the tropicalization maps. Len, Y., Markwig, H.: Lifting tropical bitngents. \textit{preprint}arXiv:1708.04480 (2017) Stacks and moduli problems, Torelli problem, Families, moduli of curves (algebraic), Jacobians, Prym varieties, Algebraic moduli of abelian varieties, classification, Tropicalizing vs. Compactifying the Torelli morphism | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians The article is devoted to the noncommutative geometry over a 4-dimensional Sklyanin algebra \(S\) and one of the main topics is \(\mathbb{P}^3_{\mathrm{skly}}=\mathrm{Proj}S\). Apart from quantum \(\mathbb{P}^2\)s substantially less was known about quantum \(\mathbb{P}^3\)s. The authors consider a two-sided noetherian connected graded \(k\)-algebra \(A\), where \(k\) denotes a field. In the category \(\mathrm{Gr}A\) of graded right \(A\)-modules with degree zero module homomorphism they write \(\mathrm{Ext}^i_{\mathrm{Gr}A}(M,N)\) for the extension groups in \(\mathrm{Gr}A\) and define \(\underline{\mathrm{Ext}}^*_{\mathrm{Gr}A}(M,N)=\bigoplus_{i\in \mathbb Z} \mathrm{Ext}^i_{\mathrm{Gr}A}(M,N(i))\). Then a connected Koszul algebra \(A\) and its quadratic dual \(A^{!}\) are taken. In particular, the Auslander property is studied over a connected graded Gorenstein Koszul algebra with Hilbert series \((1-t)^{-4}\). It is shown that a pencil of noncommutative quadric surfaces is similar to that of a generic pencil of quadrics in the commutative projective 3-space. There were found exactly four singular quadrics in the pencil. Their Picard groups and rulings are studied by the authors [``A new Koszul duality'', In preparation]. They have found out that a situation may take place when a smooth noncommutative quadric contains an analog of a curve with a self-intersection number \(-2\). While their investigations the authors actively used maximal Cohen-Macaulay modules and noncommutative versions of methods of \textit{R.-O. Buchweitz}, \textit{D. Eisenbud} and \textit{J. Herzog} [in: Singularities, representation of algebras, and vector bundles, Proc. Symp., Lambrecht/Pfalz/FRG 1985, Lect. Notes Math. 1273, 58--95; 96--116 (1987; Zbl 0633.13008)]. The results of the paper may be helpful for further investigations of higher-dimensional noncommutative quadric hypersurfaces. noncommutative algebraic geometry; noncommutative quadric surfaces; Sklyanin algebra DOI: 10.4171/JNCG/136 Noncommutative algebraic geometry, Pencils, nets, webs in algebraic geometry, Picard groups Noncommutative quadric 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 Fix positive integers \(g\), \(r\), \(s\), \(r_i\), \(1\leq i \leq s\). Let \(^{[s]}\mathcal S^{r_1,\dots,r_s}_g\) be the moduli scheme of all \((C,L_1,\dots,L_s)\) with \(C\) a smooth curve of genus \(g\), \(L_i\in \mathrm{Pic}(C), h^0(C,L_i)=r_i+1\) and \(L_i^{\otimes 2} \cong \omega_C\) (i.e., \(L_i\) is a theta-characteristic, i.e. \((C,L_i)\) is a spin curve). Here we prove the existence of irreducible components of \(^{[s]}\mathcal {S}^{r_1,... ,r_s}_g\) with the expected dimension \(3g-3 -\sum _{i=1}^{s} \binom{r_i+1}{2}\) for some \((s,r_i)\) (either \(s\leq g-2\) and \(r_i=1\) for all \(i\) or \(r_1=2\), \(r_i=1\) for \(i>1\) and \(g\geq 2s+3\)). theta-characteristic; spin curve; Brill-Noether theory Families, moduli of curves (algebraic), Special divisors on curves (gonality, Brill-Noether theory) Curves with several theta-characteristics with a prescribed number of sections | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is 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 observation in this note, that the critical dimensions 26 and 10 which occur in the theory of strings and superstrings can be observed from the results of \textit{D. Mumford} [Enseign. Math., II. Ser. 23, 39-110 (1977; Zbl 0363.14003)] on the moduli space of stable curves, has been of fundamental importance in the by now highly developed techniques in string theory. The author suggests also in this paper that the moduli space for \(g\to \infty\) should play a role for the Virasoro algebra analogous to that of the flag manifold for a simple Lie algebra. superstrings; moduli space of stable curves; string theory; Virasoro algebra Manin, Y.I., Critical dimensions of the string theories and the dualizing sheaf on the moduli space of (super) curves, Funct. Anal. Appl., 20, 244-246, (1986) Families, moduli of curves (analytic), Constructive quantum field theory, Applications of manifolds of mappings to the sciences, Families, moduli of curves (algebraic) Critical dimensions of the string theories and the dualizing sheaf on the moduli space of (super) 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 Consider a number field \(K\) with ring of integers \(O_K\) and a smooth \(K\)-curve \(X_K\) of positive genus which extends to a smooth proper scheme \(X\) over \(B=\text{Spec }O_K\) having a section \(B\to X\). Given a vector bundle \(\mathcal E\) of rank \(r>1\) and degree \(d\) over \(X\), Arakelov theory teaches one how to metrise it in a canonical fashion and how to define Chern classes \({\widehat c}_i(\widehat{\mathcal E})\) for the metrised bundle \(\widehat{\mathcal E}\). Miyaoka, Moriwaki and Soulé have shown independently that if the restriction of \(\mathcal E\) to \(X_K\) is stable, then the real number \(2r{\widehat c}_2(\widehat{\mathcal E})-(r-1)({\widehat c}_1(\widehat{\mathcal E}),{\widehat c}_1(\widehat{\mathcal E}))\) is always non-negative [see e.g. \textit{C. Soulé}, Invent. Math. 116, No. 1-3, 577-599 (1994; Zbl 0834.14013)]. In the present paper, the author shows that in the case when \(r\) and \(d\) are coprime, the above expression can be used to define a height function on the (fine) moduli space \({\mathcal M}_{X_K}(r, {\mathcal F}_K)\) of stable vector bundles over \(X_K\) of rank \(r\), degree \(d\) and determinant isomorphic to a fixed line bundle \({\mathcal F}_K\), by a procedure analogous to the Faltings-Hriljac interpretation of the canonical height over the jacobian of \(X_K\). He also includes a detailed account of a direct approach to the construction of the moduli space of stable vector bundles over the arithmetic surface \(X\), needed for obtaining a \(B\)-model of \({\mathcal M}_{X_K}(r, {\mathcal F}_K)\) over which the Arakelov-theoretic construction can be carried through. height functions; Arakelov theory; arithmetic surfaces; fine moduli spaces; vector bundles on curves C. Gasbarri, Hauteurs canoniques sur l'espace de modules des fibrés stables sur une courbe algébrique , Bull. Soc. Math. France 125 (1997), 457--491. Arithmetic varieties and schemes; Arakelov theory; heights, Families, moduli of curves (algebraic), Algebraic moduli problems, moduli of vector bundles Canonical heights on the moduli space of stable vector bundles over an algebraic 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 \(C\) be a smooth projective curve of genus \(g\) over an algebraically closed field and \(E\) a vector bundle on \(C\) of rank \(r\). The paper considers two different compactifications of the space of vector bundle quotients of \(E\) of fixed rank and degree. The first is Grothendieck's Quot scheme \(\text{Quot}_{k,d}(E)\) of coherent quotients of \(E\) of rank \(k\) and degree \(d\), while the second is the Kontsevich moduli space \(\overline{\mathcal M}_g (G (E,k), \beta_d)\) of stable maps into the relative Grassmannian \(G (E,k)\) of \(k\)-dimensional quotient spaces of \(E\). Here \(\beta _d \in H_2(G (E,k), \mathbb{Z})\) denotes the homology class of a section of the projection \(\pi : G (E,k) \rightarrow C\) corresponding to such a quotient of \(E\). Since the quotient bundles \(F\) of \(E\) of rank \(k\) are in one to one correspondence with sections \(\sigma _F\) of \(\pi\), it is possible to compactify the space of quotient bundles by compactifying the space of sections. A point of \(\overline{\mathcal M}_g (G(E,k), \beta_d)\) corresponds to the data of an isomorphism class of a stable map \((C',f)\), where \(C'\) is a complete connected curve of arithmetic genus \(g\), with at most nodes as singularities, and \(f: C' \rightarrow G (E,k)\) is a map with \(f_* [C'] = \beta _d\) [For the involved definitions see e.g. \textit{W. Fulton} and \textit{R. Pandharipande}, Proc. Symp. Pure Math. 62, 45--96 (1997; Zbl 0898.14018), and \textit{J. Harris} and \textit{I. Morrison}, ``Moduli of curves'' Graduate Texts in Mathematics. 187. New York, NY: Springer (1998; Zbl 0913.14005)].
The first main result is: \(\dim \overline{\mathcal M}_g (G(E,k), \beta_d) \leq k(r-k) + (d-d_k)r\) for all \(d \geq d_k\), where \(d_k\) denotes the minimal degree of a quotient bundle of rank \(k\). As consequence of that, the authors obtain the analogous statement about Quot schemes: \(\dim \text{Quot}_{k,d}(E) \leq k(r-k) + (d-d_k)r\) for all \(d \geq d_k\).
The authors obtain the following relevant results about the component structure of the moduli spaces \(\overline{\mathcal M}_g (G(E,k), \beta_d)\) and \(\text{Quot}_{k,d}(E)\):
(1) For all large \(d\) there is a unique component of \(\overline{\mathcal M}_g (G(E,k), \beta_d)\) whose generic point corresponds to a smooth section (i.e. a vector bundle quotient); this component is of the expected dimension \(rd-ke-k(r-k)(g-1)\).
(2) In the case of the Quot scheme the best possible result holds: For any vector bundle \(E\) on \(C\), there is an integer \(d_Q\) such that for all \(d \geq d_Q\) the scheme \(\text{Quot}_{k,d}(E)\) is irreducible, generically smooth, of the expected dimension \(rd-ke-(k(r-k)(g-1)\) and its generic point corresponds to a vector bundle quotient.
(3) In contrast with the Quot scheme situation, the irreducibility of \(\overline{\mathcal M}_g (G(E,k), \beta_d)\) is obtained only for stable bundles \(E\): For any vector bundle \(E\) the spaces \(\overline{\mathcal M}_g (G(E,k), \beta_d)\) are connected for all large \(d\). They are irreducible for all \(k\) and all large \(d\) if and only if \(E\) is a generic stable bundle (\(g \geq 2\)).
The dimension of the schemes \(\text{Quot}_{k,d}(E)\) was studied for generic stable bundles \(E\) on a curve \(C\) with genus \(g \geq 2\) also by \textit{B. Russo} and \textit{M. Teixidor i Bigas} [J. Algebr. Geom. 8, 483--496 (1999; Zbl 0942.14013)]. The methods proposed in the paper under review indicate that sometimes it is more convenient to work with stable maps and reducible domain rather than with quotients having torsion. But the authors prove that in general there is no morphism from \(\overline{\mathcal M}_g (G(E,k), \beta_d)\) to \(\text{Quot}_{k,d}(E)\) extending the identification on the locus corresponding to vector bundle quotients. Such a morphism exists for \(k= r-1\) or \(d \in \{ d_k, d_k+1 \}\).
In the last section the authors apply their results to study linear series on the moduli spaces of vector bundles on curves. Let \(C\) be a smooth curve of genus \(g \geq 2\), let \(SU_C(r)\) denote the moduli space of semistable rank \(r\) vector bundles on \(C\) of trivial determinant and let \(\mathcal L\) be the determinant bundle. The authors prove that for \(p \geq [\frac{r^2}{4}]\) the series \(|\mathcal L ^p|\) on \(SU_C(r)\) is base point free. This is used to study the moduli space \(U_C(r,0)\) of semistable bundles of rank \(r\) and degree \(0\) and the generalized theta divisor \(\Theta _L\) associated to a line bundle \(L \in \text{Pic}^{g-1}\): If \(r \leq 5\) and \(k \geq r+1\), then the linear series \(|k \Theta _L|\) is base point free on \(U_C(r,0)\) [see also \textit{M. Popa}, Duke Math. J. 107, 469--495 (2001; Zbl 1064.14032), and Trans. Am. Math. Soc. 354, 1869--1898 (2002; Zbl 0996.14015)].
This article is an important contribution towards the understanding of the geometry of moduli spaces of curves, moduli spaces and \text{Quot} schemes of vector bundles over a curve. The paper is very well written, lucid, detailed, rigorous and enlightening. quot schemes; moduli spaces of curves; moduli spaces of vector bundles on curves; stable curves; stable maps; stable vector bundles on curves; relative Grassmannian M. Popa, M. Roth, \textit{Stable maps and Quot schemes}, Invent. Math. \textbf{152} (2003), no. 3, 625-663. Vector bundles on curves and their moduli, Algebraic moduli problems, moduli of vector bundles, Homogeneous spaces and generalizations, Geometric invariant theory, Families, moduli of curves (algebraic) Stable maps and Quot schemes | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians On a stack of stable maps, the cotangent line classes are modified by subtracting certain boundary divisors. These modified cotangent line classes are compatible with forgetful morphisms, and are well-suited to enumerative geometry: tangency conditions allow simple expressions in terms of modified cotangent line classes. Topological recursion relations are established among their top products in genus 0, yielding effective recursions for characteristic numbers of rational curves in any projective homogeneous variety. In higher genus, the obtained numbers are only virtual, due to contributions from spurious components of the space of maps. For the projective plane, the necessary corrections are determined in genus 1 and 2 to give the characteristic numbers in these cases. rational curves in homogeneous varieties; cotangent line classes; enumerative geometry T. Graber, J. Kock \& R. Pandharipande,Descendant invariants and characteristic numbers, math. AG/0102017. Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects), Families, moduli of curves (algebraic), Intersection theory, characteristic classes, intersection multiplicities in algebraic geometry Descendant invariants and characteristic numbers. | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians A projective space curve \(C\subset\mathbb P^3\) has \textit{maximal rank} if all the natural restriction maps \(H^0(\mathcal O_{\mathbb P^3}(m))\to H^0(\mathcal O_C(m))\) are either injective or surjective. A long-standing conjecture predicts the existence of components of the Hilbert scheme whose general element corresponds to a smooth, irreducible, maximal rank space curve of genus \(g\) and degree \(d\), when \(g\) is not too big with respect to \(d\). Here the notion of \textit{``not too big''} is related with the fact that if \(d,g\) grow and \(g\sim O(d^\alpha)\) with \(\alpha>(3/2)\), then the conjecture cannot hold asymptotically as soon as \(h^1(N_C)=0\), i.e. as soon as the Hilbert scheme is smooth of the expected dimension around \(C\). Thus, the authors concentrate in looking for a constant \(K\) such that the conjecture holds in the range \(g\leq Kd^{3/2}\). Indeed, they find constants \(K, \epsilon\) such that the conjecture holds (and moreover the curves satisfy \(h^1(N_C(-1))=0\)) in the range \(g\leq Kd^{3/2}-6\epsilon d\). The proof is obtained by induction, by smoothing reducible curves whose components satisfy suitable cohomological properties. The procedure is particularly devoted to solve the case \(d< g+3\), which was missing in the previous literature, the delicate point being that for \(d< g+3\) the Hilbert scheme of smooth curves often turns out to be reducible. projective space curves Families, moduli of curves (algebraic), Special algebraic curves and curves of low genus, Secant varieties, tensor rank, varieties of sums of powers Maximal rank of space curves in the range A | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is 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 new sharp asymptotic with the lower order term of zeroth order on \(\mathcal{Z}_{\mathbb{F}_q(t)}(\mathcal{B})\) for counting the semistable elliptic curves over \(\mathbb{F}_q(t)\) by the bounded height of discriminant \(\Delta (X)\). The precise count is acquired by considering the moduli of nonsingular semistable elliptic fibrations over \(\mathbb{P}^1\), also known as semistable elliptic surfaces, with \(12n\) nodal singular fibers and a distinguished section. We establish a bijection of \(K\)-points between the moduli functor of semistable elliptic surfaces and the stack of morphisms \(\mathcal{L}_{1,12n} \cong\Hom_n(\mathbb{P}^1, \overline{\mathcal{M}}_{1,1})\) where \(\overline{\mathcal{M}}_{1,1}\) is the Deligne-Mumford stack of stable elliptic curves and \(K\) is any field of characteristic \(\ne 2,3\). For \(\operatorname{char}(K)=0\), we show that the class of \(\Hom_n(\mathbb{P}^1,\mathcal{P}(a,b))\) in the Grothendieck ring of \(K\)-stacks, where \(\mathcal{P}(a,b)\) is a 1-dimensional \((a, b)\) weighted projective stack, is equal to \(\mathbb{L}^{(a+b)n+1}-\mathbb{L}^{(a+b)n-1}\). Consequently, we find that the motive of the moduli \(\mathcal{L}_{1,12n}\) is \(\mathbb{L}^{10n + 1}-\mathbb{L}^{10n - 1}\) and the cardinality of the set of weighted \(\mathbb{F}_q\)-points to be \(\#_q(\mathcal{L}_{1,12n}) = q^{10n + 1}-q^{10n - 1}\). In the end, we formulate an analogous heuristic on \(\mathcal{Z}_{\mathbb{Q}}(\mathcal{B})\) for counting the semistable elliptic curves over \(\mathbb{Q}\) by the bounded height of discriminant \(\Delta\) through the global fields analogy. Families, moduli of curves (algebraic), Elliptic curves, Applications of methods of algebraic \(K\)-theory in algebraic geometry, Positive characteristic ground fields in algebraic geometry, Elliptic surfaces, elliptic or Calabi-Yau fibrations Arithmetic of the moduli of semistable elliptic 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 Lax operator algebras are infinite-dimensional Lie algebras of current type which are associated to higher genus Riemann surfaces with marked points. They were introduced by Krichever and Sheinman and are examples of almost-graded Lie algebras. By \textit{M. Schlichenmaier} and \textit{O. K. Sheinman} in [Russ. Math. Surv. 63, No. 4, 727--766 (2008); translation from Usp. Mat. Nauk 63, No. 4, 131--172 (2008; Zbl 1204.17016)] almost-graded central extensions of these algebras were studied and classified. The article under review gives the definition of the Lax operator algebras associated to the finite-dimensional Lie algebras for \(g\) one of the algebras \(\text{gl}(n), \text{sl}(n), \text{so}(n), \text{sp}(2n)\). The classification results of the above mentioned article on almost-graded central extensions are reviewed. As in the classical current algebra case there is essentially only one nontrivial central extension if the finite-dimensional algebra is simple. Lax operator algebras are related to Lax operators on Riemann surfaces. In the article also \(g\)-valued Lax equations and their corresponding phase spaces are discussed. Certain properties are shown. In particular their consistency is checked. infinite-dimensional Lie algebras; current algebras; gauge algebra; conformal algebra; central extensions; integrable systems O. K. Sheinman, ''On Certain Current Algebras Related to Finite-Zone Integration,'' in Geometry, Topology, and Mathematical Physics: S.P. Novikov's Seminar 2006--2007, Ed. by V. M. Buchstaber and I. M. Krichever (Am. Math. Soc., Providence, RI, 2008), AMS Transl., Ser. 2, 224, pp. 271--284. Kac-Moody (super)algebras; extended affine Lie algebras; toroidal Lie algebras, Virasoro and related algebras, Loop groups and related constructions, group-theoretic treatment, Vector bundles on curves and their moduli, Infinite-dimensional groups and algebras motivated by physics, including Virasoro, Kac-Moody, \(W\)-algebras and other current algebras and their representations, Differentials on Riemann surfaces, Families, moduli of curves (algebraic), Families, moduli of curves (analytic), Riemann surfaces; Weierstrass points; gap sequences, Two-dimensional field theories, conformal field theories, etc. in quantum mechanics, Lie algebras of vector fields and related (super) algebras On certain current algebras related to finite-zone 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 Author's abstract: We study linear series on a general curve of genus \(g\), whose images are exceptional with regard to their secant planes. Working in the framework of an extension of Brill-Noether theory to pairs of linear series, we prove that a general curve has no linear series with exceptional secant planes, in a very precise sense, whenever the total number of series is finite. Next, we partially solve the problem of computing the number of linear series with exceptional secant planes in a one-parameter family in terms of tautological classes associated with the family, by evaluating our hypothetical formula along judiciously-chosen test families. As an application, we compute the number of linear series with exceptional secant planes on a general curve equipped with a one-dimensional family of linear series. We pay special attention to the extremal case of \(d\)-secant \((d - 2)\)-planes to (\(2d - 1\))-dimensional series, which appears in the study of Hilbert schemes of points on surfaces. In that case, our formula may be rewritten in terms of hypergeometric series, which allows us both to prove that it is nonzero and to deduce its asymptotics in \(d\). Cotterill E.: Geometry of curves with exceptional secant planes: linear series along the general curve. Math. Zeit. 267(3), 549--582 (2011) Special divisors on curves (gonality, Brill-Noether theory), Families, moduli of curves (algebraic) Geometry of curves with exceptional secant planes: Linear series along the general 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 \(U_{d,g}\to H_{d,g}\) be the universal family of connected smooth curves of \({\mathbb{P}}^ 3\) of genus \(g\) and degree \(d.\) The relative tangent bundle and the hyperplane divisor of \({\mathbb{P}}^ 3\) define on \(U_{d,g}^ a \)divisor whose (relative) degree is \(\delta =g.c.d.(2g- 2,d)\). Assume \(d\geq \frac{5}{4}g+1\), so that \(H_{d,g}\) is connected. The author proves that if d is large enough with respect to g, then the degree of every divisor on \(U_{d,g}\) is a multiple of \(\delta\). This result provides an analog for \({\mathbb{P}}^ 3\) to a theorem of Enriques and Chisini stating, in modern terms, that on the universal family of curves of \({\mathbb{P}}^{g-1}\) of genus \(g\geq 3\) and degree \(2g-2\) every divisor has degree multiple of 2g-2. degree of divisor; projective space curve; universal family of curves MESTRANO (N.) . - Degré des diviseurs sur les familles de courbes de \Bbb P3 . Math. Ann., vol. 270, 1985 , p. 461-465. MR 86h:14020 | Zbl 0612.14026 Families, moduli of curves (algebraic), Divisors, linear systems, invertible sheaves, Projective techniques in algebraic geometry Degré des diviseurs sur les familles de courbes de \({\mathbb{P}}^ 3\). (Degree of divisors on families of curves of \({\mathbb{P}}^ 3)\) | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians In this paper, we prove that the moduli space of low degree rational curves on a low-degree complete intersection passing several suitable points is a complete intersection. The proof involves some relationships among some divisors on the Kontsevich spaces, advanced techniques of deformations of rational curves, and classical projective geometry. Families, moduli of curves (algebraic), Fano varieties Spaces of low-degree rational curves on Fano complete intersections | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians We survey three results on syzygies of curves beyond Green's conjecture, with a particular emphasis on drawing connections between the study of syzygies and other topics in moduli theory. Families, moduli of curves (algebraic), Syzygies, resolutions, complexes and commutative rings, Algebraic moduli problems, moduli of vector bundles, Special divisors on curves (gonality, Brill-Noether theory), Families, moduli, classification: algebraic theory, Rational and ruled surfaces, \(K3\) surfaces and Enriques surfaces, Jacobians, Prym varieties Syzygies of curves beyond Green's conjecture | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians Main topic: Let \(H^ d\) be the Hilbert scheme of \(P^ 2_{{\mathbb{C}}}\) of length \(d\) and \(H_{\geq \phi}\) its subscheme of \(p\in H^ d\) with the Hilbert function \(h(p)\geq \phi\), \(\phi\) any function \(\phi: N\to N\). Then \(H_{\geq \phi}\) is simple connected. The proofs needs punctual Hilbert schemes. - A more or less direct corollary of this theorem is: The Hilbert scheme of curves of degree \(d\) and genus \(g\) in the projective space \(P^ 3_{{\mathbb{C}}}\) is simple connected. connectivity of Hilbert scheme of curves; punctual Hilbert schemes Parametrization (Chow and Hilbert schemes), Topological properties in algebraic geometry, Families, moduli of curves (algebraic) Einfacher Zusammenhang der Hilbertschemata von Kurven im komplex-projektiven Raum. (Simple connectedness of the Hilbert schemes of curves in complex projective 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 The author describes the connection between the compactifications of the moduli space of curves and of line bundles on curves and the moduli spaces of polyhedral objects which arises from the the recursive, combinatorial properties of the boundary of the compactifications.
In the first part, a clear picture of the recursive and combinatorial properties of the stratification of the moduli space of stable curves by stable graphs is given, mostly based on a previous work of the author together with \textit{D. Abramovich} and \textit{S. Payne}, [Ann. Sci. Éc. Norm. Supér. (4) 48, No. 4, 765--809 (2015; Zbl 1410.14049)].
In the second part, the author discusses the combinatorial structure of compactified Jacobians of a stable curve \(X\) of genus \(g,\) with a focus on compactified Jacobians of degree \(d=g\) and \(d=g-1\). For \(d=g\), the compactified Jacobian is of Néron type and thus it has a recursive structure in terms of Néron models of all the connected partial normalizations of \(X\), described by the author in [Am. J. Math. 130, No. 1, 1--47 (2008; Zbl 1155.14023)]. The author then shows that the compactified Jacobian admits another recursive graded stratification, governed by rooted orientations, which is a refinement of the first one with connected strata. On the other hand, when \(d=g-1,\) the compactified Jacobian is never of Néron type, however it is still possible to construct a stratification having similar recursive behaviour to the case \(d=g.\) moduli spaces; stable curves; stable graphs; Jacobians; Néron models; orientations Families, moduli of curves (algebraic), Fibrations, degenerations in algebraic geometry, Algebraic moduli problems, moduli of vector bundles Compactifying 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 This paper originates in part from the Habilitation mémoire (Université de Bourgogne, Dijon, 2016) of the author and in part from an introductory talk he gave at the RAQIS'16 conference held at Geneva, Switzerland, in August 2016. It deals with a novel construction that associates an integrable, tau-symmetric hierarchy and its quantization to a cohomological field theory on the moduli space of stable curves, without the semi-simplicity assumption which is needed for the Dubrovin-Zhang hierarchy. It is inspired by Eliashberg, Givental and Hofer's symplectic field theory [\textit{Y. Eliashberg} et al., in: GAFA 2000. Visions in mathematics---Towards 2000. Proceedings of a meeting, Tel Aviv, Israel, August 25--September 3, 1999. Part II. Basel: Birkhäuser. 560--673 (2000; Zbl 0989.81114)] and is the fruit of a joint project of the author with A. Buryak, B. Dubrovin and J. Guéré (see e.g. [\textit{A. Buryak}, et al. ``Tau-structure for the double ramification hierarchies'', \url{arXiv:1602.05423}; ``Integrable systems of double ramification type'', \url{arXiv:1609.04059}]).
After a self contained introduction to the language of integrable systems in the formal loop space and the needed notions from the geometry of the moduli space of stable curves the author explains the double ramification hierarchy construction and presents its main features, with an accent on the quantization procedure, concluding with a list of examples worked out in detail. This paper does not contain new results and most of this material originates from an ongoing collaboration with A. Buryak, B. Dubrovin and J. Guéré. It is however a complete reorganization and, in part, a rephrasing of those results with the aim of showcasing the power of these methods and making them more accessible to the mathematical physics community. moduli space of stable curves; integrable systems; cohomological field theories; double ramification cycle; double ramification hierarchy Rossi, P., Integrability, quantization and moduli spaces of curves, SIGMA, 13, (2017) Families, moduli of curves (algebraic), 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.) Integrability, quantization and 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 For any complex projective variety \(Y\) and any class \(\beta\in H_2(Y,\mathbb Z)\), consider the moduli space \(M_g^{-}(Y,\beta)\) of all stable maps \(f:C\to Y\) with \(C\) a reduced connected nodal curve of arithmetic genus \(g\) and \(f_*([C])=\beta\). The expected dimension of the algebraic stack \(M_g^{-}(Y,\beta)\) is \(\text{dim}(Y)(1-g)+3g-3-b\cdot\omega_Y\). In this paper under review, the author focuses in the case when \(Y=\mathbb P^1\times\mathbb P^r\) and he looks for irreducible components \(V\) of \(M_g^{-}(\mathbb P^1\times\mathbb P^r,\beta)\) which are {\textit{good}}, that is, such that \(V\) is generically smooth and with the expected dimension. For all integers \(g,r,d\), set \(\rho(g,r,d):=g-(r+1)(g-d+r)\)(the so-called Brill-Noether number). The main theorem of this paper is the following:
\noindent Fix positive integers \(g,r,d,k\) such that \(\frac{g+2}{2}\geq k\geq r+3\geq 6, \rho(g,r,d)\geq 0\), and \(g\leq(r+1)\lfloor d/r \rfloor-r-3\). Then there exists a good component of \(M_g^{-}(\mathbb P^1\times\mathbb P^r,(k,d))\).
As a byproduct, the author obtain the existence of a generically smooth component of dimension \(\rho(g,r,d)\) for the Brill-Noether variety \(W^r_d(C)\) of a general \(k\)-gonal curve of genus \(g\). Line bundle; Brill-Noether theory; moduli space of curves; stable maps; moduli space of stable maps Families, moduli of curves (algebraic), Special divisors on curves (gonality, Brill-Noether theory), Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects) Regular components of moduli spaces of stable maps and \(k\)-gonal curves | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians Let \(X_\lambda\) and \(X_\lambda^\prime\) be monomial deformations of two Delsarte hypersurfaces in weighted projective spaces. In this paper we give a sufficient condition so that their zeta functions have a common factor. This generalises results by Doran, Kelly, Salerno, Sperber, Voight and Whitcher [\textit{C. F. Doran} et al., Isr. J. Math. 228, No. 2, 665--705 (2018; Zbl 1403.14055), see also \url{arxiv:1612.09249}], where they showed this for a particular monomial deformation of a Calabi-Yau invertible polynomial. It turns out that our factor can be of higher degree than the factor found in the paper (loc. cit.). monomial deformation of Delsarte surfaces; zeta functions Zeta functions and related questions in algebraic geometry (e.g., Birch-Swinnerton-Dyer conjecture), Varieties over finite and local fields, Picard groups, \(K3\) surfaces and Enriques surfaces, Hypersurfaces and algebraic geometry, Computational aspects of algebraic surfaces Zeta functions of monomial deformations of Delsarte 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 Let \(M_ g\) be the moduli space of curves of genus g. If \(A_ g\) is the space of principally polarized d-dimensional abelian varieties then there is a natural morphism \(j: M_ g\to A_ g\) defined by sending a curve to its jacobian. It is a classical problem to give a set of equations for the closure \(J_ g\) of the image of the morphism j. When \(g=4\) this problem was solved by Igusa utilizing a result of Schottky. In this paper the author extends Schottky's result to \(g>4\), and in so doing makes a major contribution toward solving the general problem. To be more precise, let \(\bar A_ g\) denote the Satake compactification of \(A_ g\), and let \(\bar J_ g\) denote the closure of \(J_ g\) in \(\bar A_ g\). The author defines an ideal \(S(\Gamma_ g)\) of the graded ring \(A(\Gamma_ g)\) generated by Siegel modular forms on \(\Gamma_ g=Sp(2g,{\mathbb{Z}})\). He then carries out an induction argument that shows that \(S(\Gamma_ g)\) cuts out a subset of \(\bar A_ g\) containing \(\bar J_ g\) as an irreducible component. Torelli theorem; Schottky problem; theta constants; moduli space of curves; Satake compactification; Siegel modular forms B. van Geemen,Siegel modular forms vanishing on the moduli space of curves, Inv. Math.78 (1984), 329--349. Families, moduli of curves (analytic), Families, moduli of curves (algebraic), Theta functions and abelian varieties, Algebraic moduli problems, moduli of vector bundles, Jacobians, Prym varieties Siegel modular forms vanishing on the moduli space of curves | 0 |
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