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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 Over \(\mathcal{M}_{g,n}\), the moduli space of genus \(g\) curves with \(n\) marked points, let \(\mathcal{J}^d_{g,n}\) denote the universal Jacobian of degree \(d\) line bundles. In the article under review, the authors study the question of extending the moduli space \(\mathcal{J}^d_{g,n}\) to a family over \(\overline{\mathcal{M}}_{g,n}\), the Deligne-Mumford moduli space of stable \(n\)-pointed genus \(g\) curves.
In addressing this question, the authors extend earlier work of \textit{T. Oda} and \textit{C. S. Seshadri} [Trans. Am. Math. Soc. 253, 1--90 (1979; Zbl 0418.14019)]. The idea is to first construct an affine space of stability conditions for the universal stable pointed curve.
Among other results, the authors determine the structure of such stability spaces \(V^d_{g,n}\). For example, they prove that if \(g \geq 2\), \(n \geq 1\) and \(N = N(g,n)\) is the number of boundary divisors of \(\overline{\mathcal{M}}_{g,n}\), then the stability space \(V^{g-1}_{g,n}\) is isomorphic to \(\mathbb{R}^{N-1}\times \mathbb{R}^n\) as an affine space. They also prove that, under this isomorphism, the decomposition of \(V^{g-1}_{g,n}\) into stability polytopes is the product decomposition of \(\mathbb{R}^{N-1}\) and \(\mathbb{R}^n\) into integer translates of finitely many hyperplanes.
In terms of the extent to which the compactified Jacobians \(\mathcal{J}_{g,n}(\phi)\) depend on the given stability parameter \(\phi\), the authors prove that if \(\overline{\mathcal{M}}_{g,n}\) is of general type, then there exist nondegenerate stability parameters \(\phi_1\) and \(\phi_2\) for which the compactified Jacobians \(\mathcal{J}_{g,n}(\phi_1)\) and \(\mathcal{J}_{g,n}(\phi_2)\) are non-isomorphic as Deligne-Mumford stacks.
Finally, given a nondegenerate stability parameter \(\phi \in V^d_{g,n}\), together with an integer vector \((k;d_1,\dots,d_n)\) which satisfies the condition that \[ k(2-2g)+d_1+\dots+d_n = d, \] the authors describe the locus of indeterminacy of the corresponding rational map \[ \sigma_{k,\mathbf{d}} : \overline{\mathcal{M}}_{g,n} \dashrightarrow \overline{\mathcal{J}}_{g,n}(\phi) \text{.} \] In doing so, they address a question which was raised by \textit{S. Grushevsky} and \textit{D. Zakharov} [Duke Math. J. 163, No. 5, 953--982 (2014; Zbl 1302.14039)]. compactified Jacobian; universal Jacobian; stability polytopes Families, moduli of curves (algebraic), Jacobians, Prym varieties, Algebraic moduli of abelian varieties, classification, Picard groups The stability space of compactified universal Jacobians | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians Let \(L\) be a line bundle of positive degree on a smooth projective curve \(C\) of genus \(g\). Let \({\mathcal M}_{2,L}\) denote the moduli space of semistable rank-2 vector bundles on \(C\) with determinant \(L\). The extension map \(\phi_ L:\mathbb{P}_ L = P(\text{Ext}^ 1(L,{\mathcal O}_ C)^*) \to {\mathcal M}_{2,L}\) is the rational map defined by associating to an extension \(0\to {\mathcal O}_ C \to E \to L\to 0\) the vector bundle \(E\). The main theorem of the paper gives explicitly a sequence of blow- ups \(\sigma: \widetilde{\mathbb{P}}_ L \to {\mathbb{P}}_ L\) which resolves the extension map \(\phi_ L\) into a morphism \(\widetilde{\phi}_ L = \phi_ L\circ \sigma\).
The main application of the theorem concerns the spaces \(H^ 0({\mathcal M}_{2,L},{\mathcal O}(k\theta))\) where \(\theta\) denotes the theta divisor of \({\mathcal M}_{2,L}\), namely if \(\text{deg }L = 2g\) then there is a natural identification
\[
H^ 0({\mathcal M}_{2,L},{\mathcal O}(k\theta)) = H^ 0(\mathbb{P}_ L,{\mathcal O}(kg) \otimes J_ C^{k(g-1)})
\]
, where \(J_ C\) denotes the ideal of \(C\) in \(\mathbb{P}_ L\). Similarly for \(\text{deg }L = 2g - 1\). Other applications are a new simple proof that \(\text{Pic}({\mathcal M}_{2,L}) \simeq \mathbb{Z}\), that the class of \(\theta\) in \(\text{Pic}({\mathcal M}_{2,\omega_ C})\) is irreducible and moreover, that if \(C\) is not hyperelliptic, then \(\theta \subset {\mathcal M}_{2,L}\) is birationally very ample for odd \(\text{deg }L\). line bundle on a smooth projective curve; moduli space of semistable rank-2 vector bundles; theta divisor A. Bertram: Moduli of rank \(2\) vector bundles, theta divisors, and the geometry of curves in projective space, J. Diff. Geom. 35, 1992, 429-469. Families, moduli of curves (algebraic), Theta functions and abelian varieties, Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20], Picard groups, Vector bundles on curves and their moduli Moduli of rank-2 vector bundles, theta divisors, and the geometry of curves in 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 Answering a question of \textit{V. Balaji} and \textit{P. A. Vishwanath} [Duke Math. J. 76, 773--792 (1994; Zbl 0844.14005)] it is proved that the Picard bundle on the moduli space of stable vector bundles of rank two, on a Riemann surface of genus at least three, with fixed determinant of odd degree is stable. DOI: 10.1007/BF02829596 Picard groups, Vector bundles on curves and their moduli, Families, moduli of curves (algebraic) Stability of Picard bundle over moduli space of stable vector bundles of rank two over a curve | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians We study the stack \(\mathcal B_{h,g,n}\) of uniform cyclic covers of degree \(n\) between smooth curves of genus \(h\) and \(g\) and, for \(h\gg g\), present it as an open substack of a vector bundle over the universal Jacobian stack of \(\mathcal M_g\). We use this description to compute the integral Picard group of \(\mathcal B_{h,g,n}\), showing that it is generated by tautological classes of \(\mathcal B_{h,g,n}\). Poma, Flavia; Talpo, Mattia; Tonini, Fabio: Stacks of uniform cyclic covers of curves and their Picard groups. Algebr. geom. 2, No. 1, 91-122 (2015) Stacks and moduli problems, Picard groups, Families, moduli of curves (algebraic) Stacks of uniform cyclic covers of curves and their 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 [For the entire collection see Zbl 0611.00006.]
The present paper is an account of recent results due to E. Arbarello, M. Cornalba, J. Harris and D. Mumford on the Picard group of the functors \(M_{g,h}\) and \(\bar M_{g,h}\); here \(M_{g,h}\) is the functor of smooth curves of genus \(g\) with h punctures, while \(\bar M_{g,h}\) is the corresponding functor of stable curves. The union of these results provides an explicit description of a basis for the (torsion free) group \(Pic(M_{g,h})\) and \(Pic(\bar M_{g,h})\). moduli of algebraic curves; Picard group E. Arbarello and M. Cornalba, The Picard groups of the moduli spaces of curves , preprint, Università di Pavia, 1985. Picard groups, Families, moduli of curves (algebraic) On the Picard group of the moduli space of algebraic 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 every prime \(p\) let denote by \(M_g^-(p)\) the moduli scheme of stable curves of genus \(g\) over an algebraically closed field of characteristic \(p\). In this paper it is proved the following theorem:
Fix an integer \(g\geq 5\) and a prime \(p> 84(g-1)\), then the Néron-Severi group \(\text{NS} (M_g^-(p))\) of \(M_g^-(p)\) is freely generated by the \([g/2]+2\) classes \(\lambda\) and \(\Delta_i\), \(0\leq i\leq [g/2]\).
The proof uses a reduction to the characteristic 0 case (i.e. to the Harer's topological theorem) and a result by \textit{M. Pikaart} and \textit{A. J. de Jong} [The moduli space of curves, Proc. Conf. Texel Island 1994, Prog. Math. 129, 483-509 (1995; Zbl 0860.14024)]. moduli scheme of stable curves; characteristic \(p\); Néron-Severi group Families, moduli of curves (algebraic), Picard groups, Finite ground fields in algebraic geometry On the Picard group of the moduli scheme of stable curves in positive characteristic | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians We study the fixed singularities imposed on members of a linear system of surfaces in \(\mathbb P^3_{\mathbb C}\) by its base locus \(Z\). For a 1-dimensional subscheme \(Z \subset \mathbb P^3\) with finitely many points \(p_i\) of embedding dimension three and \(d \gg 0\), we determine the nature of the singularities \(p_i \in S\) for general \(S \in |H^0 (\mathbb P^3, I_Z (d))|\) and give a method to compute the kernel of the restriction map from \(\mathrm{Cl}S \to \mathrm{Cl}\mathcal O_{S,p_{i}}\). One tool developed is an algorithm to identify the type of an \(\mathbf A_n\) singularity via its local equation. We illustrate the method for representative \(Z\) and use Noether-Lefschetz theory to compute \(\mathrm{Pic}S\). Brevik, J.; Nollet, S., Picard groups of normal surfaces, J. singul., 4, 154-170, (2012) Picard groups, Deformations of singularities, Families, moduli of curves (algebraic), Plane and space curves Picard groups of normal 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 \(M_{g,h}\) (\({\mathcal M}_{g,h})\) denote the moduli space (functor) of smooth h-pointed curves of genus \(g\) over \({\mathbb{C}}\), and \(\bar M_{g,h}\) (\(\bar {\mathcal M}_{g,h})\) its natural compactification by means of stable curves. - It is known that the Picard group of \(M_{g,h}\), \(Pic(M_{g,h})\), is a free abelian group on \(h+1\) generators when \(g\geq 3\) [\textit{J. Harer}, Invent. Math. 72, 221-239 (1983; Zbl 0533.57003) and ``The cohomology of the moduli space of curves'', C.I.M.E. notes (Montecatini (1985)], and it is observed by \textit{D. Mumford} [Enseign. Math., II. Sér. 23, 39-110 (1977; Zbl 0363.14003)] that the Picard group of \({\mathcal M}_{g,h}\), Pic(\({\mathcal M}_{g,h})\), has no torsion and contains \(Pic(M_{g,h})\) as a subgroup of finite index (a proof of the latter is sketched in the appendix).
Explicit bases for Pic(\({\mathcal M}_{g,h})\) and Pic(\(\bar {\mathcal M}_{g,h})\), which is also free, are exhibited: Theorem \((h=0):\) For any \(g\geq 3\), Pic(\(\bar {\mathcal M}_{g,0})\) is freely generated by \(\lambda,\delta_ 0,\delta_ 1,...,\delta_{[g/2]}\); while Pic(\({\mathcal M}_{g,0})\) is freely generated by \(\lambda\). Here \(\lambda,\delta_ 0,\delta_ 1,...,\delta_{[g/2]}\) denote, respectively, the Hodge class, and the boundary classes. - Theorem \((h>0).\) For every \(g\geq 3\), Pic(\(\bar {\mathcal M}_{g,h})\) is freely generated by \(\lambda,\psi_ 1,...,\psi_ h\) and \(\delta_ 0,\delta_{\alpha;i_ 1,...,i_ a}\) (0\(\leq \alpha \leq [g/2]\), \(0\leq a\leq h\) with \(\alpha\geq 2\) if \(a=0\), \(i_ 1<...<i_ a)\); while Pic(\({\mathcal M}_{g,h})\) is freely generated by \(\lambda,\psi_ 1,...,\psi_ h\). Here the \(\delta\) 's are the boundary classes in Pic(\(\bar {\mathcal M}_{g,h})\) and the \(\psi\) 's are the classes in Pic(\(\bar {\mathcal M}_{g,h})\) defined for a family \(F=(\pi: X\to S,\;\sigma_ 1,...,\sigma_ h)\) of h-pointed stable curves of genus \(g\) by setting \((\psi_ i)_ F=\sigma^*_ i(\omega_{\pi})\), \(i=1,...,h\), where \(\omega_{\pi}\) is the relative dualizing sheaf.
The proof of the theorem for \(h=0\) proceeds as follows: One knows that \(\lambda\) and \(\delta\) 's are linearly independent and that any class \(\xi\) in Pic(\(\bar {\mathcal M}_{g,0})\) is a linear combination of \(\lambda\) and \(\delta\) 's with coefficients in \({\mathbb{Q}}\), i.e., \(\xi =a\lambda +\sum b_ i\delta_ i.\) The proof is completed if one can show that \(a,b_ i\in {\mathbb{Z}}\), and this is done by constructing two different sets of families of \(k+2\) stable curves \(G_ 1,...,G_{k+2}\) \((k=[g/2])\) with the property that the corresponding values of the determinant \(\det(\eta)\) of the matrix
\[
\eta = \eta(G_ 1,...,G_{k+2})= \begin{pmatrix} \deg_{G_ 1}\lambda & \deg_{G_ 1}\delta_ 1 & ... & \deg_{G_ 1}\delta_ k \\ \vdots &&&\vdots \\ \deg_{G_{k+2}}\lambda & ... & ... & \deg_{G_{k+2}}\delta_ k \end{pmatrix}
\]
are relatively prime.
The proof of the theorem for \(h>0\) proceeds as follows: First it is proved on the basis of Harer's theorem that the classes \(\lambda,\psi_ 1,...,\psi_ h\), \(\delta_ 0,\delta_{\alpha;i_ 1,...,i_ a}\) form a basis of Pic(\(\bar {\mathcal M}_{g,h})\otimes {\mathbb{Q}}\), and that the classes \(\lambda,\psi_ 1,...,\psi_ h\) form a basis of \(Pic({\mathcal M}_{g,h})\otimes {\mathbb{Q}}\). Then the proof is completed by the following proposition and on induction on h: Let L be a line bundle on \(\bar {\mathcal M}_{g,h+1}\). If L is trivial on smooth curves there exists a line bundle \({\mathcal L}\) on \(\bar {\mathcal M}_{g,h}\) such that \(cl(L)\equiv \vartheta (cl({\mathcal L}))\) modulo boundary classes. Conversely, if there is \({\mathcal L}\) on \(\bar {\mathcal M}_{g,h}\) such that cl(L)- \(\vartheta(cl({\mathcal L}))\) is an integral linear combination of boundary classes other than the \(\delta_{0;i,h+1}\), then L is trivial on smooth curves.
As an application of the second theorem, a conjecture of Franchetta is proved: Let \((M_{g,h})^ 0\) be the open subset of \(M_{g,h}\) consisting of all genus g\ h-pointed curves without non-trivial automorphisms. Let \({\mathcal C}\to (M_{g,0})^ 0\) be the universal family of genus g curves, S a Zariski open subset of \((M_{g,0})^ 0\) and \(\pi: X\to S\) the restriction of the universal family to S. Then for any line bundle L on X, the restriction of L to any fiber of \(\pi\) is an integral multiple of the canonical bundle. Finally, a result on \(Pic(\bar M_{g,h})\) is proved: If \(g\geq 3\), \(A_{3g+h-4}(\bar M_{g,h})\) is the index-two subgroup of \(Pic(\bar {\mathcal M}_{g,h})\) generated by \(\psi_ 1,...,\psi_ h,2\lambda,\lambda +\delta_ 1\), and the boundary classes different from \(\delta_ 1\). moduli space of curves of genus g; Picard group Enrico Arbarello and Maurizio Cornalba. The {P}icard groups of the moduli spaces of curves. {\em Topology}, 26(2):153--171, 1987 Families, moduli of curves (algebraic), Picard groups The Picard groups of the moduli spaces of curves | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians Let \(G\) be a simple simply-connected algebraic group and \(C\) be a smooth irreducible projective curve over \(\mathbb{C}\) with \(g\geq 2\) and \(M\) be the moduli space of \(G\)-bundles on \(C\). The main result of the paper is: \(\text{Pic} M =\mathbb{Z}\). Also it is proven that \(M\) is a Gorenstein variety and that \(H^i(M,\Theta (V))=0\) for \(i\geq 1\) where \(\Theta (V)\) is the theta bundle on \(M\) made by a finite dimensional representation \(V\) of \(G\). The key ingredient of the proof is to study the morphism \(\psi: X^s\to M\) from an open subset in the generalized flag variety \(X\) of the Kac-Moody group associated to \(G\) [\(\psi\) was defined in an earlier paper: \textit{S. Kumar}, \textit{M. S. Narasimhan} and \textit{A. Ramanathan}, Math. Ann. 300, No. 1, 41-75 (1994; Zbl 0803.14012)] and to prove that the lifting of line bundles can be extended to a map \(\overline {\psi^*}: \text{Pic} M\to \text{Pic} X\). moduli space; Gorenstein variety; theta bundle; Kac-Moody group Shrawan Kumar and M. S. Narasimhan, Picard group of the moduli spaces of \?-bundles, Math. Ann. 308 (1997), no. 1, 155 -- 173. Picard groups, Group actions on varieties or schemes (quotients), Kac-Moody (super)algebras; extended affine Lie algebras; toroidal Lie algebras, Families, moduli of curves (algebraic), Theta functions and abelian varieties Picard group of the moduli spaces of \(G\)-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 For a proper and vertical logarithmic curve \(X\) over a logarithmic base scheme \(S\), the authors introduce a logarithmic Picard group \(\mathrm{LogPic}(X/S)\) and a tropical Picard group \(\mathrm{TroPic}(\mathcal{X}/S)\), which are both -- depending on the version considered -- sheaves or stacks on an appropriate site of log schemes over \(S\). The relation between the two is an appealing example of the interplay between logarithmic geometry and tropical geometry.
Given \(X/S\), first they introduce the \textit{tropicalization} \(\mathcal{X}/S\), which is a family of tropical curves varying over the logarithmic base \(S\) -- an idea occurring already in the earlier work [\textit{R. Cavalieri} et al., Forum Math. Sigma 8, Paper No. e23, 93 p. (2020; Zbl 1444.14005)]. The family \(\mathcal{X}/S\) has, for every geometric point \(\bar s \in S\), a tropical curve \(\mathcal{X}_{\bar s}\) metrized by \(\overline{\mathcal{M}}_{S,\bar s}\), compatible with generization maps. Each \(\mathcal{X}_{\bar s}\) is actually a site with a structure group \(L_{\bar s}\). Then \(\mathbf{TroPic}(\mathcal{X}/S)\) is a stack whose objects are compatible families of \(L_{\bar s}\)-torsors satisfying an important technical condition called \textit{bounded monodromy}. \(\mathrm{TroPic}(\mathcal{X}/S)\) is the associated sheaf of isomorphism classes.
If \(S\) is an algebraically closed field, then the bounded monodromy condition also makes sense for \(\mathcal{M}_X^{gp}\)-torsors. In general, a \textit{logarithmic line bundle} is an \(\mathcal{M}_X^{gp}\)-torsor whose fibers over geometric points have bounded monodromy. Then \(\mathbf{LogPic}(X/S)\) is the stack whose objects are logarithmic line bundles, and \(\mathrm{LogPic}(X/S)\) is the associated sheaf.
While neither the study of \(\mathcal{M}_X^{gp}\)-torsors nor the bounded monodromy condition is new, the study of the stack \(\mathbf{LogPic}(X/S)\) as a \textit{(log-)geometric object} is: The authors show that there is a log scheme \(U\) together with a morphism \(U \to \mathbf{LogPic}(X/S)\) which is representable by logarithmic (algebraic) spaces (with their definition of them), log smooth, and universally surjective. The diagonal of \(\mathbf{LogPic}(X/S)\) is representable by logarithmic spaces. They show furthermore that \(\mathbf{LogPic}(X/S)\) is log smooth; they introduce a degree decomposition into \(\mathbf{LogPic}^d(X/S)\) and show that each component \(\mathbf{LogPic}^d(X/S)\) is proper.
These results show that \(\mathbf{LogPic}(X/S)\) is a logarithmic stack in a sense that goes beyond the standard definition. So far, a log stack has either been an algebraic stack endowed with a log structure, i.e., a sheaf of monoids, or a stack over a category of log schemes but which admits a \emph{strict} log smooth morphism from a log scheme. Under rather mild assumptions, these two definitions are equivalent, see [\textit{M. C. Olsson}, Ann. Sci. Éc. Norm. Supér. (4) 36, No. 5, 747--791 (2003; Zbl 1069.14022)] and [\textit{J. Shentu} and \textit{D. Wang}, Int. J. Math. 28, No. 10, Article ID 1792002, 7 p. (2017; Zbl 1410.14004)] with its erratum, both under slightly different hypotheses than the article under review. However, the map \(U \to \mathbf{LogPic}(X/S)\) is log smooth but not strict, and, as the authors point out, \(\mathbf{LogPic}(X/S)\) cannot be represented by an algebraic stack with a log structure.
Every logarithmic line bundle can be tropicalized; this yields an exact sequence \[0 \rightarrow \mathbf{Pic}^{[0]}(X/S) \rightarrow \mathbf{LogPic}(X/S) \rightarrow \mathbf{TroPic}(X/S) \rightarrow 0\] of commutative group stacks. The term on the left is the multidegree \(0\)-component of the Picard group. For unmarked stable curves -- which can always be enhanced to a proper vertical logarithmic curve -- this object is well-behaved in families and proper for smooth and slightly degenerate curves but only separated in more degenerate cases. Many compactifications of this object have been attempted in the literature but none of them is a proper group scheme -- this is not possible. With \(\mathbf{LogPic}(X/S)\), the authors offer a compactification as a proper group stack over logarithmic schemes.
The authors also show that the \textit{logarithmic Jacobian} \(\mathrm{LogPic}^0(X/S)\) is a logarithmic abelian variety in the sense of Kajiwara-Kato-Nakayama. logarithmic geometry; tropical geometry; Picard group; Jacobian; algebraic curves Logarithmic algebraic geometry, log schemes, Picard groups, Jacobians, Prym varieties, Foundations of tropical geometry and relations with algebra, Divisors, linear systems, invertible sheaves, Algebraic moduli problems, moduli of vector bundles, Stacks and moduli problems, Families, moduli of curves (algebraic), Picard schemes, higher Jacobians The logarithmic Picard group and its tropicalization | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is 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 classification of compact, complex surfaces amounts to the classification of minimal surfaces. This is first of all a classification according to Kodaira dimension. A refinement of this very coarse classification is the Enriques-Kodaira classification, a description of which is a first purpose of this book. Apart from the Enriques-Kodaira classification, this book is mainly devoted to a deeper study of some of these classes, namely K3-surfaces, Enriques surfaces and surfaces of general type.
The content of the different chapters is as follows.
In chapter I we collect - practically without proofs - most of the definitions and results from topology, algebra, differential geometry, analytic geometry and algebraic geometry which we shall need.
Chapter II is devoted to (possibly non-reduced) curves on (not necessarily compact) surfaces: dualising sheaf, Picard variety, singularities and their resolution. The Riemann-Roch theorem is reduced to the smooth case and Serre duality derived from the reduced projective case. Simple curve singularities are classified. Analytic intersection numbers are defined for divisors and shown to be the same as the topological ones.
The first part of chapter III deals with surface singularities, their resolution and the converse of this process, the blowing down of exceptional curves. The results are applied to study bimeromorphic maps and minimal models. Rational double points and their relations with simple curve singularities are treated with care. The second part of chapter III is devoted to (proper) curve fibrations of surfaces over curves. The main achievement here is a proof of Iitaka's conjecture \(C_{2,1}\) about the Kodaira dimension of such fibrations. We base it on properties of the period map for stable curves, the Satake compactification and the Torelli theorem for curves.
Chapter IV is not very homogeneous. We have collected in this chapter several general theorems about surfaces which will play an important role later on in the book. The first sections deal with special features of the transcendental theory (differential forms) on compact surfaces. The main point is that for a compact surface the Fröhlicher spectral sequence always degenerates. Combining the consequences of this fact with the topological index theorem we find, following Kodaira, relations between topological and analytic invariants which are crucial in handling non-algebraic surfaces. We also prove the important signature theorem (known as algebraic index theorem in the case of algebraic surfaces). From the other subjects treated in this chapter we mention projectivity criteria (with an application to almost-complex surfaces without any complex structure) and the vanishing theorems of Ramanujam and Mumford. As to chapter V (examples), we have included this chapter as a preparation for the next one.
In chapter VI we present the Enriques-Kodaira classification. At the end we apply classification to deformations of surfaces.
Chapter VII is about surfaces of general type.
Chapter VIII deals with K3-surfaces and Enriques surfaces. We fully prove the Torelli theorem for marked K3-surfaces, the surjectivity of the period map for K3-surfaces, and the bijectivity of the period map for Enriques surfaces. curve singularities; surface singularities; Iitaka's conjecture; Kodaira dimension; differential forms; vanishing theorems; Enriques-Kodaira classification; surfaces of general type; K3-surfaces; Enriques surfaces; Torelli theorem for marked K3-surfaces W. Barth, C. Peters and A. Van de Ven, \textit{Compact complex surfaces}, Springer, Germany (1984). Families, moduli, classification: algebraic theory, Research exposition (monographs, survey articles) pertaining to algebraic geometry, Research exposition (monographs, survey articles) pertaining to several complex variables and analytic spaces, Moduli, classification: analytic theory; relations with modular forms, Compact complex surfaces, Transcendental methods of algebraic geometry (complex-analytic aspects), Picard groups, Singularities of surfaces or higher-dimensional varieties, Families, moduli of curves (algebraic), Algebraic moduli problems, moduli of vector bundles, Fine and coarse moduli spaces Compact complex 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 In this paper we describe the Picard group of the variety \({\mathcal U} (r,d)\) which parametrizes semistable vector bundles of rank \(r\) and degree \(d\) on the fibers of the universal curve \({\mathcal C}_g\). The bundle \({\mathcal U} (r,d)\) lies over the moduli space \({\mathcal M}^0_g\) of smooth curves of genus \(g\) \((g \geq 3)\) without automorphisms. universal curve; Picard group of semistable vector bundles; moduli space Kouvidakis, A., On the moduli space of vector bundles on the fibers of the universal curve, J. Differential Geom., 37, 3, 505-522, (1993) Algebraic moduli problems, moduli of vector bundles, Families, moduli of curves (algebraic), Picard groups, Vector bundles on curves and their moduli On the moduli space of vector bundles on the fibers of the universal 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 \(X\) be a connected projective nodal curve of genus \(g \geq 3\) over an algebraically closed field \(k\). If \(Z\) is a proper subcurve of \(X\), one denotes by \(k_Z\) the number of intersection points of \(Z\) with its complement \(\overline{X\setminus Z}\). Recall that \(X\) is called \textit{stable} (resp., \textit{semistable}) if, for every smooth rational component \(E\) of \(X\) one has \(k_E \geq 3\) (resp., \(k_E \geq 2\)). If \(X\) is semistable and \(k_E = 2\) then \(E\) is called an \textit{exceptional component} of \(X\). A line bundle \(L\) of degree \(d\) on a semistable curve \(X\) is called \textit{balanced} if it satisfies the following inequalities [considered by \textit{D. Gieseker}, Lectures on moduli of curves (Lectures on Mathematics and Physics. Mathematics, 69. Tata Institute of Fundamental Research, Bombay. Springer Verlag, Berlin-Heidelberg-New York) (1982; Zbl 0534.14012)]:
\[
\frac{\text{deg}({\omega}_X\, | \, Z)}{2g-2}d - \frac{k_Z}{2}\;\leq \;\text{deg}(L\, | \, Z)\;\leq \;\frac{\text{deg}({\omega}_X\, | \, Z)}{2g-2}d + \frac{k_Z}{2}
\]
for every connected proper subcurve \(Z\) of \(X\) and if \(\text{deg}(L\, | \, E) = 1\) for every exceptional component \(E\) of \(X\). If \(X\) admits a balanced line bundle then it must be \textit{quasistable} in the sense that any two exceptional components of it never meet.
\textit{L. Caporaso} [J. Am. Math. Soc. 7, No. 3, 589--660 (1994; Zbl 0827.14014)] constructed a compactification \(\overline{P}_{d,g} \rightarrow \overline{M}_g\) of the universal Picard variety \(\text{Pic}^d_g \rightarrow M^0_g\), where \(\overline{M}_g\) is the moduli scheme of stable curves of genus \(g\) and \(M^0_g \subset \overline{M}_g\) is the locus of automorphism-free nonsingular curves. More precisely, for \(d \gg 0\), \(\overline{P}_{d,g}\) is the GIT-quotient \(H_d/\text{PGL}(r+1)\), where \(H_d\) is the locus of GIT-semistable points of \(\text{Hilb}^{dt-g+1}_{{\mathbb P}^r}\), \(r = d-g\). The points of \(H_d\) correspond to nondegenerate quasistable curves in \({\mathbb P}^r\) embedded by a balanced line bundle.
Let \(\overline{\mathcal G}_{d,g}\) be the category whose objects are pairs \((f:{\mathcal C} \rightarrow Y,{\mathcal L})\), where \(f\) is a family of quasistable curves of genus \(g\) and \(\mathcal L\) a balanced line bundle of relative degree \(d\) over \(Y\). A morphism between two such pairs is a cartesian diagram:
\[
\begin{tikzcd} \mathcal C \rar["h"]\dar & \mathcal C^\prime\dar \\Y \rar & Y^\prime \end{tikzcd}
\]
toghether with an isomorphism \({\mathcal L} \simeq h^{\ast}{\mathcal L}^{\prime}\). In the paper under review, the author shows that, for \(d >> 0\), \(\overline{\mathcal G}_{d,g}\) is isomorphic to the quotient stack \([H_d/\text{GL}(r+1)]\), hence it is an algebraic (Artin) stack. She also shows that the quotient stack \(\overline{\mathcal P}_{d,g} := [H_d/\text{PGL}(r+1)]\) is a stackification of the prestack whose sections over a scheme \(Y\) are given by pairs \((f:{\mathcal C} \rightarrow Y, {\mathcal L})\) as above and whose arrows between two such pairs are cartesian diagrams as above together with an isomorphism \({\mathcal L} \simeq h^{\ast}{\mathcal L}^{\prime} \otimes f^{\ast}M\), for some \(M \in \text{Pic}\, Y\).
Moreover, let \(U_d\) denote the open subset of \(H_d\) corresponding to stable curves \(X\) which are \(d\)-\textit{general} in the sense that for any balanced line bundle of degree \(d\) on \(X\) the first inequality of Gieseker is strict for all \(Z \subset X\). The author shows that the quotient stack \(\overline{\mathcal P}^{\text{Nér}}_{d,g} := [U_d/\text{PGL}(r+1)]\) is a Deligne-Mumford stack and it is strongly representable over the open set \(\overline{\mathcal M}^d_g\) of the moduli stack \(\overline{\mathcal M}_g\) corresponding to \(d\)-general stable curves. She also gives a geometric description of the set \({\Sigma}^d_g \subset \overline{M}_g\) of \(d\)-special stable curves. These last results generalize some results of \textit{L. Caporaso} [Am. J. Math. 130, No. 1, 1--47 (2008; Zbl 1155.14023)] who considered the case \(\text{gcd}(d-g+1,2g-2) = 1\) which implies \(U_d = H_d\). stable curve; moduli stack; Picard variety Melo, M., Compactified Picard stacks over \(\overline{\mathcal{M}}_g\), Math. Z., 263, 4, 939-957, (2009) Families, moduli of curves (algebraic), Stacks and moduli problems, Picard groups Compactified Picard stacks over \({\overline{\mathcal M}_g}\) | 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 In a previous paper [Invent. Math. 72, 221-239 (1983; Zbl 0533.57003)], the author computed the second homology group, and as a consequence the Picard group, of the moduli space \(M_ g\) of Riemann surfaces of genus \(g\). In this paper, he decides the homology groups for the moduli space \(M_ g[\varepsilon]\) of Riemann surfaces with spin structures, and similarly as a consequence he shows that the rational Picard group has rank 1 for \(g \geq 9\). This is a continuation of a previous paper [Math. Ann. 287, No. 2, 323-334 (1990; Zbl 0715.57004)]. moduli space of curves; Picard group; Riemann surfaces; spin structures John L. Harer, The rational Picard group of the moduli space of Riemann surfaces with spin structure, Mapping class groups and moduli spaces of Riemann surfaces (Göttingen, 1991/Seattle, WA, 1991) Contemp. Math., vol. 150, Amer. Math. Soc., Providence, RI, 1993, pp. 107 -- 136. Families, moduli of curves (algebraic), Picard groups, Topology of Euclidean 2-space, 2-manifolds, Specialized structures on manifolds (spin manifolds, framed manifolds, etc.) The rational Picard group of the moduli space of Riemann surfaces with spin structure | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is 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 geometry of the space \(W^ r_{n,d}=\{E \in U(n,d) \mid h^ 0(X,E) \geq r+1\}\) where \(U(n,d)\) is the moduli space of stable bundles of rank \(n\) and degree \(d\) on a smooth complex projective curve \(X\).
In the first part of the paper, there is a review of the background material in the theory of determinantal varieties, in the second part, a clear recall of the construction of \(U(n,d)\), in section 3, the author defines the analytic structure of \(W^ r_{n,d}\), and in section 4 he gives the description of the ideal in \(H^ 1(X,{\mathcal E}nd(E))\) of the tangent space \(T_ E(W^ r_{n,d})\) of \(W^ r_{n,d}\) at a point \(E\), computes its degree and gives a set theoretical description of \(T_ E(W^ r_{n,d})\). Brill-Noether theory; moduli space of stable bundles; determinantal varieties; tangent space Fine and coarse moduli spaces, Determinantal varieties, Families, moduli of curves (algebraic), Picard groups A singularity theorem in Brill-Noether theory of higher rank | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians The object of this paper is the classification of stable \(n\)-pointed trees. Stable n-pointed trees arise in a natural way if one tries to find moduli for totally degenerate curves. The authors prove in particular the existence of a fine moduli space \(B_ n\) of stable n-pointed trees.
A connected projective variety C over a field k together with an n-tuple \(\phi =(\phi_ 1,...,\phi_ n)\) of distinct k-rational points of C is called a \textit{stable n-pointed tree} \((C,\phi)\) of projective lines over k if
(1) every component of C is isomorphic to the projective line over k;
(2) every singular point of C is k-rational and an ordinary double point;
(3) the intersection graph of the components of C is a tree;
(4) the set \(\{\phi_ 1,...,\phi_ n\}\cup \{\text{singular points of C}\}\) has at least 3 points on every component of C, and
(5) \(\phi_ 1,...,\phi_ n\) are regular points on \(C.\)
\(B_ n\) is a closed subscheme of a product of projective lines over \({\mathbb{Z}}\) given by a multihomogeneous ideal which is written down in the paper. The k-valued points of \(B_ n\) are in 1-1-correspondence with the isomorphy classes of stable n-pointed trees of projective lines over k. \(B_ n\) is smooth and of relative dimension 2n-3 over \({\mathbb{Z}}\). The canonical projection \(B_{n+1}\to B_ n\) is the universal family of stable n-pointed trees. The Picard group of \(B_ n\) is free of rank \(2^{n-1}-(n+1)-n(n-3)/2.\) The authors give a method to compute the Betti numbers of B(\({\mathbb{C}})\) (see the following review). Further, it turns out that \(B_ n\) is a blow-up of the quotient of semi-stable points in \({\mathbb{P}}^ n_ 1\) for the action of fractional linear transformations in every component. The authors describe the blow-up in several steps where at each stage the obtained space is interpreted as a solution to a certain moduli problem. blow-up of semi-stable points; stable n-pointed trees; fine moduli space; Picard group; Betti numbers L. Gerritzen, F. Herrlich, and M. van der Put, Stable \?-pointed trees of projective lines, Nederl. Akad. Wetensch. Indag. Math. 50 (1988), no. 2, 131 -- 163. Families, moduli of curves (algebraic), Topological properties in algebraic geometry, Picard groups, Fine and coarse moduli spaces Stable \(n\)-pointed trees of projective lines | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians Let \(G\) be an affine reductive algebraic group over an algebraically closed field \(k\) of any characteristic. Denote by \({\mathcal M}_G\) the moduli stack of principal G-bundles over a smooth projective curve \(C\) of genus \(g_C \geq 0\). The connected components of \(\mathcal{M}_G\) are indexed by \(\pi _1(G)\), so let \({\mathcal M}_G^d\) be, for \(d\in \pi _1(G)\), the corresponding connected component.
The authors determine \(\text{Pic}({\mathcal M}^d_G)\), based on the results in the case \(G\) simple connected, over complex numbers and generalizations, due to Faltings, in positive characteristic (see the \textit{Introduction} for the necessary references). Taking a natural scheme structure \(\underline{\text{Pic}}({\mathcal M}^d_G)\) on \(\text{Pic}({\mathcal M}^d_G)\), one shows (see Th. 5.3.1) that \(\underline{\text{Pic}}({\mathcal M}^d_G)\) contains the scheme of homomorphisms from \(\pi _1(G)\) to the Jacobian \(J_C\) as an open subgrup; the quotient is a group \(NS({\mathcal M}_G^d)\), called \textit{Néron-Severi group}, which is abelian, finitely generated. In fact the Néron-Severi group is introduced previously (see 5.2). The main result of the paper is Theorem 5.2 which treats the map of Picard groups induced by a map of groups \(G\to H\). principal bundle; moduli stack; Picard group; Picard functor; affine Grassmanian Biswas I., Hoffmann N., The line bundles on moduli stacks of principal bundles on a curve, Doc. Math., 2010, 15, 35--72 Picard groups, Algebraic moduli problems, moduli of vector bundles, Families, moduli of curves (algebraic) The line bundles on moduli stacks of principal bundles on a curve | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians Consider the family \(M\) of nondegenerate plane cuspidal cubics in \({\mathbb{P}^3}\). A compactification \(\bar{M}\) can be constructed as the projectivization \(\bar{M}=\mathbb{P}({\mathcal E})\) of a suitable vector bundle \({\mathcal E}\). The author
constructs a resolution of \(\bar{M}\) and uses this to compute the Chow ring \(\text{CH}(\bar{M})\) (and hence, in particular, the Picard group \(\text{Pic}(\bar{M})\)). (Equivariant) Chow groups and rings; motives, Picard groups, Plane and space curves, Families, moduli of curves (algebraic) The Picard group of the variety of plane cuspidal cubics 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 This article treats the Picard group of the moduli (stack) of \(r\)-spin curves and its compactification. Generalized spin curves, or \(r\)-spin curves are a natural generalization of \(2\)-spin curves (algebraic curves with a theta-characteristic), and have been of interest lately because they are the subject of a remarkable conjecture of E. Witten, and because of the similarities between the intersection theory of these moduli spaces and that of the moduli of stable maps. We generalize results of Cornalba, describing and giving relations between many of the elements of the Picard group of the stacks. These relations are important in the proof of the genus-zero case of Witten's conjecture given by \textit{T. J. Jarvis, T. Kimura} and \textit{A. Vaintrob} [Compos. Math. 126, No. 2, 157-212 (2001; Zbl 1015.14028)]. We use these relations to show that when \(2\) or \(3\) divides \(r\), the Picard group has non-zero torsion. And finally, we work out some specific examples. Witten conjecture; Picard group; \(r\)-spin curves Jarvis, T.: The Picard group of the moduli of higher spin curves. New York J. Math. 7, 23--47 (2001) Families, moduli of curves (algebraic), Picard groups, Moduli of Riemann surfaces, Teichmüller theory (complex-analytic aspects in several variables), Two-dimensional field theories, conformal field theories, etc. in quantum mechanics, Relationships between algebraic curves and physics The Picard group of the moduli of higher spin 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 After having fixed two positive integers \(g\geq 3\) and \(d\), one can consider the functor of isomorphism classes of smooth, proper and connected complex curves of genus \(g\), admitting a finite morphism of degree \(d\) to \(\mathbb{P}_{\mathbb{C}}^1\), whose branch divisor is supported at \(2g+2d-2\) distinct points. This moduli problem is known to have a coarse moduli space, denoted by \(\mathcal{H}_{d,g}\), which is a normal, \(\mathbb{Q}\)-factorial and irreducible quasi-projective complex variety of dimension \(2g+2d-5\). The Picard rank conjecture predicts that \(\mathrm{Pic}(\mathcal{H}_{d,g})\otimes \mathbb{Q}=0\). One consequence of this conjecture is the expectation that a certain partial compactification \(\widetilde{\mathcal{H}_{d,g}}\) of \(\mathcal{H}_{d,g}\), obtained by allowing nodal, irreducible curves and non simply branches, should have rational Picard group generated by boundary classes. It is known that the boundary classes can be expressed in terms of ``tautological classes''. In particular the expectation is that \(\mathrm{Pic}(\widetilde{\mathcal{H}_{d,g}})\otimes \mathbb{Q}\) is generated by tautological classes. The validity of the conjecture was known before the paper under review for \(d=2,3\) and for large \(d\), namely \(d>2g-2\). In the paper under review the authors prove this conjecture under the assumption that \(3\leq d\leq 5\) (Theorem A). The strategy of the proof consists in first showing that for \(d\geq 3\) (resp. \(d\geq 4\)) there are at least two (resp. at least three) divisorial components supported on \(\widetilde{\mathcal{H}_{d,g}}\setminus\mathcal{H}_{d,g}\) whose classes are linearly independent in \(\mathrm{Pic}( \widetilde{\mathcal{H}_{d,g}})\otimes \mathbb{Q}\) (Proposition 2.15). The most delicate part is then to show that \(\mathrm{rk}(\mathrm{Pic}( \widetilde{\mathcal{H}_{d,g}})\otimes \mathbb{Q})\leq 2\) (resp. \(\leq 3\)) for \(d=3\) (resp. for \(d=4,5\)). In order to produce these upper bounds the authors find a suitable open \(U\subset \widetilde{\mathcal{H}_{d,g}}\) which can be expressed as successive quotients of an open in a projective space by the action of explicit linear algebraic groups and whose number of divisorial components in \(\widetilde{\mathcal{H}_{d,g}}\setminus U\) can be explicitly computed. The authors associate to each smooth curve \(C\) of genus \(g\), equipped with a degree \(d\) morphism \(C\rightarrow \mathbb{P}_{\mathbb{C}}^1\) an embedding of \(C\) into the projectification of the associated Tschirnhausen bundle over \(\mathbb{P}_{\mathbb{C}}^1\). A resolution of the structure sheaf of the curve via this embedding is computed by using theorem 2.1 in [\textit{G. Casnati} and \textit{T. Ekedahl}, J. Algebr. Geom. 5, No. 3, 439--460 (1996; Zbl 0866.14009)]. The open \(U\subset \widetilde{\mathcal{H}_{d,g}}\) is obtained via a suitable mixing (depending on \(d\)) of the loci of curves where the associated Tschirnausen bundle and the first bundle in the Casnati-Ekedahl resolution are most generic. The loci are compared with certain Severi varieties as considered in [\textit{A. Ohbuchi}, J. Math., Tokushima Univ. 31, 7--10 (1997; Zbl 0938.14011)]. These Severi varieties are defined as follows. Fix \(m\) a positive integer, consider the Hirzebruch surface \(\mathbb{F}_m\) and \(\tau\subset \mathbb{F}_m\) a section with self-intersection equal to \(m\). Define \(\mathcal{V}_g(\mathbb{F}_m,d\tau)\) as the closure in the linear series \(|d\tau|\) of the locus \(\mathcal{U}_g(\mathbb{F}_m,d\tau)\) parametrizing irreducible, nodal curves of genus \(g\) in \(|d\tau|\). Ohbuchi gives a way to produce from a smooth curve of genus \(g\) equipped with a degree \(d\) morphism to \(\mathbb{P}_{\mathbb{C}}^1\) and a positive integer \(m\) a point in the Severi variety \(\mathcal{V}_g(\mathbb{F}_m,d\tau)\).
\noindent As consequence of this association the authors can show that for any \(d\) and \(m\) larger than an explicit number (depending on \(d\) and \(g\)) the Picard rank conjecture is equivalent to \(\mathrm{Pic}(\mathcal{U}_g(\mathbb{F}_m,d\tau))\otimes\mathbb{Q}=0\) (Theorem B). Hurwitz space; Picard group A. Deopurkar and A. Patel, The Picard rank conjecture for the Hurwitz spaces of degree up to five. Available at http://arxiv.org/pdf/1402.1439v2, 2014. Families, moduli of curves (algebraic), Picard groups The Picard rank conjecture for the Hurwitz spaces of degree up to five | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is 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 sequel to part I of this paper [Trans. Am. Math. Soc. 309, No. 1, 1--34 (1988; Zbl 0677.14003)], the authors establish the linear independence, over \(\mathbb Q\), of their intrinsic divisor classes A, B, C and \(\Delta\). For the proof, the authors construct a restriction homomorphism from \(\operatorname{Pic}(W(d,\delta))\) to \(\operatorname{Pic}(W(d,\delta-1))\), where \(W(d,\delta)\) denotes the Severi variety (in the author's sense) of plane curves of degree \(d\) which are singular at exactly \(\delta\) nodes. The restriction homomorphism reduces the verification of independence to small values of the genus \(g\); the conclusion follows by inspecting five explicit families of curves. The paper concludes with a series of useful examples.
[For the entire collection see Zbl 0635.00006.] Picard group; divisor classes; Severi variety Harris, J. and Diaz, S., The Geometry of the Severi Variety II: Independence of Divisor Classes and Examples, Algebraic Geometry (Sundance, UT, 1986), Lecture Notes in Math., Springer- Verlag, 1311 (1988), 23-50. Families, moduli of curves (algebraic), Divisors, linear systems, invertible sheaves, Picard groups Geometry of Severi varieties. II: Independence of divisor classes and example | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians Let \(V \subset \mathbb P H^0 ({\mathcal O}_{\mathbb P^2} (d))\) be the Severi variety parametrizing curves \(C \subset \mathbb P^2\) of degree \(d\) having exactly \(n\) nodes. The closure \(\overline V\) being complicated, \textit{S. Diaz} and \textit{J. Harris} defined a partial compactification by adding the codimension one equisingular strata in \(\overline V\) to \(V\) to obtain \(\tilde V\) and then normalizing \(\tilde V\) to produce a smooth variety \(W\) [Trans. Am. Math. Soc. 309, 1--34 (1988; Zbl 0677.14003); Trans. Am. Math. Soc. 309, 433--468 (1988; Zbl 0707.14022)]. They also compute many intrinsic divisor classes in \(\text{Pic} (W)_{\mathbb Q}\) in terms of five divisors \(CP, NL, TN, TR\) and \(\Delta\). For example, \(CP\) corresponds to the locus of integral curves containing a fixed point.
The authors consider the morphism \(f:W \to {\mathbb P^{2 [n]}}\) extending the rational map \(V \to {\mathbb P^{2 [n]}}\) taking a curve to its set of nodes. It is known from work of \textit{R. Treger} that \(f\) is birational if \(6n=d^2+3d\) (i.e. \(\dim V_{d,n} = 2n = \dim {\mathbb P^{2 [n]}}\)) except when \((d,n) = (6,9)\) [Canad. J. Math. 41, 193--212 (1989; Zbl 0666.14013)] and it was shown by \textit{J. Fogarty} that \(\text{Pic} \; {\mathbb P^{2 [n]}}\) is freely generated by the class \(H\) corresponding to subschemes whose support intersects a fixed line and \(B\) corresponding to non-reduced subschemes of length \(n\) [Am. J. Math. 95, 660--687 (1973; Zbl 0299.14020)]. Assuming that \(f\) is birational, the authors compute the push-forward of the divisors \(CP, NL, TN, TR\) and \(\Delta\) described above in terms of \(H\) and \(B\).
In particular, the canonical divisor \(K_W\) is not effective, answering a question posed by Diaz and Harris. Hilbert scheme of points; Severi variety; nodal plane curves Families, moduli of curves (algebraic), Picard groups, Plane and space curves On the position of nodes of 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 Let \(\mathcal M_{g,n}\) be the moduli space of smooth genus \(g\) curves with \(n\) ordered marked points. The symmetric group \(S_n\) acts naturally on \(\mathcal M_{g,n}\) by permuting the marked points. The main result of the paper is a formula of the generating function of the \(S_n\) equivariant Euler characteristics of \(\mathcal M_{g,n}\). The author first derives a formula of the equivariant Euler characteristics of a configuration space with a finite group action, and then uses it to prove the main result. The coefficients involved in the main formula correspond to the orbifold Euler characteristics of moduli spaces of curves with a specific type of automorphism. These Euler characteristics are computed using a result of \textit{J. Harer} and \textit{D. Zagier} [Invent. Math. 85, 457--485 (1986; Zbl 0616.14017)]. moduli spaces; equivariant Euler characteristic; orbifold Euler characteristic DOI: 10.1016/j.aim.2013.10.003 Families, moduli of curves (algebraic), Fine and coarse moduli spaces The equivariant Euler characteristic of moduli spaces of curves | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians Let \({\mathcal M}_ 2\) be the moduli space of curves of genus 2 and \({\mathcal S}\) be the moduli space of ''nodal cubics'' i.e. of cubic surfaces with one ordinary double point. We produce a birational map \({\mathcal S}\to {\mathcal M}_ 2\) as follows: to a nodal cubic S with note P we associate a genus 2 curve \(\Gamma_ S\) which is the double cover of the conic \({\mathbb{P}}roj(\tan gent\quad cone\quad to\quad S\quad at\quad P)\) branched at the six points which are the lines through P lying on S; conversely, the bicanonical image of a genus 2 curve \(\Gamma\) is a conic in \({\mathbb{P}}^ 2\) with six distinguished points on it and this determines a nodal cubic. By using the classical Sylvester's theorem which gives a unique ''canonical'' equation for a generic (even nodal) cubic surface we construct an explicit ''birational model'' W for \({\mathcal S}\). W is a hypersurface of \({\mathbb{P}}^ 4/\sigma_ 5\) where \(\sigma_ 5\) is the symmetric group on 5 letters. The rationality of W is then proved by some computations, thus yielding the rationality of \({\mathcal S}\) and \({\mathcal M}_ 2\) over an algebraically closed field of characteristic 0. rationality of moduli space of curves of genus 2; rationality of moduli space of nodal cubics Rational and unirational varieties, Families, moduli of curves (algebraic), Families, moduli, classification: algebraic theory, Algebraic moduli problems, moduli of vector bundles Nodal cubic surfaces and the rationality of the moduli space of curves of genus two | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians Hurwitz spaces are spaces of pairs \((S,f)\) where \(S\) is a Riemann surface and \(f:S\to\widehat{\mathbb C}\) a meromorphic function. In this work, we study one-dimensional Hurwitz spaces \({\mathcal H}^{D_p}\) of meromorphic \(p\)-fold functions with four branched points, three of them fixed; the corresponding monodromy representation over each branched point is a product of \((p-1)/2\) transpositions and the monodromy group
is the dihedral group \(D_p\). We prove that the completion \(\overline{{\mathcal H}^{D_p}}\) of the Hurwitz space \({\mathcal H}^{D_p}\) is uniformized by a non-normal index \(p+1\) subgroup of a triangular group with signature \((0;[p,p,p])\). We also establish the relation of the meromorphic covers with elliptic functions and show that \({\mathcal H}^{D_p}\) is a quotient of the upper half plane by the modular group \(\Gamma(2)\cap\Gamma_0(p)\). Finally, we study the real forms of the Belyi projection \(\overline{{\mathcal H}^{D_p}}\to\widehat{\mathbb C}\) and show that there are two nonbicoformal equivalent such real forms which are topologically conjugated. Costa, AF; Izquierdo, M.; Riera, G., One-dimensional Hurwitz spaces, modular curves, and real forms of Belyi meromorphic functions, Int. J. Math. Math. Sci., 2008, 1-18, (2008) Families, moduli of curves (algebraic), Coverings of curves, fundamental group, Fuchsian groups and automorphic functions (aspects of compact Riemann surfaces and uniformization) One-dimensional Hurwitz spaces, modular curves, and real forms of Belyi meromorphic functions | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians The moduli space \(\overline{M}_{0,n}\) may be embedded into the product of projective spaces \(\mathbb{P}^1\times\mathbb{P}^2\times\cdots \times\mathbb{P}^{n-3}\), using a combination of the Kapranov map \(|\psi_n|:\overline{M}_{0,n}\to\mathbb{P}^{n-3}\) and the forgetful maps \(\pi_i:\overline{M}_{0,i}\to\overline{M}_{0,i-1}\). We give an explicit combinatorial formula for the multidegree of this embedding in terms of certain parking functions of height \(n-3\). We use this combinatorial interpretation to show that the total degree of the embedding (thought of as the projectivization of its cone in \(\mathbb{A}^2\times\mathbb{A}^3\cdots\times\mathbb{A}^{n-2})\) is equal to \((2(n-3)-1)!!=(2n-7)(2n-9)\cdots(5)(3)(1)\). As a consequence, we also obtain a new combinatorial interpretation for the odd double factorial.
Editorial remark: This is an extended version of [\textit{R. Cavalieri} et al., Sémin. Lothar. Comb. 84B, 84B.32, 12 p. (2020; Zbl 1447.05003)]. parking functions; moduli of curves; multidegree; intersection theory Permutations, words, matrices, Exact enumeration problems, generating functions, Families, moduli of curves (algebraic), Combinatorial aspects of algebraic geometry, Combinatorial aspects of representation theory Projective embeddings of \(\overline{M}_{0,n}\) and parking functions | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians This paper develops some fundamental intersection theory of Deligne-Mumford stacks, building on earlier work of \textit{A. Vistoli} [Invent. Math. 97, No. 3, 613--670 (1989; Zbl 0694.14001)] and \textit{A. Kresch} [Invent. Math. 136, No. 3, 483--496 (1999; Zbl 0923.14003), by providing a more global framework which properly encodes global geometric information such as Chern classes and Chow ring structures. This machinery is applied, for instance, to certain weighted blow-ups, and more specifically, to the setting of moduli of stable maps, building on earlier work of the authors.
More precisely, the paper begins by developing machinery to in essence transform a proper local embedding of DM stacks into an actual embedding, via a fiber-product arising from a universally closed morphism that they construct. This construction provides the technical backbone of all intersection-theoretic work that follows, the main results of which are nicely described in the introduction. The first application is to Chern classes of weighted blow-ups. This is further applied to moduli of stable maps and intermediate weighted stable maps, which are spaces introduced in earlier work of the Mustatas as modular interpretations of the intermediate stages of an iterated blow-up occurring in a construction of the usual Kontsevich moduli space. Finally, the paper includes an appendix on the Euler sequence of a weighted projective bundle. Deligne-Mumford stack; local embedding; etale lift; Chern classes; weighted blow-ups; moduli space of stable maps A. M. Mustaţă and A. Mustaţă, The structure of a local embedding and Chern classes of weighted blow-ups , J. Eur. Math. Soc. 14 (2012), no. 6, 1739-1794. Intersection theory, characteristic classes, intersection multiplicities in algebraic geometry, Algebraic cycles, Families, moduli of curves (algebraic) The structure of a local embedding and Chern classes of weighted blow-ups | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is 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 quadratic stable range property is discussed. The ring \(A\) has quadratic stable range \(1\) (\(\text{qsr}(A) = 1\)) if every primitive quadratic form over \(A\) represents a unit. The property is motivated by the ring of holomorphic functions on a connected noncompact Riemann surface. The authors prove the following results:
(1) if \(\text{qsr} (A)=1\) then the stable range of \(A\) equals \(1\) and \(\text{Pic} (A)=1\).
(2) \(\text{qsr} (A)=1\) iff \(\text{Pic}(T)=1\) for every quadratic \(A\)-algebra \(T\).
They also classify quadratic forms over Bezout domains of characteristic not 2 satisfying a very strong approximation property (defined in the paper). This classification applies to the ring of holomorphic functions mentioned above. quadratic stable range; Picard group; quadratic form; holomorphic function; Riemann surface; Bezout domain D. R. Estes and R. M. Guralnick, ''A stable range for quadratic forms over commutative rings,'' J. Pure Appl. Algebra, 120, No. 3, 255--280 (1997). Quadratic forms over local rings and fields, Quadratic forms over global rings and fields, Quadratic and bilinear forms, inner products, Picard groups, Rings and algebras of continuous, differentiable or analytic functions A stable range for quadratic forms over commutative 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 These are the lecture notes of the lectures on Siegel modular forms at the Nordfjordeid Summer School on Modular Forms and their Applications. We give a survey of Siegel modular forms and explain the joint work with \textit{Carel Faber} on vector-valued Siegel modular forms of genus 2 [C. R., Math., Acad. Sci. Paris 338, No. 5, 381--384 (2004; Zbl 1062.14034), 338, No. 6, 467--470 (2004; Zbl 1055.14026)] and present evidence for a conjecture of Harder on congruences between Siegel modular forms of genus 1 and 2. conjecture of Harder; congruences between Siegel modular forms of genus 1 and 2; Eisenstein cohomology Gerard van der Geer, ``Siegel Modular Forms'', , 2007 Siegel modular groups; Siegel and Hilbert-Siegel modular and automorphic forms, Congruences for modular and \(p\)-adic modular forms, Cohomology of arithmetic groups, Families, moduli of curves (algebraic), Algebraic moduli of abelian varieties, classification Siegel modular forms and their 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 Here we give an existence theorem for integral curves \(C\) contained in a smooth projective surface \(S\) and with as only singularities prescribed ordinary multiple points at general points of \(S\). The proof heavily use the proof of the case \(S = \mathbb{P}^2\) given by \textit{T. Mignon} [J. Algebr. Geom. 10, No. 2, 281--297 (2001; Zbl 0987.14019)]. Plane and space curves, Families, moduli of curves (algebraic), Surfaces and higher-dimensional varieties Curves in a projective surface with prescribed ordinary 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 \(X\) be a projective variety, over an algebraically closed field, that is covered by rational curves (e.g. a Fano manifold over \({\mathbb Z}\)). In [J. Algebr. Geom. 11, 245--256 (2002; Zbl 1054.14035)] the first author considered the question of when there exists a unique minimal degree rational curve containing two given points. This paper was motivated by the following infinitesimal analogue: Are there natural conditions that guarantee that a minimal degree rational curve is uniquely determined by a tangent vector? Following Miyaoka's approach, the authors show that in characteristic 0, if \(H \subset \text{RatCurves}^n(X)\) is a proper, covering family of rational curves such that none of the associated curves has a cuspidal singularity, and if \(x \in X\) is a general point, then all curves associated with the closed subfamily \( H_x := \{ \ell \in H \mid x \in \ell \} \subset H\) are smooth at \(x\), and no two of them share a common tangent direction at \(x\). For positive characteristic an additional hypothesis is needed to obtain this conclusion. A central element of the proof is the study of families of dubbies, that is, reducible curves that consist of touching rational curves. Three applications are given. First, under certain hypotheses, \(H_x\) is irreducible. For the second application, let \(X\) be a complex variety satisfying the above hypotheses, and assume also that \(b_2(X) =1\). Let \(\text{Aut}_0(X)\) denote the maximal connected subgroup of the group of automorphisms of \(X\), and let \(\text{Aut}_0(H)\) be the analogous automorphism group for \(H\). Then these groups coincide. The third application is that the main results apply also to contact manifolds. Fano manifold; rational curve of minimal degree; dubbies Kebekus, S.; Kovács, S. J., Are rational curves determined by tangent vectors?, Ann. Inst. Fourier (Grenoble), 54, 1, 53-79, (2004) Families, moduli of curves (algebraic), Fano varieties, Special algebraic curves and curves of low genus Are minimal degree rational curves determined by their tangent vectors? | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is 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 0621.00009.]
This note is supplementary to the authors' paper in Math. Ann. 279, No.3, 435-448 (1988; Zbl 0657.14003) and consists largely of applications of a paper by \textit{E. D. Davis} [Ann. Univ. Ferrara, Nuova Ser., Sez. VII 32, 93-107 (1986; Zbl 0639.14033)].
Let S be the surface obtained by blowing-up a finite set of points Z of \({\mathbb{P}}^ 2\). The theorem of the title asserts that under certain conditions on generality and size of Z a certain divisor on S is very ample. It is shown that a generalization and sharpening of that theorem is a rather simple consequence of a characterization of ``punctured complete intersections'' in \({\mathbb{P}}^ 2\) related to the classical Cayley-Bacharach theorem. That term means ``scheme of \(colength\quad 1\) in a complete intersection'' - so to speak, ``a complete intersection minus one point''. linear systems of curves; generality on pointsets in projective 2-space; very ample divisor; blowing-up a finite set of points; Cayley-Bacharach theorem Divisors, linear systems, invertible sheaves, Projective techniques in algebraic geometry, Families, moduli of curves (algebraic) Bese's very ampleness theorem and punctured 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 Hassett's moduli spaces of weighted stable curves form an important class of alternate modular compactifications of the moduli space of smooth curves with marked points. In this article, we define a tropical analog of these moduli spaces, and show that the naive set-theoretic tropicalization map can be identified with a natural deformation retraction onto the non-Archimedean skeleton. This result generalizes work of Abramovich, Caporaso, and Payne [\textit{D. Abramovich} et al., Ann. Sci. Éc. Norm. Supér. (4) 48, No. 4, 765--809 (2015; Zbl 1410.14049)] treating the Deligne-Knudsen-Mumford compactification of the moduli space of smooth curves with marked points. We also study tropical analogs of the tautological maps, investigate the dependence of the tropical moduli spaces on the weight data, and consider the example of Losev-Manin spaces. Ulirsch, M, Tropical geometry of moduli spaces of weighted stable curves, J. Lond. Math. Soc., 92, 427-450, (2015) , Families, moduli of curves (algebraic), Moduli of Riemann surfaces, Teichmüller theory (complex-analytic aspects in several variables), Formal methods and deformations in algebraic geometry, Non-Archimedean analysis Tropical geometry of moduli spaces of weighted 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 main result of this article (Theorem 5.1) gives lower bounds on \(n\) such that the compactified moduli space \(\overline{{\mathcal M}_{g,n}}\) of genus \(g\) curves with \(n\) markings is of general type i.e., has maximal Kodaira-dimension. For \(g\geq 24\) earlier work of \textit{D. Eisenbud, J. Harris} and \textit{D. Mumford} [Invent. Math. 67, 23--86 (1982; Zbl 0506.14016) and ibid. 90, 359--387 (1987; Zbl 0631.14023)] had established that \(\overline{\mathcal M}_{g,n}\) is indeed of general type for all \(n\geq 0\). In this article the author deals with the case \(4\leq g\leq 23\). The proof is based on explicit constructions of certain divisors on \(\overline{{\mathcal M}_{g,n}}\). A. Logan, ''The Kodaira dimension of moduli spaces of curves with marked points,'' Amer. J. Math., vol. 125, iss. 1, pp. 105-138, 2003. Families, moduli of curves (algebraic), Singularities of curves, local rings, Special divisors on curves (gonality, Brill-Noether theory) The Kodaira dimension of moduli spaces of curves with marked 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 In the article under review, the authors establish a number of important results pertaining to syzygies of algebraic curves.
The authors' first result concerns the Prym-Green Conjecture, as formulated in [\textit{A. Chiodo} et al., ibid. 194, No. 1, 73--118 (2013; Zbl 1284.14006)], which asserts that the minimal free resolution of the section ring \(\Gamma_C(K_C \otimes \eta)\) for \([C,\eta] \in \mathcal{R}_g\) a general Prym curve is natural; here \(C\) is a smooth curve of genus \(g\), \(K_C\) denotes the canonical line bundle of \(C\), \(\eta\) is a non-trivial two torsion line bundle on \(C\) and \(\mathcal{R}_g\) denotes the moduli space of genus \(g\) Prym curves.
The authors' first result is that the Prym-Green conjecture holds true for a general Prym curve \([C,\eta] \in \mathcal{R}_g\) of odd genus. Equivalently, if \(K_{p,q}(C,K_C\otimes \eta)\) denotes the Koszul cohomology groups of \(p\)-th syzygies of weight \(q\), for \([C,\eta] \in \mathcal{R}_g\) a general Prym curve of odd genus, then the authors prove that for all \(q\) there is at most one \(p\) for which the Koszul cohomology group \(K_{p,q-p}(C,K_C \otimes \eta)\) is nonzero.
One aspect to the proof of this result involves studying syzygies of certain Prym curves of odd genus \(g \geq 11\) lying on general \(K3\) surfaces \(X\) equipped with a double cover \(\widetilde{X} \rightarrow X\) branched along eight disjoint rational curves.
The authors then proceed to establish many instances of the Green-Lazarsfeld Secant Conjecture, as stated in [\textit{M. Green} and \textit{R. Lazarsfeld}, ibid. 83, 73--90 (1986; Zbl 0594.14010)]. As one example, the authors prove that the Green-Lazarsfeld Secant Conjecture holds true for a general curve \(C\) of genus \(g\) and a general line bundle \(L\) of degree \(d\) on \(C\).
Important to the authors' proof of their results is \textit{C. Voisin}'s proof of Green's Conjecture on syzygies [J. Eur. Math. Soc. 4, No. 4, 363--404 (2002; Zbl 1080.14525); Compos. Math. 141, No. 5, 1163--1190 (2005; Zbl 1083.14038)], in addition to Theorem 1.6 of the paper under review which gives a formula for the class of the syzygy divisor in \(\mathrm{CH}^1(\mathcal{M}_{g,2g})\), the Chow ring of the moduli space of genus \(g\) curves with \(2g\) marked points. 10.1007/s00222-015-0595-7 Families, moduli of curves (algebraic), Divisors, linear systems, invertible sheaves, \(K3\) surfaces and Enriques surfaces, Jacobians, Prym varieties, Syzygies, resolutions, complexes and commutative rings The generic Green-Lazarsfeld secant conjecture | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians Let \(k\) be an algebraically closed field of characteristic 0. Consider a smooth variety \(B\) over \(k\) together with a closed subset \(\Delta\subset B\). An algebraic family \(f: X\to B\) of varieties over the base \(B\) is called admissible (with respect to the pair \((B,\Delta)\)) if it is not isotrivial and \(\Delta\) contains the discriminant locus of the morphism \(f\), that is, the restricted map \(f: X\setminus f^{-1}(\Delta)\to B\setminus\Delta\) is smooth. In moduli theory, the study of admissible families of certain classes of polarized algebraic varieties plays a fundamental role, as it is intimately related to the question of what kind of proper subvarieties may appear in the moduli stack of the respective moduli problem.
In the survey article under review, the author gives an overview on some recent developments in this context. In fact, the topic of this article can be traced back to the origins of algebraic deformation and moduli theory, but has undergone an anormous transformation during the last decade, parts of which are to be depicted in the present survey.
The material is organized in seven sections. After an enlightening introduction to the subject, including historical remarks and some classical results concerning moduli of curves and Abelian varieties, the starting point of the discussion in Section 2 is the famous Shafarevich conjecture formulated in 1962. This conjecture states that, for a smooth projective curve \(B\) of genus \(q\) and a finite subset \(\Delta\subset B\), there exist only finitely many isomorphism classes of \((B,\Delta)\)-admissible families of curves of genus \(g\). In this setting, Shafarevich's conjecture was confirmed by \textit{A. N. Parshin} [Izv. Akad. Nauk SSSR, Ser. Mat. 32, 1191--1219 (1968; Zbl 0181.23902)] and \textit{S. J. Arakelov} [Izv. Akad. Nauk SSSR, Ser. Mat. 35, 1269--1293 (1971; Zbl 0238.14012)], together with a refined version of it. Also, the interrelation between the Shafarevich conjecture and the Mordell conjecture, the fundamental contributions by \textit{Y. I. Manin} and \textit{G. Faltings}, the De Franchis theorem (1913), and some more recent results in the number field case are desribed. Section 3 discusses hyperbolicity and boundedness properties for admissible families of curves over a base curve \(B\), mainly from the viewpoint of Parshin's approach and under the aspect of moduli of curves. Section 4 turns to families of higher-dimensional varieties over a base curve \(B\) culminating in a higher-dimensional variant of the Shafarevich conjecture, together with the explanation of a number of more recent results concerning this generalization.
In this context, the works of S. Kovacs, Viehweg-Zuo, Bedulev-Viehweg, Kovacs-Lieblich, Jorgenson-Todorov, and others are briefly mentioned. Section 5 is devoted to a further generalization, namely to the case of a higher-dimensional base scheme \(B\) which leads to the so-called \textit{E. Viehweg} conjecture [Positivity of direct image sheaves and applications to higher-dimensional families of manifolds, ICTP Lect. Notes 6, 249--284 (2001; Zbl 1092.14044)] and its very recent refinement by \textit{S. Kebekus} and \textit{S. Kovacs} [Invent. Math. 172, No. 3, 657--682 (2008; Zbl 1140.14031)].
Section 6 briefly discusses some recent results on uniform and effective bounds for the number of deformation types of admissible families of canonically polarized varieties with prescribed Hilbert polynomial [cf.: \textit{S. Kovacs} and \textit{M. Lieblich}, Boundedness of families of canonically polarized manifolds: A higher-dimensional analogue of Shafarevich's congecture; Ann. Math. (to appear), \url{arXiv:math AG/0611672}]. In Section 7, some of the most important methods and techniques used in the proofs of the afore-mentioned results are depicted, with special emphasis on positivity properties of direct image sheaves, cohomological vanishing theorems, and kernels of Kodaira-Spencer maps. Finally, Section 8 provides some hints to further results and current directions of research in this context, thereby completing the overall panoramic picture. stacks; moduli spaces; families of algebraic varieties; fibrations; degenerations; Shafarevich conjecture; deformation theory; Kodaira-Spencer maps S. J. Kovács, ''Subvarieties of moduli stacks of canonically polarized varieties: generalizations of Shafarevich's conjecture,'' in Algebraic Geometry-Seattle 2005. Part 2, Providence, RI: Amer. Math. Soc., 2009, vol. 80, pp. 685-709. Generalizations (algebraic spaces, stacks), Families, moduli, classification: algebraic theory, Stacks and moduli problems, Algebraic moduli problems, moduli of vector bundles, Fine and coarse moduli spaces, Fibrations, degenerations in algebraic geometry, Formal methods and deformations in algebraic geometry, Families, moduli of curves (algebraic), Deformations of complex structures Subvarieties of moduli stacks of canonically polarized varieties: generalizations of Shafarevich's conjecture | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians The authors outline the proof of a remarkable formula for Hodge integrals conjectured by \textit{M.~Marino} and \textit{C.~Vafa} [in: Orbifolds in mathematics and physics. Contemp. Math. 310, 185--204 (2002; Zbl 1042.81071)] based on large \(N\) duality. The proof appeared in the authors' paper in [J. Differ. Geom. 65, No. 2, 289--340 (2003; Zbl 1077.14084)]. Chiu-Chu Melissa Liu, Kefeng Liu, and Jian Zhou, On a proof of a conjecture of Mariño-Vafa on Hodge integrals, Math. Res. Lett. 11 (2004), no. 2-3, 259 -- 272. Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects), Families, moduli of curves (algebraic) On a proof of a conjecture of Mariño-Vafa on 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 The workshop focused on Severi varieties on \(K3\) surfaces, hyperkähler manifolds and their automorphisms. The main aim was to bring researchers in deformation theory of curves and singularities together with researchers studying hyperkähler manifolds for mutual learning and interaction, and to discuss recent developments and open problems. Collections of abstracts of lectures, Proceedings of conferences of miscellaneous specific interest, Families, moduli of curves (algebraic), Singularities of curves, local rings, Special divisors on curves (gonality, Brill-Noether theory), Divisors, linear systems, invertible sheaves, \(K3\) surfaces and Enriques surfaces, Automorphisms of surfaces and higher-dimensional varieties, Proceedings, conferences, collections, etc. pertaining to algebraic geometry, Hyper-Kähler and quaternionic Kähler geometry, ``special'' geometry Mini-workshop: Singular curves on \(K3\) surfaces and hyperkähler manifolds. Abstracts from the mini-workshop held November 8--14, 2015 | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians Let $k$ be a finite field of order $q$. For $C$ a projective, non-singular,
geometrically irreducible algebraic curve over $k$ of genus $g$, a basic problem is to
compute $N:=\# C(k)$, the number of $k$-rational points of $C$. We have that $N=q+1-t$,
where $t$ is the trace of the Frobenius map acting on the first étale cohomology group
$H^1(J_{\text{ét}},{\mathbb Z}_\ell)$ of the Jacobian $J$ of $C$ being $\ell$ a prime different
of the characteristic of $k$; moreover, $|t|\leq g\lfloor 2\sqrt{q}\rfloor$ (the
Hasse-Weil-Serre bound). One is often interesting in computing $N_q(g)$, the maximal
possible number of $k$-rational points on curves of genus $g$.
\textit{J.-P. Serre} [Sémin. Théor. Nombres, Univ. Bordeaux I 1982--1983, Exp. No. 22, 8 p. (1983; Zbl 0538.14016)] computed
$N_q(1)$ and $N_q(2)$, and consider the possibility to obtain a similar result for $N_q(3)$.
He noticed that this task is more involved as here we have the so-called \textit{Serre
obstruction}, namely there are Jacobians of genus 3 curves over $\bar k$ which are not the
Jacobian of curves over $k$; a reason for this obstruction is the existence of
non-hyperelliptic curves of genus bigger than $2$. Although there have been many
computational and conceptual approaches to Serre's obstruction (see e.g. \textit{T. Ibukiyama} [Tohoku Math. J. (2) 45, No. 3, 311--329 (1993; Zbl 0819.14007)], \textit{G. Lachaud} and \textit{C. Ritzenthaler} [Ser. Number Theory Appl. 5, 88--115 (2008; Zbl 1151.14321)]), there is no so
far a closed formula for $N_q(g)$, $g\geq 3$.
In the paper under review, among other things, $N_q(3)$ is investigated via the possible
values of the trace of Frobenius: for large negative value of $t$ it is shown that there are
more curves with trace $t$ than with trace $-t$. Indeed, let ${\mathcal N}_{q,3}(t)$ be the
number of non-hyperelliptic genus $3$ curves over $k$ of trace $t$ up to $k$-isomorphism.
Then the function ${\mathcal M}_{q,3}(t):={\mathcal N}_{q,3}(t)-{\mathcal N}_{q,3}(-t)$ is
considered for $0\leq t\leq 6\sqrt{q}$. It is shown that ${\mathcal M}_{q,3}(t)\leq 0$ for
$t$ large enough. For instance this is enough to show that there is a genus $3$-curve $C$
over $k$ such that $\#C(k)\geq q+1+3\lfloor\sqrt{q}\rfloor-3$ (cf. \textit{K. Lauter} and \textit{J.-P. Serre} [Compos. Math. 134, No. 1, 87--111 (2002; Zbl 1031.11038)]). See \textit{R. Lercier} et al. [LMS J. Comput. Math. 17A, 128--147 (2014; Zbl 1333.14060)] for computations on prime fields of small order. genus 3 curves; plane quartics; moduli; families; enumeration; finite fields Computational aspects of algebraic curves, Families, moduli of curves (algebraic), Arithmetic ground fields for curves, Automorphisms of curves, Special algebraic curves and curves of low genus, Plane and space curves Distributions of traces of Frobenius for smooth plane curves over finite fields | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians The author discusses new developments in the theories of operads and algebraic varieties which arise from their mutual interplay. Operads will be recalled and their importance in algebraic geometry will be discussed. In particular the description of known algebraic structures in terms of operads (``universal algebra approach'') and the algebro-geometric examples like the operads derived from the moduli stack of stable pointed curves with certain genus are related by homology. Also the combinatorics of trees and operads are closely related. For instance it has been shown by \textit{V. Ginzburg} and the author that the tree parts of graph complexes which are important in the quasiclassical description of Chern-Simons invariants and for the cohomology of infinite-dimensional Lie algebras can be generalized in terms of operads. To be more detailed, for a family of differential-graded (dg) vector spaces the free operad generated by this family as well as the suspension (with twisted \(S_n\)-actions) can be defined. Then to any dg-operad \(\mathcal P\) with \({\mathcal P}(0)=0\) and \({\mathcal P}(1)= {\mathbb C}\) a free operad generated by the suspension of the duals of \({\mathcal P}(n)\), \(n>1\) exists which has a natural structure of a dg-operad \({\mathbb D}({\mathcal P})\), the so-called cobar-dual of \(\mathcal P\). The double cobar-dual is quasiisomorphic to \(\mathcal P\). These results may be interpreted as some sort of cohomology theory on the category of (dg-)operads.
For quadratic operads a natural Koszul dual operad can be defined in analogy to Koszul duals of algebras. Any quadratic operad \(\mathcal P\) admits a natural morphism of dg-operads of the cobar-dual of \(\mathcal P\) into the Koszul dual of \(\mathcal P\), and \(\mathcal P\) is called Koszul if this morphism is a quasi-isomorphism. In particular the operads \textit{As, Com, Lie} are Koszul.
Next, an operad \(\mathcal P\) will be called cyclic if there is an action of the symmetric category of permutations on \(\mathcal P\) which is compatible with composition in a certain sense. Cyclic operads provide an invariant scalar product on their \(\mathcal P\)-algebras. The above mentioned algebro-geometric operads over the moduli stack of stable pointed curves with certain genus are cyclic. Using the notion of cyclic operad so-called (twisted) modular operads can be defined, and in turn a more general definition of cobar-duality on these sort of operads again covers the graph complexes in rather full generality. Then an Euler characteristic on these cobar-dual twisted modular dg-operads \(F({\mathcal P})\) can be derived solely in terms of the underlying modular dg-operad \(\mathcal P\). In particular for \({\mathcal P}=As\) this result relates the integral of the Fourier transform, the topological Euler characteristic, the Euler, Möbius and Riemann zeta functions.
Eventually, the author explains the usefulness of operads in the theory of characteristic classes in geometry. Then he discusses (heuristic) generalizations of these results to noncommutative \(\mathcal P\)-geometries based on ``arbitrary'' operads for which these generalized notions are meaningful. operads; superpositions; moduli spaces; stacks , Families, moduli of curves (algebraic), Relational systems, laws of composition, Monads (= standard construction, triple or triad), algebras for monads, homology and derived functors for monads, Other classes of algebras Operads and algebraic geometry | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians Let \(\rho: X\to S\) be a holomorphic contraction where \(X\) is a smooth complex surface and \(S\) is a normal surface. Assume that \(\varphi\) is a biholomorphism \(X\backslash C @>\sim>> S \backslash \{x_0\}\) where \(C\) is a divisor and \(x_0\) is a rational singularity.
The author proves the following: \(\rho^*; \text{Pic} (S)\to \text{Pic}(X)\) is injective; if \(C\) is irreducible, smooth and \((C^2) < 2-2g(C)\) then
\[
0\to\text{Pic} (S) @>\rho^*>> \text{Pic} (X) @>r>> \text{Pic} (C)
\]
is an exact sequence \((r\) is the morphism obtained by \({\mathcal O}_C)\). Picard group; normal surface Deformations of complex structures, Picard groups On the Picard group of some normal 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 trigonal curve of genus \(g\geq 5\) defined over an algebraically closed field of characteristic zero. The canonical model of \(C\) lies on a unique rational ruled surface \(R_{m,n}:=\mathbb{P}({\mathcal O}(m)\oplus{\mathcal O} (n))\) in \(\mathbb{P}^{g-1}\), where \(0<m\leq n\), \(\;m+n=g-2\). The integer \(m\) is known as the Maroni invariant of \(C\). Denote by \(M^3_g(m)\) the moduli space of genus \(g\) trigonal curves of Maroni invariant \(m\). The authors first prove that if \(P\in C\) is an unramified Weierstrass point then there exist integers \(r\) and \(\alpha\geq 1\) such that the gap-sequence \(G_P\) at \(P\) is \((1,2, \dots,r,r+1+ \alpha,r+2+2 \alpha,\dots, g+\alpha)\), i.e. \(G_P\) splits into two sequences of consecutive integers, where \(r\) and \(\alpha\) must satisfy the conditions \(n+1\leq r\leq g-1\), \(\;1\leq\alpha\leq 2r+1-g\). Say this point \(P\) is of type \((r,\alpha)\). Then the authors define a stratification of \(M^3_g(m)\) as follows: For any set of integers \(t,t_E,s_E,r,\alpha\), all \(\geq 0\), let \({\mathfrak C}^m_{t,t_E,s_E}\) and \({\mathfrak C}^m_{t,t_E,s_E, (r,\alpha)}\) be the subsets of \(M^3_g (m)\) consisting of the curves containing \(t\) points of total ramification on \(E\), \(t_E\) points of total ramification on \(E\), \(s_E\) points of simple ramification on \(E\) and one unramified point of type \((r,\alpha)\) respectively, where \(E\) denotes the unique unisecant curve of degree \(m\) on \(R_{m,n}\) when \(m<n\).
The main result in the paper is: Assume that \(t+t_E+ s_E\leq m+2\) (respectively \(t+t_E+s_E\leq \min(2m+3-r,2r-g- \alpha+1))\) then \({\mathfrak C}^m_{t,t_E,s_E}\) (respectively \({\mathfrak C}^m_{t,t_E,s_E,(r,\alpha)})\) is an irreducible subvariety of \(M^3_g(m)\) of codimension \(t+2t_E+s_E\) (respectively \(t+2t_E+s_E+g-r+ \alpha_1)\); moreover its general element is a curve having, in addition to the imposed Weierstrass points, only simply ramified points not on \(E\).
This paper continues previous works by \textit{M. Coppens} [Indag. Math. 47, 245-276 (1985; Zbl 0592.14025) and J. Pure Appl. Algebra 43, 11-25 (1986; Zbl 0616.14012)] and \textit{S. J. Kim} [J. Pure Appl. Algebra 63, No. 2, 171-180 (1990; Zbl 0712.14019)] on Weierstrass points on trigonal curves. trigonal curve; Maroni invariant; Weierstrass point; gap-sequence; ramification Brundu, M; Sacchiero, G, On the varieties parametrizing trigonal curves with assigned Weierstrass points, Commun. Algebra, 26, 3291-3312, (1998) Special divisors on curves (gonality, Brill-Noether theory), Riemann surfaces; Weierstrass points; gap sequences, Families, moduli of curves (algebraic), Divisors, linear systems, invertible sheaves On the varieties parametrizing trigonal curves with assigned 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 Let \(\pi:X\to C\) be a triple covering of curves where \(C\) is Brill-Noether general. We prove the existence of a base point free pencil of degree \(d=g-[{3h+1\over 2}]-2\) on the curve \(X\) which is not composed with \(\pi\). Brill-Noether curve; divisor; covering of curves; base point free pencil Coverings of curves, fundamental group, Pencils, nets, webs in algebraic geometry, Divisors, linear systems, invertible sheaves, Families, moduli of curves (algebraic) On the construction of a special divisor of some special 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 \({\mathcal M}_{g,n}\) denote the moduli stack of genus \(g\) smooth algebraic curves with \(n\) marked points, and \(\overline{\mathcal M}_{g,n}\) its Deligne-Mumford compactification. In the paper under review, the author exploits the Grothendieck-Riemann-Roch theorem of \textit{B. Toen} [K-Theory 18, 33--76 (1999; Zbl 0946.14004)] to give explicit formulas for the Chern classes of the tangent bundle to \(\overline{\mathcal M}_{g,n}\). These formulas show that the Chern classes lie in the tautological ring of \(\overline{\mathcal M}_{g,n}\), thus supporting the general philosophy that all classes of natural geometrical significance are tautological classes. Chern classes; moduli stack; stable curves; tautological ring Bini, G., Chern classes of the moduli stack of curves, Math. res. lett., 12, 5-6, 759-766, (2005) Families, moduli of curves (algebraic), Intersection theory, characteristic classes, intersection multiplicities in algebraic geometry Chern classes of the moduli stack 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 We prove by degeneration to Prym-canonical binary curves that the first Gaussian map \(\mu _A\) of the Prym canonical line bundle \(\omega _C \otimes A\) is surjective for the general point \([C,A] \in \mathcal{R }_g\) if \(g \geq 12\), while it is injective if \(g \leq 11\). Gaussian maps; binary curves; Prym-canonical curves Families, moduli of curves (algebraic), Special algebraic curves and curves of low genus, Projective techniques in algebraic geometry On the first Gaussian map for Prym-canonical line 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 We present a survey on congruences of lines of low order. We
start by recalling the main properties of the focal locus, and use it to reobtain the classification of congruences of order one in \(\mathbb{P}^3\). We then explain the main ideas of a work in progress with S. Verra, outlining how to classify congruences of lines of order two in \(\mathbb{P}^3\). We end by stating the main problems on this topic. Arrondo, E., \textit{line congruences of low order}, Milan J. Math., 70, 223-243, (2002) Research exposition (monographs, survey articles) pertaining to algebraic geometry, Configurations and arrangements of linear subspaces, Grassmannians, Schubert varieties, flag manifolds, Families, moduli of curves (algebraic) Line congruences of low order | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians Let \((K,v)\) be a discrete valuation field with ring of integers \(\mathcal O\) and residue field \(k\). If \(Y\) is a smooth projective irreducible curve over \(K\) of genus \(g(Y) \geq 1\), then after replacing \(K\) by a finite field extension there is a semistable model \(\mathcal Y\) over \(\mathrm{Spec} \; \mathcal O\) [\textit{P. Deligne} and \textit{D. Mumford}, Publ. Math., Inst. Hautes Étud. Sci. 36, 75--109 (1969; Zbl 0181.48803)] and if \(g(Y) \geq 2\), then \(\mathcal Y\) is unique, called the \textit{stable model}. The \textit{stable reduction} of \(Y\) is the special fiber \(\overline Y\) of \(\mathcal Y\), whose \textit{reduction type} is a description of the graph of irreducible components together with the genus of the normalizations of the components. When \(g(Y)=2\), \textit{Q. Liu} determined the reduction type in terms of Igusa invariants, including the \(j\)-invariants of the elliptic components when \(\overline Y\) is singular [Math. Ann. 295, 201--222 (1993; Zbl 0819.14010)]. When \(g(Y)=3\), the general model is a plane quartic with hyperelliptic curves at the boundary. Here \textit{R. Lercier} et al. characterized when the stable reduction \(\overline Y\) is a smooth hyperelliptic curve in terms of the Dixmier-Ohno invariants and in this case computed the Shioda invariants of \(\overline Y\) [Algebra Number Theory 15, 1429--1468 (2021; Zbl 1494.13006)].
The authors ask whether it is possible to characterize the reduction type of a smooth plane quartic in terms of the Dixmier-Ohno invariants when \(\overline Y\) is singular. They succeed for the stratum of \(\mathcal M_{3,V} \subset \mathcal M_3\) consisting of curves \(Y\) for which \(\mathrm{Aut}_{\overline K} (Y)\) contains a subgroup isomorphic to \(V = \mathbb Z_2 \times \mathbb Z_2\) with \(g(Y/V)=0\) and such that all degree two subcovers of \(Y \to Y/V\) have genus one. These curves were studied long ago by \textit{Edg. Ciani} [Palermo Rend. 13, 347--373 (1899; JFM 30.0527.07)] and more recently by \textit{E. W. Howe} et al. [Forum Math. 12, 315--364 (2000; Zbl 0983.11037)] and \textit{G. Lachaud} and \textit{C. Ritzenthaler} [Ser. Number Theory Appl. 5, 88--115 (2008; Zbl 1151.14321)]. All curves \(Y\) parametrized by \(\mathcal M_{3,V}\) admit a \(V\)-Galois cover \(f: Y \to \mathbb P^1_K\), which allows the authors to calculate the possible reduction numbers by extending the method of \textit{I. I. Bouw} and \textit{S. Wewers} [Glasg. Math. J. 59, No. 1, 77--108 (2017; Zbl 1430.11090)]. They find \(13\) reduction types in all. plane quartic curves; Dixmier-Ohno invariants; stable reduction; reduction type Families, moduli of curves (algebraic), Plane and space curves, Arithmetic ground fields for curves, Curves over finite and local fields, Computational aspects of algebraic curves Reduction types of genus-3 curves in a special stratum of their 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 A congruence of lines is a \((n - 1)\)-dimensional family of lines in \(\mathbb{P}^n \) (over \(\mathbb{C})\), i.e. a variety \(Y\) of dimension (and hence of codimension) \(n - 1\) in the Grassmannian Gr\((1, \mathbb{P}^n)\). A fundamental curve for \(Y\) is a curve \(C \subset \mathbb{P}^n\) which meets all the lines of \(Y\). In this paper the authors classify all smooth congruences with fundamental curve \(C\) generalizing a paper by \textit{E. Arrondo} and \textit{M. Gross} [Manuscr. 79, No. 3-4, 283-298 (1993; Zbl 0803.14019)], where the case \(n = 3\) was treated. An explicit construction for all possible congruences that they found is also given. congruence of lines; Grassmannian; fundamental curve Arrondo, E., M. Bertolini and C. Turrini: Classi cation of smooth congruences with a fundamental curve. Projective Geometry with applications. Number 166 in LN. Marcel Dekker, 1994 Families, moduli of curves (algebraic), Grassmannians, Schubert varieties, flag manifolds Classification of smooth congruences with a fundamental curve | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians The author proves the following result for curves in positive characteristic. Fix integers \(g,d,h,\delta\) with \(g \geq 2\), \(d\geq 3g\) and \(0<\delta \leq h+\delta<d-g\). Fix a smooth curve \(X\) with genus \(g\) and \(L \in \text{Pic}^ d(X)\) such that the complete linear system has gap sequence (1,2,3,...). Then there is no base point free linear subspace \(V\) of \(H^ 0(X,L)\) such that \(h^ 0(X,L)-\dim (V)=\delta\) and gap sequence \(b_ i\) such that \(b_ h=h+\delta\). characteristic \(p\); gap sequence; base point free linear subseries Arithmetic ground fields for curves, Finite ground fields in algebraic geometry, Divisors, linear systems, invertible sheaves, Picard groups, Riemann surfaces; Weierstrass points; gap sequences On the general osculating flag of a projection of a curve in char\( p\) | 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 Here we consider the postulation of general unions in \(\mathbb P^r\) of a smooth non-special curve \(C\) and several secant lines of \(C\). postulation; secant line; maximal rank Plane and space curves, Families, moduli of curves (algebraic), Singularities of curves, local rings Unions of curves and secant lines with maximal rank | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians We suggest a general method of computation of the homology of certain smooth covers \(\hat{\mathcal{M}}_{g, 1}(\mathbb{C})\) of moduli spaces \(\mathcal{M}_{g, 1}(\mathbb{C})\) of pointed curves of genus \(g\). Namely, we consider moduli spaces of algebraic curves with level \(m\) structures. The method is based on the lifting of the Strebel-Penner stratification of \(\mathcal{M}_{g, 1}(\mathbb{C})\). We apply this method for \(g \leq 2\) and obtain Betti numbers; these results are consistent with \textit{R. C. Penner} [Commun. Math. Phys. 113, 299--339 (1987; Zbl 0642.32012); J. Differ. Geom. 27, No. 1, 35--53 (1988; Zbl 0608.30046)] and \textit{J. Harer} and \textit{D. Zagier} [Invent. Math. 85, 457--485 (1986; Zbl 0616.14017)] results on Euler characteristics. moduli spaces of algebraic curves; Betti numbers; dessins d'enfants Dunin-Barkowski, P.; Popolitov, A.; Shabat, G.; Sleptsov, A., On the homology of certain smooth covers of moduli spaces of algebraic curves, Differential Geom. Appl., 40, 86-102, (2015) Families, moduli of curves (algebraic), Dessins d'enfants theory, Coverings of curves, fundamental group, Coverings in algebraic geometry, Homotopy theory and fundamental groups in algebraic geometry, Stratifications in topological manifolds On the homology of certain smooth covers of moduli spaces of algebraic 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 moduli space of five lines in \(\mathbb{P}^2\) can be described by a quintic Del Pezzo surface in \(\mathbb{P}^5\). Given five fixed lines in \(\mathbb{P}^3\) and a fixed plane, we define a map from \(\mathbb{P}^3\) to the quintic Del Pezzo surface by projecting the lines to the fixed plane, and taking the point on the Del Pezzo surface defined by the image lines as the image of the point of projection. We show that the fibers of this map are twisted cubic curves. Conversely, we show that the moduli space of curves in \(\mathbb{P}^3\) with the five fixed lines as secants can be seen as isomorphic to the quintic Del Pezzo surface. moduli space of five lines; projection; quintic Del Pezzo surface Projective techniques in algebraic geometry, Families, moduli of curves (algebraic) Projection of five lines in 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 [For the entire collection see Zbl 0594.00010.]
Let C be a complex projective algebraic plane curve of degree n of equation \(F(x)=0\) \((x=(x_ 0,x_ 1,x_ 2))\). A determinantal representation of C is a matrix \(U(x)\) of order n with linear entries in \(x_ 0,x_ 1,x_ 2\), satisfying \(\det (U(x))=c\cdot F(x)\), \(c\neq 0\). It is known that a general complex curve admits a determinantal representation. For real curves, i.e. when F has real coefficients, one can consider s.a. (self-adjoint) determinantal representations, by requiring that \(U^*(x)=U(x)\) and that \(\det(U(x)) = c\cdot F(x)\), \(c>0\). Two s.a. determinantal representations \(U_ 1, U_ 2\) of degree \(n\) are called equivalent if there exists a matrix \(P\in GL(n,{\mathbb{C}})\) such that \(U_ 2=PU_ 1P^*\). The author classifies the s.a. determinantal representations of real elliptic cubics, up to equivalence. The case of rational cubics was already studied by the author in a previous paper [Math. Theory of Network and Systems, Proc. Int. Symp., Beer-Sheva/Israel 1983, Lect. Notes Control Inf. Sci. 58, 882-898 (1984; Zbl 0577.14040)]. By reducing the cubics to their Weierstraß canonical forms, he shows that the complete set \({\mathcal S}\) of s.a. canonical represenations is equivalent either to a punctured circle or to a disjoint union of a circle and a punctured circle. Furthermore he investigates elementary transformations which allow him to obtain from a given s.a. determinantal representation all the non-equivalent ones in the same component of \({\mathcal S}\). Finally he introduces and studies a notion of definiteness for s.a. representations and relates this concept to the components of \({\mathcal S}.\)
[See also the paper announced below.] determinantal representations of real elliptic cubics V. VINNIKOV, \textit{Self-adjoint determinantal representations of real irreducible cubics}, Operator Theory: Advances and Applications, 19 (1986), 415--442. Determinantal varieties, Matrices over function rings in one or more variables, Families, moduli of curves (algebraic) Self-adjoint determinantal representations of real irreducible cubics | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians We compute the Artin invariant of the minimal resolutions of supersingular weighted Delsarte K3 surfaces. Consequently, we construct K3 surfaces with Artin invariant 10. It is one of the two values for the invariant that were not realized in the paper by \textit{T. Shioda} [J. Reine Angew. Math. 381, 205-210 (1987; Zbl 0618.14014)]. Néron-Severi group; Artin invariant; minimal resolutions; supersingular weighted Delsarte K3 surfaces \(K3\) surfaces and Enriques surfaces, Singularities of surfaces or higher-dimensional varieties, Picard groups Supersingular \(K3\) surfaces with Artin invariant 10 | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is 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 \(d\geq n\geq 2\) let \(C(d,n)\) be the bound of Guido Castelnuovo (1865-1952) on the genus of a curve in \(\mathbb{P}_n\), i.e. \(C(d,n)= m(2d-2- n(n-1) (m+1)/2\), where \(d-1= m(n-1)+\varepsilon\) with \(0\leq \varepsilon <n-1\). The problem of finding a non-degenerate irreducible curve of degree \(d\) in \(\mathbb{P}_n\) with \(s\) nodes as its only singularities, for any \(s\leq C(d,n)\), was solved over the complex field in case \(n=2\) by Francesco Severi (1879-1961) [\textit{F. Severi}, ``Vorlesungen über algebraische Geometrie'' (Leipzig 1921; JFM 48.0687.01); Anhang F], and in case \(n\geq 3\) by \textit{A. Tannenbaum} [Math. Ann. 240, 213-221 (1979; Zbl 0385.14008) and Compos. Math. 41, 107-126 (1980; Zbl 0399.14018)].
The analysis of the real case distinguishes between real nodes with real branches, isolated real nodes and imaginary nodes. The author proves the following theorem:
Given integers \(\alpha, \beta, \gamma\) such that \(\alpha+ \beta+ 2\gamma\leq C(d,n)\), there exists a nondegenerate irreducible nodal curve of degree \(d\) in \(\mathbb{P}_n\), having \(\alpha\) real nodes with real branches, \(\beta\) isolated real nodes and \(2\gamma\) imaginary nodes.
The case \(n=2\) was solved by \textit{E. Shustin} [Topology 32, No. 4, 845-856 (1993; Zbl 0845.14017)], who extended methods due to \textit{O. Ya. Viro} [``Constructing real algebraic varieties with prescribed topology'' (Russian) (Thesis, LOMI, Leningrad); English translation: ``Patchworking real algebraic varieties'' (Uppsala Univ. 1985)]; see also \textit{J.-J. Risler} [Séminaire Bourbaki 1992/93, Exposé No. 763, Astérisque 216, 69-86 (1993; Zbl 0824.14045)]. Some particular cases have been already considered by \textit{D. Pecker} in Compos. Math. 87, 1-4 (1993; Zbl 0783.14013) and in Bull. Sci. Math., II. Sér. 118, No. 5, 475-484 (1994; Zbl 0829.14027). The method consists in constructing some rational curves with the maximal number of double points on ruled smooth surfaces and then in simplifying them by means of arguments which go back to the paper of \textit{L. Brusotti} [Atti Reale Accad. Lincei, Rend. Cl. Sci. Fis. Mat. Natur., 30, No. 1, 375-379 (1921; JFM 48.0729.02)]. real space curves; real nodes; Castelnuovo bound; JFM 48.0687.01; JFM 48.0729.02 Plane and space curves, Topology of real algebraic varieties, Singularities of curves, local rings, Families, moduli of curves (algebraic) Note about the reality of double points of space curves | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians Let X be a smooth projective complex curve of genus \(\geq 3\). It is known that when n and d are two relatively prime integers, the second intermediate Jacobian of the moduli space \(M(n,d,X)_ L\) of semi-stable vector bundles of rank \(n\) and degree \(d\) on X with fixed determinant \(L,\) is isomorphic to the principally polarized Jacobian J(X) of X.
This article studies the case \(n=2\) and \(d=0\). The moduli space is no longer smooth, but Seshadri constructed a natural desingularization N(X) as a moduli space of parabolic semi-stable vector bundles of rank 4 on X. It turns out that J(X) and \(J^ 2(N(X))\) are isomorphic. More precisely, the (3,1) component of the second Chern class of the universal rank 4 vector bundle on \(N(X)\times X\) induces a map \(\phi: J(X)\to J^ 2(N(X))\) which is twice an isomorphism. The proof uses clever geometric constructions to reduce this result to the one by Mumford and Newstead that the similar map \(J(X)\to J^ 2(M(2,-1,X)_ L)\) induced by the second Chern class of the universal bundle is an isomorphism. Then, the author proves that there is a canonical polarization of \(J^ 2(N(X))\). Unfortunately, contrary to what the theorem in the introduction asserts, its pull-back under \(\phi\) cannot be the canonical principal polarization on J(X), since \(\phi\) has a nontrivial kernel. What the author actually proves is the weaker statement that this pull-back is a (unknown) multiple of the principal polarization of J(X). intermediate Jacobian; principally polarized Jacobian; moduli space V. Balaji, Intermediate Jacobian of some moduli spaces of vector bundles on curves , Amer. J. Math. 112 (1990), no. 4, 611-629. JSTOR: Jacobians, Prym varieties, Families, moduli of curves (algebraic), Picard schemes, higher Jacobians, Vector bundles on curves and their moduli Intermediate Jacobian of some moduli spaces 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 We study the connection between characteristics of moduli spaces of Riemann surfaces with marked points aud matrix models. The Kontsevich matrix model describes intersection indices on continuous moduli spaces, and the Kontsevich-Penner matrix model describes intersection indices on discretized moduli spaces. Analyzing the constraint algebras satisfied by various generalized Kontsevich matrix models, we derive time transformations that establish exact relations between different models appearing in mathematical physics. We solve the Hermitian one-matrix model using the moment technique in the genus expansion and construct a recursive procedure for solving this model in the double scaling limit. Families, moduli of curves (algebraic), String and superstring theories; other extended objects (e.g., branes) in quantum field theory, Continuum limits in quantum field theory, Relationships between algebraic curves and physics Matrix models: Geometry of moduli spaces and exact solutions | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians Frobenius manifolds are proved to have lots of applications in enumerative geometry, singularity theory, integrable hierarchies, and string theory. Givental constructed a group action on the space of Frobenius manifolds allowing to transfer known results from a Frobenius manifold to another in the same orbit. Losev proposed a reconstruction of a part of the structure of a Frobenius manifold from a germ of a pencil of flat connections. It involves a commutativity equation reflecting the topology of the Losev-Manin compactification of the moduli space of genus 0 curves. This is analogous to the famous WDVV equation arising from the topology of the Deligne-Mumford compactification of the moduli space of genus 0 curves.
The paper under review studies an analog of Givental's group action on the space of solutions of the commutativity equation. Four equivalent formulations of this group action are presented: {\parindent=0.7cm\begin{itemize}\item[i)] in terms of the cohomology classes on Losev-Manin moduli spaces, \item[ii)] in terms of differential operators acting on formal matrix Gromov-Witten potential, \item[iii)] in terms of a linear algebraic interpretation of the descendant of the commutativity equation, \item[iv)] in terms of \(\tau\)-functions of multi-component Kadomtsev-Petviashvili hierarchies.
\end{itemize}} It is shown that the last formulation is equivalent to the Losev-Polyubin classification of solutions of the commutativity equation obtained via dressing transformation technique. Frobenius manifolds; cohomological field theory; commutativity equation; Losev-Manin compactification; Givental's group action; Kadomtsev-Petviashvili hierarchy Shadrin, S., Zvonkine, D.: A group action on Losev--Manin cohomological field theories. arXiv:0909.0800v1, 1--21 Gromov-Witten invariants, quantum cohomology, Frobenius manifolds, Families, moduli of curves (algebraic) A group action on Losev-Manin cohomological field theories | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians See the preview in Zbl 0521.14009. semistable degenerate orthogonal bundles; semistable symplectic bundles; vector bundles on curve; ramified covering; moduli spaces; quadratic forms; symplectic forms Bhosle U, Degenerate symplectic and orthogonal bundles on \(\mathbb{P}\)1,Math. Ann. 267 (1984) 347--364 Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20], Algebraic moduli problems, moduli of vector bundles, Families, moduli of curves (algebraic), General binary quadratic forms, Coverings in algebraic geometry Degenerate symplectic and orthogonal bundles on \({\mathbb{P}}^ 1\) | 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 \(\tilde{\mathcal{M}}_{g,n}\) denote the moduli space of stable \(n\)-pointed curve. It is defined over a prime field and a twisted form of it over a field \(K\) is a variety \(Y\) defined over \(K\) such that \(\tilde{\mathcal{M}}_{g,n}\) and \(Y\) are isomorphic over the algebraic closure of \(K\). This papers that any \(F\)-form of \(\tilde{\mathcal{M}}_{g,n}\) is rational (characteristic \(\ne 2\)) if either \(g=1\) and \(3\le n\le 4\) or \(g=2\) and \(2\le n\le 3\) or \(g=3\) and \(1\le n\le 14\) or \(g=4\) and \(1\le n\le 9\) or \(g=5\) and \(1\le n\le 12\). Earlier [\textit{M. Florence} and \textit{Z. Reichstein} Bull. Lond. Math. Soc. 50, No. 1, 148--158 (2018; Zbl 1428.14084)] it was proved (char \(\ne 2\)) that all \(F\)-forms of \(\tilde{\mathcal{M}}_{0,n}\) are unirational, for \(n\) odd all are rational, but for \(n\) even there are fields \(F\) for which there are \(F\)-forms which are not retract rational or stably rational). The paper also contains the relevant literature and tools (in particular for the authomorphisms over the field \(F\) of \(\tilde{\mathcal{M}}_{g,n}\) [\textit{A. Massarenti}, J. Lond. Math. Soc., II. Ser. 89, No. 1, 131--150 (2014; Zbl 1327.14133); \textit{B. Fantechi} and \textit{A. Massarenti}, Int. Math. Res. Not. IRMN 2017, No. 8, 2431--2463 (Zbl 1428.14084)]. F-form; moduli space of pointed stable curves; rationality; twisted form of moduli spaces. Rationality questions in algebraic geometry, Families, moduli of curves (algebraic), Other nonalgebraically closed ground fields in algebraic geometry, Special algebraic curves and curves of low genus On the rationality problem for forms of moduli spaces of stable marked curves of positive genus | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians A complex perturbation of an analytic circle diffeomorphism is a biholomorphic map whose fundamental domain is an elliptic curve. V. I. Arnold conjectured that the modulus of this curve tends to the rotation number the of originating diffeomorphism as the perturbation parameter tends to zero. In this note, Arnold's conjecture is proved for circle diffeomorphisms that are analytically equivalent to a rotation. The proof is carried out by rising the theory of quasiconformal mappings. V. S. Moldavskiĭ, Moduli of elliptic curves and rotation numbers of diffeomorphisms of the circle, Funct. Anal. Appl., 35, 234, (2001) Families, moduli of curves (algebraic), Elliptic curves, Dynamical systems involving maps of the circle, Dynamical systems involving smooth mappings and diffeomorphisms, Rotation numbers and vectors Moduli of elliptic curves and rotation numbers of circle diffeomorphisms | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is 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 \(l\) be a prime number. In the paper, we study the outer Galois action on the profinite and the relative pro-\(l\) completions of mapping class groups of pointed orientable topological surfaces. In the profinite case, we prove that the outer Galois action is faithful. In the pro-\(l\) case, we prove that the kernel of the outer Galois action has certain stability properties with respect to the genus and the number of punctures. Also, we prove a variant of the above results for arbitrary families of curves. mapping class group; outer Galois representation; hyperbolic curve Iijima, Yu, Galois action on mapping class groups, Hiroshima Math. J., 45, 2, 207-230, (2015) Coverings of curves, fundamental group, Families, moduli of curves (algebraic) Galois action on mapping class groups | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians The fundamental theorems of classical Brill-Noether theory of smooth projective curves have not been extended yet to stable curves, due to the technical difficulties of combinatorial nature. The goal of this paper is to extend some of them to binary curves. A binary curve is a stable nodal curve having two irreducible rational components, intersecting at \(g+1\) points. Their moduli space \(B_g\subset \overline{M_g}\) is irreducible of dimension \(2g-4\). The analogue of theorems of Riemann, Clifford and Martens are proved to hold for any binary curve and for line bundles parametrized by the compactified Jacobian scheme. An analogue of Brill-Noether theorem is proved for general binary curves and for \(r\leq 2\). More precisely, let
\[
B_d(g)=\{(d_1,d_2) \mid d_1+d_2=d, \frac{d-g-1}{2}\leq d_i\leq \frac{d-g+1}{2}, i=1,2\}
\]
be the set of balanced multidegrees.
If \(\underline{d}\in B_d(g)\), put \(W^r_{\underline{d}}(X)= \{L\in Pic^{\underline{d}}(X)\mid h^o(L)\geq r+1\}\), where \(X\) is a binary curve. Then it is proved that, for a general binary curve \(X\) and for \(r\leq 2\), \(\dim W^r_{\underline{d}}(X)\leq \rho_d^r(g)\), the usual Brill-Noether number, and equality holds for some \(\underline{d}\). Moreover \(\dim \overline{W^r_{\underline{d}}(X)}= \rho_d^r(g)\). stable curve; moduli space; Clifford theorem; Picard scheme; Brill-Noether Caporaso L.: Brill-Noether theory of binary curves. Math. Res. Lett. 17(2), 243--262 (2010) Families, moduli of curves (algebraic), Divisors, linear systems, invertible sheaves, Special divisors on curves (gonality, Brill-Noether theory) Brill-Noether theory of binary 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 gives a very nice survey on the geometric theory of theta characteristics. Special attention is given to the historical development of the subject. The first 2 sections deal with the classical theory, interpreting a theta characteristic as a quadric in a vector space over the field of 2 elements. In the third section, Cornalba's compactification of the moduli spaces of stable spin curves as well as an important result of Ludwig concerning their resolution of singularities are outlined. In the fourth section some ways for construction effective divisors of these moduli spaces and the computation of their cohomology classes are discussed. The fifth section presents unirational parametrizations of these moduli spaces in small genus, using Mukai's models of special \(K3\) surfaces. The results of these 2 sections imply the main result of the paper, namely the birational classification of the moduli spaces of even and odd spin curves, due to the author and Verra. The last section gives some open problems related to syzygies and Brill-Noether stratification of these moduli spaces. Moreover an application to string theory is explained. theta characteristics; spin curves; moduli of spin curves Kusner, R., Schmitt, N.: The spinor representation of minimal surfaces. arXiv:dg-ga/9512003v1 (1995) Jacobians, Prym varieties, Families, moduli of curves (algebraic) Theta characteristics and their 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 A plane quadrangle \(ABCD\) has three diagonal points \(X\cong BC{}_\wedge AD\), \(Y\cong CA{}_\wedge BD\), \(Z\cong AB{}_\wedge CD\). \(AB, AC, AD\) have, in the pencil of lines through \(A\), a Hessian pair \(a, a'\) which cut \(BC\) at the Hessian duad of the triad \(B,C,X\), this duad being also on the Hessian pair \(d,d'\) of \(DA, DB, DC\); and so on. The pencil of conics through \(A, B, C, D\) includes members \(\Omega, \Omega'\) with tangents \(a,a'\); the notation is chosen so that \(b, c, d\) touch \(\Omega\) while \(b', c', d'\) touch \(\Omega'\). There is a pencil \(P\) of four-nodal plane sextics having nodes at \(A, B, C, D\) with nodal tangents \(a, a', b, b', c, c', d, d'\), and having for their further intersections with \(BC, CA, AB, AD, BD, CD\) the Hessian duads alluded to above. The two composite sextics \(\Omega a'b'c'd'\), \(\Omega' abcd\) belong to \(P\), as do a pair \(R, R'\) of ten-nodal (and so rational) curves. But the most interesting member of \(P\) is a curve \(W\), first encountered by \textit{A. Wiman} [Math. Ann. 48, 195--240 (1897; JFM 30.0600.01)] admitting a group of 120 Cremona self-transformations. Wiman did not give the above geometrical derivation of \(W\), nor did he mention \(P\). Equations for the curves on \(P\), and of some quadratic transformations, are found.
But the true significance of this geometry appears only when the investigation broadens to involve the del Pezzo quintic surface \(F\), in [\textit{F. Bath}, Proc. Camb. Philos. Soc. 24, 48--55 (1928; JFM 54.0718.04)], that is mapped on the plane by the cubics through \(A,B,C, D\). Each of the ten lines on \(F\) meets three others; the triad of intersections has a Hessian duad \(\delta\) and \(P\) maps a pencil \(P^*\) of curves of order ten on \(F\), with the ten \(\delta\) for its twenty base points. One member of \(P^*\) consists of the ten lines, two other members each of five conics, two others are rational, each having six nodes. Every other member is non-singular, and so canonical of genus 6, one of them, \(\Gamma\), being the canonical model of \(W\). \(\Gamma\) is thus uniquely determined when \(F\) is given and so admits, with \(F\), a group of 120 self-projectivities. This group is isomorphic to the symmetric group of degree five, and indeed permutes the five pencils of conics on \(F\). While a curve of genus 6 has, in general, fifteen moduli \(\Gamma\) has no free modulus at all. pencil of lines; Hessian pair; Hessian duad; Cremona self-tansformations; del Pezzo quintic surface Edge, WL, A pencil of four-nodal plane sextics, Math. Proc. Cambridge Philos. Soc., 89, 413-421, (1981) Projective analytic geometry, Projective techniques in algebraic geometry, Enumerative problems (combinatorial problems) in algebraic geometry, Families, moduli of curves (algebraic), Line geometries and their generalizations, Euclidean geometries (general) and generalizations A pencil of four-nodal plane sextics | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is 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 provide a uniform vanishing result for the graded components of the finite length Koszul module associated to a subspace \(K\subseteq \bigwedge^2 V\) as well as a sharp upper bound for its Hilbert function. This purely algebraic statement has interesting applications to the study of a number of invariants associated to finitely generated groups, such as the Alexander invariants, the Chen ranks, and the degree of growth and virtual nilpotency class. For instance, we explicitly bound the aforementioned invariants in terms of the first Betti number for the maximal metabelian quotients of (1) the Torelli group associated to the moduli space of curves, (2) nilpotent fundamental groups of compact Kähler manifolds, and (3) the Torelli group of a free group. Chen ranks; 1-formal group; Koszul module; metabelian group; resonance variety; Alexander module; lower central series; virtually nilpotent group; Torelli group; Kähler group Topological methods in group theory, 2-dimensional topology (including mapping class groups of surfaces, Teichmüller theory, curve complexes, etc.), Automorphism groups of groups, Nilpotent groups, Compact Kähler manifolds: generalizations, classification, Families, moduli of curves (algebraic) Topological invariants of groups and Koszul modules | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians Let \(X\) be a smooth hypersurface of degree \(d\) in \(\mathbb{CP}^n\). The Hilbert scheme parametrizing \(k\) dimensional linear subspaces contained in \(X\) is denoted \(F_k(X)\). In particular, \(F_1(X)\) parametrizes lines contained in \(X\) (commonly know as Fano variety of lines on \(X\)). A well known conjecture in the field, due to Debarre and de Jong, states that the variety of lines \(F_1(X)\) has the expected dimension \(2n-3-d\) when \(n\geq d\). It is a classically known result in the field that the conjecture holds for \(d=3\). \textit{A. Collino} [J. Lond. Math. Soc., II. Ser. 19, 257--267 (1979; Zbl 0432.14024)] proved the conjecture for \(d=4\). In an unpublished work, Debarre showed the conjecture to be true when \(d\leq 5\). In [Duke Math. J. 95, No. 1, 125--160 (1998; Zbl 0991.14018)] the three authors proved the statement when \(d\) is small with respect to \(n\). The cases \(d\leq 6\) was settled in [\textit{R. Beheshti}, J. Reine Angew. Math. 592, 1--21 (2006; Zbl 1094.14029); [\textit{J. M. Landsberg} and \textit{C. Robles}, J. Lond. Math. Soc., II. Ser. 82, No. 3, 733--746 (2010; Zbl 1221.14058); \textit{J. Landsberg} and \textit{O. Tommasi}, Mich. Math. J. 59, No. 3, 573--588 (2010; Zbl 1209.14035)]. The cases \(d\leq 8\) were proven by \textit{R. Beheshti} [Math. Ann. 360, No. 3--4, 753--768 (2014; Zbl 1304.14065)].
In the paper under review, the two authors prove the Debarre-de Jong conjecture when \(n\geq 2d-4\). The key ingredient is a result (Lemma 2.1) that provides an upper bound to the dimension of the locus of tangency of a smooth hypersurface with varieties cut out by lower degree equations. The authors also prove a similar result for the space \(F_k(X)\). Namely, they show that if \(n\geq 2\binom{d+k-1}{k}+k\) then \(F_k(X)\) is irreducible of the expected dimension. As an application, it is proven the unirationality of smooth hypersurfaces for which the inequality \(n\geq 2^{d!}\) holds. birational geometry; hypersurfaces; unirationality Rationality questions in algebraic geometry, Families, moduli of curves (algebraic) Linear subspaces of 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 [For the entire collection see Zbl 0744.00029.]
Each Siegel modular form \(f(\tau)\) of degree \(g\) and weight \(k\) possesses a Fourier-Jacobi expansion of the form
\[
f(\tau)=\sum^ \infty_{\mu=0} a_ \mu(\tau',z) e^{2\pi i\mu w},\quad\tau=\begin{pmatrix}\tau'&z\\t_ z&w\end{pmatrix},
\]
where \(w\) belongs to the upper half-plane in \(\mathbb{C}\). It is well-known that there exists a constant \(\rho_ g\) such that \(f\) vanishes identically whenever \(a_ \mu(\tau',z)\equiv 0\) for all \(\mu\leq k/\rho_ g\). The precise values of \(\rho_ g\) were known for \(g=1,2,3\). The author demonstrates \(\rho_ 4=8\) as an application of results of \textit{J. Harris} and \textit{I. Morrison} [Invent. Math. 99, 321-355 (1990; Zbl 0705.14026)]. Moreover he shows that there essentially exists only one modular form with \(a_ \mu(\tau',z)\not\equiv 0\) for \(8\mu=k\), namely Schottky's polynomial \(J(\tau)\). The geometric meaning of \(\rho_ g\) as a slope in a particular moduli space is explained. theta series; slopes; Siegel modular form; Fourier-Jacobi expansion; Schottky's polynomial; moduli space Salvati Manni, R.: Modular forms of the fourth degree. (Remark on a paper of Harris and Morrison). Proc. Conf., Trento/Italy 1990, Lect. Notes Math., vol. 1515, pp. 106--111 (1992) Siegel modular groups; Siegel and Hilbert-Siegel modular and automorphic forms, Families, moduli of curves (algebraic), Divisors, linear systems, invertible sheaves Modular forms of the fourth degree. (Remark on a paper of Harris and Morrison) | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians Curves of genus \( g \geq 3\) over \( C\) with a level-2 structure are parametrized by \( M_g(2)= H_g / \Gamma_g(2) \) where \(H_g\) is Siegel upper half space and \(\Gamma_g(2)\) is the congruence level-2 group. \(M_g(2) \) parametrizes the set of pairs \((C, \sigma)\) where \(C\) is a curve of genus \(g\) and \( \sigma: (\mathbb{Z}/2\mathbb{Z})^g \rightarrow H_1(C, \mathbb{Z}/2\mathbb{Z})\) is a symplectic isomorphism. For \(g \leq 5 \) the hyperelliptic locus of \(M_g(2)\) has been studied as e.g.~ for \(g= 5\), \textit{R. D. M. Accola} [in: Contributions to Analysis, Academic Press, New York, 11--18 (1974; Zbl 0313.32029)] has established that with a condition on the three characteristics the intersection of the corresponding divisors is a union of irreducible hyperelliptic components. The author states and claims to prove that this fails for genus \(6\) in theorem 1.1 which states that each subvariety of \(M_6(2)\) intersection of four thetanull divisors contains an element which is not hyperelliptic and in theorem 1.2 he states and proves that every irreducible component of the hyperelliptic locus in \(M_g(2)\) is a connected component of the intersection of \( g-2\) thetanull divisors; both are the main theorems of this paper.
The paper is organized as follows.~In section two, the main definitions on the theory of abelian varieties and theta divisors and basic notation is introduced. In section three, the basic notions of theta characteristics and properties of symplectic torsors are given. In section four, the authors states and proves various lemmas used to prove theorem 1.1. In section 4.1, the author states and proves proposition 4.2 which states that for a curve \(C\) of genus \(6\) the action of \(Sp(J_2(C))\) on the sets of four even distinct theta characteristics on \(C\) has at most four orbits defined by four conditions precisely stated in this proposition.~Section 4.2 ends with proposition 4.6 which assumes the existence of a bi-elliptic curve \textit{a priori} , does not prove the statement given as theorem 1.1 very clearly.~In section five, the author prove s theorem 1.2 for which he proves that for each \((C, \sigma )\) a hyperelliptic element of \(M_g(2)\) with \( g \geq 3 \) the \(g-2\) thetanull divisors associated to the theta characteristics intersect transversally at \((C, \sigma)\) in proposition 5.2.~In the course of the proof of theorem 1.2 the author describes each irreducible component of \(H_g(2)\) for each open subvariety \(U\) of \(C^{2g + 2}\) of points with distinct coordinates by defining it very explicitly in terms of each point \( \xi \in U\) in terms of remark 3.6 stated in section three.~The author concludes the proof using proposition 5.2. families; algebraic moduli; theta functions; Schottky problem; special curves and curves of low genus Siegel modular groups; Siegel and Hilbert-Siegel modular and automorphic forms, Families, moduli of curves (algebraic), Theta functions and curves; Schottky problem, Special algebraic curves and curves of low genus Vanishing thetanull and hyperelliptic curves | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians Let \(X\subset \mathbb{P}^n\) be a general complex hypersurface of degree \(d\). It is known that if \(d\) is sufficiently large, \(d\geq 2n - 2\), then \(X\) does not contain rational curves [see \textit{H. Clemens}, Ann. Sci. Éc. Norm. Supér., IV Sér. 19, No.~4, 629--636 (1986; Zbl 0611.14024); \textit{C. Voisin}, J. Differ. Geom. 44, 200--213 (1996; Zbl 0883.14022) and 49, No. 3, 601--611 (1998; Zbl 0994.14026)]. For \(d = 2n - 3\), \(n\geq 6\), \(X\) contains a finite number of lines but does not contain rational curves of degree \(e \geq 2\) [\textit{G. Pacienza}, J. Algebr. Geom. 12, 245--267 (2003; Zbl 1054.14057)]. In this paper the authors prove the following relevant result for low degree: For \(n\geq 2\) and \(d < (n + 1)/2\), a general complex hypersurface \(X\) in \(\mathbb P^n\) of degree \(d\) has the property that for each integer \(e\geq 1\) the scheme \(R_e(X)\) parametrizing degree \(e\) smooth rational curves on \(X\) is an integral local complete intersection scheme of ``expected'' dimension \((n+1-d)e +(n-4)\). The scheme \(R_e(X)\) is embedded as an open subscheme in the Kontsevich moduli space \(\overline{\mathcal M}_{0,0}(X,e)\) parametrizing stable maps to \(X\) and a partition of \(\overline{\mathcal M}_{0,0}(X,e)\) into locally closed subsets is used as in \textit{K. Behrend} and \textit{Yu. Manin} [Duke Math. J. 85, 1--60 (1996; Zbl 0872.14019)]. The authors use classical results about lines on hypersurfaces and include a new result about flatness of the projection map from the space of pointed lines. Moreover they use the deformation theory of stable maps, properness of the stack \(\overline{\mathcal M}_{0,r}(X,e)\) and a version of Mori's bend-and-break lemma. The authors use dual graphs associated to pointed curves and stable \(A\)-graphs as in the above article of K. Behrend and Yu. Manin (loc. cit.). dual graphs; Hilbert schemes; Kontsevich moduli spaces of stable maps; stacks Harris, J; Roth, M; Starr, J, Rational curves on hypersurfaces of low degree, J. Reine Angew. Math., 571, 73-106, (2004) Families, moduli of curves (algebraic), Algebraic moduli problems, moduli of vector bundles, Hypersurfaces and algebraic geometry, Parametrization (Chow and Hilbert schemes) Rational curves on hypersurfaces of low 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 This paper is concerned with the explicit computation of the cohomology of the tautological ring \(R^\bullet(C_g^n)\), where \(C_g^n\) is the moduli space of genus \(g\) smooth curves with \(n\) non necessarily distinct marked points. The authors are able to describe the entire cohomology, when \(g\leq 4\). There are two significative applications of these results: first, the authors provide a proof of Faber conjecture when \(g\leq 4\) and make some progress in understanding the likely failure of the conjecture when \(g\) is big; on the other side, a counterexample to a conjecture of Morita is given.
The main technical idea behind the paper is that the cohomological information on \(R^\bullet(C_g^n)\) can be packaged efficiently by considering Chow motives on \(M_g\). In fact, if one consider the forgetful map
\[
f:C_g^n\longrightarrow M_g,
\]
it is possible to decompose \(R^qf_*\mathbb{Q}\) as a direct sum of local systems \(V_\lambda\), indexed by \(\operatorname{Sp}(2g)\), and pulled back from \(A_g\) -- the moduli space of principally polarized abelian varieties -- via the Torelli map. Passing to the motivic setting, one can decompose the Chow motive \(h(C_g^n/M_g)\) into Chow motives \(\mathbf{V}_\lambda\), each one of them being a motivic lift of the corresponding local system \(V_\lambda\).
As a biproduct of their method, the author can also give precise information about the cohomology ring \(H^\bullet(C_g^n,\mathbb{Q})\). For these results, the motivic setting is an overkill, as they can already be obtained via the Leray spectral sequences by considering the decomposition of \(R^qf_*\mathbb{Q}\) in local systems. Although the paper is wirtten in the language of Chow motives, the authors provide an useful dictionary which would allow the reader unfamiliar with this setting -- but used to handling local systems -- to understand the results at the level of cohomology groups. tautological rings; moduli spaces of curves; Chow motives; twisted commutative algebras Families, moduli of curves (algebraic), Algebraic cycles, Intersection theory, characteristic classes, intersection multiplicities in algebraic geometry, Classifying spaces of groups and \(H\)-spaces in algebraic topology Tautological classes with twisted coefficients | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is 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 we prove (using reducible curves and an inductive procedure) the following results.
Theorem 1 (The maximal rank conjecture in \(\mathbb P^3)\): If \(d\geq (3g+12)/4\), \(g\geq 0\), there is a smooth connected curve \(C\subset \mathbb P^3\) of degree \(d\), genus \(g\), with general moduli and of maximal rank (i.e. for all \(k\) the restriction maps \(H^0(\mathbb P^3,\mathcal O_{\mathbb P^3}(k))\to H^0(C,\mathcal O_C(k))\) have maximal rank).
Theorem 2: There is a function \(u_3\) with \(\lim_{g\to \infty}u_3(g)= \frac12\) such that for every \(d\geq gu_3(g)\), there is a generally smooth component \(W(d,g;3)\) of \(\text{Hilb}(\mathbb P^3)\) with the right number of moduli and such that a general element of \(W(d,g;3)\) has maximal rank.
Later we extend Theorem 2 to \(\mathbb P^n\), \(n>3\), with \(\lim_{g\to \infty}u_n(g)=(n-2)/(n-1)\).
[For the entire collection see Zbl 0614.00006.] deformation; space curve; postulation; Hilbert scheme; moduli of curves; maximal rank conjecture E. Ballico and Ph. Ellia, Beyond the maximal rank conjecture for curves in \(\mathbf P^3\) , in Space curves , Lecture Notes in Math., vol. 1266, Springer-Verlag, Berlin, 1987, pp. 1-23. Special algebraic curves and curves of low genus, Projective techniques in algebraic geometry, Families, moduli of curves (algebraic) Beyond the maximal rank conjecture for curves in \(\mathbb P^3\) | 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 \textit{V. Mercat} conjectured in [Int. J. Math. 13, No. 7, 785--796 (2002; Zbl 1068.14039)] that for a smooth curve \(C\) the Clifford Index for semistable vector bundles of rank \(n \geq 2\) equals the Clifford index for line bundles on \(C\). Several counter examples have been given for ranks 2 and 3. The present paper considers this question for ranks 4 and 5. In the first theorem it is shown that the conjecture fails for rank-4 bundles for any curve of genus \(g \geq 20\) which lies on a \(K3\)-surface with cyclic Picard group. The bundles which give the counter examples are restricted Lazarsfeld-Mukai bundles on the \(K3\)-surface. With another idea the authors consider the rank-5 case. They consider the normal bundle of a canonical curve of genus 7 in projective 6-space and show that it is stable to give counter examples for rank 5. In particular the conjecture fails for a general curve of genus 7. The stability of the normal bundle is interesting also from several different points of view. The authors conjecture that it is stable for a general curve of genus \(g \geq 7\). Brill-Noether theory of vector bundles; Lazarsfeld-Mukai bundle Algebraic moduli problems, moduli of vector bundles, Families, moduli of curves (algebraic), \(K3\) surfaces and Enriques surfaces Restricted Lazarsfeld-Mukai bundles and 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 The authors study a conjecture of Raskind's, which may be thought of as an analogue of the Tate conjecture. In order to state it, let us first fix some notations: suppose \(X/K\) is a smooth and proper variety over a local field \(K/\mathbb{Q}_p\). Let \(\bar{K}\) denote an algebraic closure of \(K\), and \(G_K\) the Galois group \(\mathrm{Gal}(\bar{K}/K)\). We first consider codimension one cycles, and consider the cycle class map
\[
c: \mathrm{NS}\otimes_{\mathbb{Z}} \mathbb{Q}_p\rightarrow H^2_{\text{ét}}(X_{\bar{K}}, \mathbb{Q}_p(1))^{G_K},
\]
where \(\mathrm{NS}\) denotes the Néron-Severi group, and the target denotes the Galois invariants in (Tate-twisted) étale cohomology.
For varieties over a finite field or a number field, the map \(c\) is predicted to be an isomorphism by the Tate conjecture. Although formally similar, the current setting of \(X\) over local fields is in fact rather different; here, the map \(c\) is injective but easily shown to be not surjective, and so if one wants an isomorphism it is necessary to impose further conditions on \(X\). This is what Raskind's conjecture aims to do:
Conjecture (Raskind). The map \(c\) is surjective if \(X\) has totally degenerate reduction.
The phrase \textit{totally degenerate reduction} needs to be made precise, but essentially means that \(X\) has bad reduction, and the components of the special fiber (as well as their intersections, and the intersections of those, and so on) have Chow groups as simple as possible. It can be thought of as a maximal unipotent monodromy condition. Raskind's conjecture is in fact more general and deals with cycles of higher codimensions, but we will abusively refer to the above conjecture as Raskind's conjecture in this review.
One key difference between Tate-type conjectures and, for example, the Hodge conjecture to note is that, while the Hodge conjecture would imply that one can pin down the \(\mathbb{Q}\)-vector space of algebraic cycles inside cohomology, the Tate conjecture only allows one to do so after tensoring to \(\mathbb{Q}_{\ell}\). This observation will be relevant in what follows.
The following is one of the main results of this paper
Theorem. There exist abelian surfaces \(B/K\) for which Raskind's conjecture is false.
Let us give an outline of the proof; we will be brief here since this is nicely explained in the Introduction. Using one of Fontaine's equivalence of categories from \(p\)-adic Hodge theory, one can write
\[
H^2_{\text{ét}}(X_{\bar{K}}, \mathbb{Q}_p(1))^{G_K} \cong H \cap \mathrm{Fil}^1_{dR},
\]
where \(H\) is a \(\mathbb{Q}_p\)-vector space (a subspace of \(H^2_{\text{log-cris}}(X)\)), and the intersection is with the Hodge filtration after comparing to de Rham cohomology. On the other hand, \(H\) has a natural \textit{rational structure}, being spanned by algebraic cycles of the special fiber, and Raskind's conjecture amounts to comparing the intersection of this rational structure with \(\mathrm{Fil}^1_{dR}\), and the a priori larger intersection \(H \cap \mathrm{Fil}^1_{dR}.\) Since one has an explicit description of \(H\) and its rational structure, this gives a strategy to prove the Theorem, and the authors explicitly find very clean counterexamples in this way.
The authors also consider the analogue of Raskind's Conjecture for homomorphisms between abelian varieties and find counterexamples using the same strategy as above; they also prove positive results for abelian varieties isogenous to products of Tate elliptic curves. Throughout the paper the authors work in the more general setting of abeloid varieties. The paper under review is very well written, with the ideas clearly presented. Tate conjecture; abelian and abeloid varieties; \(p\)-adic fields and \(p\)-adic uniformisation; \(p\)-adic Hodge theory; filtered \((\varphi, N)\)-module; totally degenerate reduction \(p\)-adic cohomology, crystalline cohomology, Galois representations, Picard groups, Isogeny \(p\)-adic Tate conjectures and abeloid varieties | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians The multiplihedra are a family of polytopes related to the better-known associahedra. Where the \(n\)th associahedron is a CW-complex \(K_n\) with vertices corresponding to ways of fully parenthesizing a string of length \(n\) and edges corresponding to instances of \(a(bc)=(ab)c\), the vertices of the multiplihedron \(J_n\) corresponds to ways of applying a homomorphism piecewise to the string, and its edges correspond to instances of either \(a(bc)= (ab)c\) or \(f(ab)= f(a)f(b)\). The dimension of \(J_n\) is \(n-1\), one higher than that of \(K_n\).
In this paper, the authors consider geometric realizations of the multiplihedra as moduli spaces of stable quilted discs, colored metric ribbon trees, and as the nonnegative real part of a complex moduli space of stable scaled marked curves, while most the basic ideas can be followed by anybody familiar with the representation of an association pattern as a rooted tree, the reader who is approaching this topic from the direction of combinatorics or algebra should be aware that this paper is far from self-contained. Moreover, the terminology used in this branch of geometry is sometimes nonobvious (a ``nodal disc'' is not a disc but a tree [graph] of discs); and the literature tends to assume an insider's knowledge. Given the importance of this subdiscipline and its position on the intersection of (at least) categorical algebra, combinatorics, and conformal field theory, this is regrettable. multiplihedron; associahedron; nodal disk; metric tree; moment polytope; moduli spaces of stable quilted discs; colored metric ribbon trees; stable scaled marked curves S. Ma'u, C. Woodward, \textit{Geometric realizations of the multiplihedra}, Compos. Math. \textbf{146} (2010), no. 4, 1002-1028. , Families, moduli of curves (algebraic), Lattice polytopes in convex geometry (including relations with commutative algebra and algebraic geometry) Geometric realizations of the multiplihedra | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians In a recent paper, \textit{R. Pandharipande}, \textit{J. P. Solomon} and \textit{R. J. Tessler} [``Intersection theory on moduli of disks, open KdV and Virasoro'', Preprint, \url{arXiv:1409.2191}] initiated a study of the intersection theory on the moduli space of Riemann surfaces with boundary. The authors conjectured KdV and Virasoro type equations that completely determine all intersection numbers. In this paper, we study these equations in detail. In particular, we prove that the KdV and the Virasoro type equations for the intersection numbers on the moduli space of Riemann surfaces with boundary are equivalent. Riemann surfaces with boundary; moduli space; KdV equations A. Buryak, \textit{Equivalence of the open KdV and the open Virasoro equations for the moduli space of Riemann surfaces with boundary}, arXiv:1409.3888 [INSPIRE]. KdV equations (Korteweg-de Vries equations), Families, moduli of curves (algebraic) Equivalence of the open KdV and the open Virasoro equations for the moduli space of Riemann surfaces with boundary | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is 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 orientable cover of the moduli space of real genus zero algebraic curves with marked points is a compact aspherical manifold tiled by associahedra, which resolves the singularities of the space of phylogenetic trees. The resolution maps planar metric trees to their underlying abstract representatives, collapsing and folding an explicit geometric decomposition of the moduli space into cubes, endowing the resolving space with an interesting canonical pseudometric. Indeed, the given map can be reinterpreted as relating the \textit{real} and the \textit{tropical} versions of the Deligne-Knudsen-Mumford compactification of the moduli space of Riemann spheres. phylogenetics; configuration spaces; associahedron; tree spaces S. Devadoss and J. Morava, \textit{Navigation in tree spaces}, Adv. Appl. Math., 67 (2015), pp. 75--95. Families, moduli of curves (algebraic), Taxonomy, cladistics, statistics in mathematical biology, \(n\)-dimensional polytopes Navigation in tree spaces | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians Let \(X\) be a complex projective, irreducible, smooth curve of genus \(g\geq 2\), and let \(L\) be a line bundle on \(X\). Write \(SU_X:=SU_X(r,L)\) for the moduli space of semi-stable vector bundles \(F\to X\) such that \(\text{rank}(F)=r\) and \(\det(F)=L\). Let \(\Theta_X\) be the ample generator of \(\text{Pic}(SU_X)\simeq{\mathbb Z}\). One may also consider the relative case \(\mathcal X\to T\) of a family of curves (of genus \(g\)) over the base \(T\), with \(\mathcal X\) smooth projective over \(T\). Fixing a line bundle \(\mathcal L\) on \(\mathcal X\), one obtains the family \(\Pi:\mathcal{SU}_{\mathcal X}\to T\) of relative moduli spaces. For a line bundle \(\Theta_{\mathcal X}\) on \(\mathcal{SU}_{\mathcal X}\) that restricts to \(\Theta_X\) for each fibre \(X\) of \(\Pi\) one may define the direct image sheaves \(\mathcal V_k:=\Pi_*\Theta_{\mathcal X}^k\), \(k\in{\mathbb Z}\). Then the \(\mathcal V_k\) are in fact vector bundles. The following theorem holds: \(\mathcal V_k\) carries a natural connection \(\nabla\) with vanishing curvature. This result was obtained by several people [\textit{A. Tsuchiya, K.Ueno} and \textit{Y. Yamada} in: Integrable systems in quantum field theory and statistical mechanics, Proc. Symp., Kyoto and Kyuzeso 1988, Adv. Stud. Pure Math. 19, 459-566 (1989; Zbl 0696.17010); \textit{N. J. Hitchin}, Commun. Math. Phys. 131, No. 2, 347-380 (1990; Zbl 0718.53021); \textit{S. Axelrod, S. Della Pietra} and \textit{E. Witten}, J. Differ. Geom. 33, No. 3, 787-902 (1991; Zbl 0697.53061); \textit{G. Faltings}, J. Algebr. Geom. 2, No. 3, 507-568 (1993; Zbl 0790.14019)]. In the underlying paper an attempt is made to describe the paper by Faltings.
The set-up is complex analytic. One considers the moduli space \(\mathcal R\) of irreducible representations of the fundamental group of the Riemann surface \(\Sigma\) with (almost) complex structure \(J\) underlying \(X\), into \(SL(2,{\mathbb C})\). Then \(\mathcal R\) is independent of \(J\) and carries a holomorphic symplectic form \(\Omega\), and one shows that \(\mathcal R\) admits a holomorphic line bundle \(\mathcal L\) with holomorphic connection \(\nabla\) with corresponding curvature \(\Omega\). For \(k\in{\mathbb Z}\) one writes \(\nabla^k\) for the connection on \(\mathcal L^k\) induced by \(\nabla\). Let \(\tau_0\) denote the holomorphic volume form corresponding to \(\Omega\), and write \(\tau=e^f\tau_0\) for another volume form, where \(f\) is to be determined properly. There is an open set \(U\subset\mathcal R\) and a map \(\phi:U\to SU^s_X\), where \(SU^s_X\subset SU_X\) is the open set parametrising stable bundles. Reversing the complex structure \(J\mapsto -J\) one writes \(\widetilde{X}\), \(\widetilde{\phi}\), \(\ldots\), etc.\ for the objects, maps, \(\ldots\), etc.\ corresponding to \(-J\). Then the fibres of \(\phi\) and \(\widetilde{\phi}\) are transversal. Let \(\kappa_1\) denote the pull-back to \(U\) of the canonical bundle \(\kappa=\kappa_{SU_X^s}\) of \(SU_X^s\). Then \(\kappa=\Theta^{-c}\) with \(c=4\). Now, on the one hand, one has an isomorphism \(\kappa_1\overset\sim\rightarrow\mathcal L^{-c}\) and an induced connection \(\nabla^{-c}\) . On the other hand, for suitable \(f\) (thus \(\tau)\), one constructs a connection \({}^{\tau}\nabla\) on \(\kappa_1\) with suitable properties with respect to (the fibres of) \(\phi\) and \(\widetilde{\phi}\). The main point is now that one may choose \(\tau\) such that \(\nabla^{-c}={}^{\tau}\nabla\). connection; relative moduli spaces; vanishing curvature; fundamental group of the Riemann surface Ramadas, T. R.: Faltings construction of the K -- Z connection. Comm. math. Phys. 196, 133-143 (1998) Riemann surfaces; Weierstrass points; gap sequences, Special connections and metrics on vector bundles (Hermite-Einstein, Yang-Mills), Families, moduli of curves (algebraic), Vector bundles on curves and their moduli Faltings' construction of the K-Z connection | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is 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 Jacobians of Fermat curves are of CM type. Is there a systematic way to construct other curves with this property ? The authors have this problem in mind and ask if the canonical lifting, due to Serre and Tate, of the Jacobian of an ordinary curve over a perfect field k of characteristic \(p>0\) is again the Jacobian of some curve. In this paper it is shown that, when p is odd and the genus is \(\geq 4\), the answer is ''no'' for most curves, even if one works \(mod p^ 2.\) The same problem is independently treated by \textit{F. Oort} and \textit{T. Sekiguchi} [J. Math. Soc. Japan 38, 427-437 (1986; Zbl 0605.14031)], and the results considerably overlap in both works. But the general ideas of the arguments are quite different from each other. Our authors proceed by ''pure thought'', while the others follow a very concrete way. moduli space; principally polarized abelian variety; crystalline cohomology; canonical lifting; Jacobian of an ordinary curve Dwork, B.; Ogus, A., \textit{canonical liftings of Jacobians}, Compositio Math., 58, 111-131, (1986) Picard schemes, higher Jacobians, Jacobians, Prym varieties, Families, moduli of curves (algebraic) Canonical liftings of Jacobians | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians The paper deals with fibrations \(f:S\to B\), where \(S\) is a complex projective surface and \(B\) is a complex projective curve. \(f\) is assumed to be onto and \(S\) relatively minimal (there are no \(-1\) curves in fibers of \(f\)). The author focus on the case, when \(f\) is not locally trivial. In this case they define (after Xiao) the slope of \(f\) as \(s_f=\frac{K_f^2}{\chi_f}\). Here \(K_f=K_S-f^*K_B\) and \(\chi_f=\deg f_*\omega_{S/B}\). By Noether theorem \(s_f\) takes values between \(0\) and \(12\). The aim of this paper is to find a lower bound on \(s_f\).
The authors state a conjecture that \(s_f\geq 4\frac{g-1}{g-q_f}\), where \(g\) is the genus of a generic fiber and \(q_f\) is the so-called relative irregularity of \(f\). Then they present instances when this conjecture holds and admit that there are some counterexamples known. In the subsequent parts of the paper the authors prove a weaker bound on \(s_f\), namely \(s_f\geq 4\frac{g-1}{g-\lfloor m/2\rfloor}\). Here \(m\) is the maximum of the irregularity \(q_f\) and the Clifford index \(\text{Cliff}(f)\). The most interesting fact about this bound is that \(q_f\) and \(\text{Cliff}(f)\) are apparently unrelated.
The authors provide two different proofs of this bound: first one uses the Harder-Narashiman filtration of some subsheaf of \(f_*\omega_f\). This method was first introduced by Xiao. The other one is the application of the Cornalba--Harris method. The paper ends with some examples of mappings showing that \(q_f\) and \(\text{Cliff}(f)\) are in fact unrelated. Namely \(q_f\) may be large and \(\text{Cliff}(f)\) small and conversely. fibration; slope; relative irregularity; Clifford index; linear stability M. Á. Barja and L. Stoppino, Linear stability of projected canonical curves with applications to the slope of fibred surfaces, J. Math. Soc. Japan 60 (2008), no. 1, 171-192. Families, moduli of curves (algebraic), Fibrations, degenerations in algebraic geometry, Special divisors on curves (gonality, Brill-Noether theory), Surfaces of general type, Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20] Linear stability of projected canonical curves with applications to the slope of fibred 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 In this paper, for each \(n\geqslant g\geqslant 0\) we consider the moduli stack \(\widetilde{{\mathcal{U}}}_{g,n}^{ns}\) of curves \((C,p_1,\ldots ,p_n,v_1,\ldots ,v_n)\) of arithmetic genus \(g\) with \(n\) smooth marked points \(p_i\) and nonzero tangent vectors \(v_i\) at them, such that the divisor \(p_1+\cdots +p_n\) is nonspecial (has \(h^1=0)\) and ample. With some mild restrictions on the characteristic we show that it is a scheme, affine over the Grassmannian \(G(n-g,n)\). We also construct an isomorphism of \(\widetilde{{\mathcal{U}}}_{g,n}^{ns}\) with a certain relative moduli of \(A_{\infty }\)-structures (up to an equivalence) over a family of graded associative algebras parametrized by \(G(n-g,n)\). \(A_ \infty\)-algebra; moduli of curves; Sato Grassmannian Families, moduli of curves (algebraic), Differential graded algebras and applications (associative algebraic aspects) Moduli of curves with nonspecial divisors and relative moduli of \(A_{\infty }\)-structures | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians Let \(H_0\) be a discrete periodic Schrödinger operator on \(\ell^2(\mathbb{Z}^d)\):
\[
H_0=-\Delta +V,
\]
where \(\Delta\) is the discrete Laplacian and \(V:\mathbb{Z}^d\rightarrow \mathbb{C}\) is periodic. We prove that for any \(d\ge 3\), the Fermi variety at every energy level is irreducible (modulo periodicity). For \(d=2\), we prove that the Fermi variety at every energy level except for the average of the potential is irreducible (modulo periodicity) and the Fermi variety at the average of the potential has at most two irreducible components (modulo periodicity). This is sharp since for \(d=2\) and a constant potential \(V\), the Fermi variety at \(V\)-level has exactly two irreducible components (modulo periodicity). We also prove that the Bloch variety is irreducible (modulo periodicity) for any \(d\ge 2\). As applications, we prove that when \(V\) is a real-valued periodic function, the level set of any extrema of any spectral band functions, spectral band edges in particular, has dimension at most \(d-2\) for any \(d\ge 3\), and finite cardinality for \(d=2\). We also show that \(H=-\Delta +V+v\) does not have any embedded eigenvalues provided that \(v\) decays super-exponentially analytic variety; algebraic variety; Fermi variety; Bloch variety; irreducibility; extrema; band function; band edge; embedded eigenvalue; unique continuation; Landis' conjecture; periodic Schrödinger operator Schrödinger operator, Schrödinger equation, Projective techniques in algebraic geometry, Analytic subsets of affine space, Eigenvalue problems for linear operators, Families, moduli of curves (algebraic) Irreducibility of the Fermi variety for discrete periodic Schrödinger operators and embedded eigenvalues | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians A curve \(C \subset {\mathbb P}^n\) is a pure \(1\)-dimensional locally Cohen-Macaulay closed subscheme. Its Rao function is the sequence \(\{ h^1(\mathcal {I}_C(t))\}\), \(t \geq 0\). Here the authors gives sharp upper bounds for this functions, under the assumption that the general hyperplane section of \(C\) spans a linear subspace of dimension \(n-k\). To prove that their bound is sharp they produce examples using Gorenstein liaison. curves in a projective space; Rao module; linkage; hyperplane section:; degenerate hyperplane section; Gorenstein liaison; Gorenstein linkage Linkage, Families, moduli of curves (algebraic), Plane and space curves Even G-liaison classes of some unions 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 S. Kondo defined a birational map between the moduli space of non-hyperelliptic curves of genus four and a ball quotient, thus providing a Baily-Borel compactification for the moduli space of such curves [\textit{S. Kondō}, Adv. Stud. Pure Math. 36, 383--400 (2002; Zbl 1043.14005)].
In this paper the authors construct a GIT compactification \(\overline{M}_4^{\mathrm{GIT}}\) of the moduli space of non-hyperelliptic curves of genus four and study its relationship with Kondo's compactification. More precisely, the main theorem states that the natural period map between \(\overline{M}_4^{\mathrm{GIT}}\) and Kondo's compactification can be resolved by blowing-up one point (where the exceptional divisor parametrizes hyperelliptic curves with a \(g_2^1\)) and its resolution contracts a rational curve to one cusp.
The space \(\overline{M}_4^{\mathrm{GIT}}\) is constructed by establishing a correspondence between genus four curves and cubic threefolds with an ordinary node and applying results by D. Allcock about GIT for cubic threefolds. Moreover, the authors prove that the space \(\overline{M}_4^{\mathrm{GIT}}\) coincides with a GIT quotient of the Chow variety of canonically embedded genus four curves and identify it with the Hassett-Keel space \(\overline{M}_4(5/9)\).
Similar results for the moduli space of curves of genus three have been given by \textit{E. Looijenga} [Contemp. Math. 422, 107--120 (2007; Zbl 1126.14034)] and the reviewer [Nagoya Math. J. 196, 1--26 (2009; Zbl 1184.14060)]. Hassett-Keel program; genus 4 curves; ball quotients Casalaina-Martin S., Jensen D. and Laza R., The geometry of the ball quotient model of the moduli space of genus four curves, Compact moduli spaces and vector bundles, Contemp. Math. 564, American Mathematical Society, Providence (2012), 107-136. Families, moduli of curves (algebraic), Special algebraic curves and curves of low genus, Families, moduli of curves (analytic), Geometric invariant theory, Minimal model program (Mori theory, extremal rays), Modular and automorphic functions The geometry of the ball quotient model of the moduli space of genus four curves | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians Let \(k\) be a perfect field and \(\bar{k}\) an algebraic closure of \(k\). Let \(C\) be a smooth projective curve of genus \(> 0\) and \(\text{Aut}(C)\) the group of automorphisms of \(C\) defined over \(\bar{k}\). The set of twists of \(C\) is defined to be the set of classes
\[
\text{Twist}_k(C):=\{C^{\prime}/k\, \text{curve}|\, \text{ exists } \bar{k}\text{-isomorphism}\, \phi: C^{\prime}\rightarrow C\}/k\text{-isomorphism}.
\]
The paper under review is devoted to the explicit computation of the twists of non-hyperelliptic curves of genus 3 or equivalently of smooth plane quartic curves, in case where \(k\) is a number field. If a curve has trivial automorphism group, then the set of twists is also trivial, and so, the computation of the twists, is reduced to plane quartic curves with non-trivial automorphism group. Henn's classification provides a classification up to \({\mathbb C}\)-isomorphism of such curves. Then, the method described in [the author, Rev. Mat. Iberoam. 33, No. 1, 169--182 (2017; Zbl 1381.11050)], provides the classification of all the smooth plane quartic twists. Note that this classification is not only interesting in its own as a classifying object, but also has many applications, as for instance computing Sato-Tate groups, solving Diophantine equations or computing \({\mathbb Q}\)-curves realizing certain Galois representations. twists of curves; plane quartic curves; automorphism groups; Fermat quartic; Klein quartic Lorenzo García, E., Twists of non-hyperelliptic genus 3 curves, Int. J. Number Theory, 14, 06, 1785-1812, (2018) Curves of arbitrary genus or genus \(\ne 1\) over global fields, Families, moduli of curves (algebraic), Special algebraic curves and curves of low genus, Plane and space curves Twists of non-hyperelliptic curves of genus \(3\) | 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 Let \(X\) be a normal variety defined over an algebraically closed field. Denote by \(\text{Div}(X)\) and \(\text{Car}(X)\) the group of Weil divisors, respectively Cartier divisors, on \(X\). The main result of the article is:
Let \(X\) be a complete normal variety such that \((\text{Div}(X)/\text{Car}(X)) \otimes_{\mathbb{Z}} \mathbb{Q}\) is finite dimensional. Then \(X\) contains only finitely many maximal, under inclusion, open quasi-projective sets.
A simple consequence of the main result is the following nice generalization of the Kleiman-Chevalley criterion:
Let \(X'\) be a normal variety which can be embedded as an open subset of a variety \(X\) such that \((\text{Div}(X)/\text{Car}(X)) \otimes_{\mathbb{Z}}\mathbb{Q}\) is finite dimensional. Then \(X'\) is quasi-projective if and only if every finite subset of \(X'\) is contained in an affine open subset of \(X'\). Kleiman-Chevalley quasi-projectivity criterion; Weil divisor; Cartier divisor; normal variety; quasi-projective variety Włodarczyk, Maximal quasiprojective subsets and the Kleiman-Chevalley quasiprojectivity criterion, J. Math. Sci. Univ. Tokyo 6 pp 41-- (1999) Divisors, linear systems, invertible sheaves, Projective and enumerative algebraic geometry, Picard groups, (Equivariant) Chow groups and rings; motives Maximal quasiprojective subsets and the Kleiman-Chevalley quasiprojectivity criterion | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians We show that various loci of stable curves of sufficiently large genus admitting degree \(d\) covers of positive genus curves define non-tautological algebraic cycles on \(\overline{\mathcal{M}}_{g,N}\), assuming the non-vanishing of the \(d\)-th Fourier coefficient of a certain modular form. Our results build on those of
\textit{T. Graber} and \textit{R. Pandharipande} [Mich. Math. J. 51, No. 1, 93--109 (2003; Zbl 1079.14511)] and
\textit{J. van Zelm} [Pac. J. Math. 294, No. 2, 495--504 (2018; Zbl 1388.14085)] for degree 2 covers of elliptic curves; the main new ingredient is a method to intersect the cycles in question with boundary strata, as developed recently by
\textit{J. Schmitt} and \textit{J. van Zelm} [Sel. Math., New Ser. 26, No. 5, Paper No. 79, 69 p. (2020; Zbl 1461.14037)] and
the author [Sel. Math., New Ser. 27, No. 5, Paper No. 96, 74 p. (2021; Zbl 1480.14021)]. Algebraic cycles, Intersection theory, characteristic classes, intersection multiplicities in algebraic geometry, Families, moduli of curves (algebraic), Coverings of curves, fundamental group Non-tautological Hurwitz cycles | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians We analyze morphisms from pointed curves to \(K3\) surfaces with a distinguished rational curve, such that the marked points are taken to the rational curve, perhaps with specified cross ratios. This builds on work of Mukai and others characterizing embeddings of curves into \(K3\) surfaces via non-abelian Brill-Noether theory. Our study leads naturally to enumerative problems, which we solve in several specific cases. These have applications to the existence of sections of del Pezzo fibrations with prescribed invariants. \(K3\) surfaces; Brill-Noether theory \(K3\) surfaces and Enriques surfaces, Special divisors on curves (gonality, Brill-Noether theory), Families, moduli of curves (algebraic), Families, moduli, classification: algebraic theory Embedding pointed curves in \(K3\) 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 The object of the paper under review is to compute the essential dimension of various stacks of curves and abelian varieties over fields.
Let \(k\) be a field, and let \(\mathcal X\) be an algebraic stack over \(k\). Given a field extension \(L/k\) and an object \(a\) in \(\mathcal X(L)\), the \textit{essential dimension \(\text{ed}(a)\)} of \(a\) is the minimal transcendence degree \(\text{trdeg}_k(K)\) of an intermediate field extension \(k \to K \to L\) such that \(a\) is in the essential image of the functor \(F(K) \to F(L)\). The \textit{essential dimension \(\text{ed}(\mathcal X)\)} of \(\mathcal X\) is the supremum of the values \(\text{ed}(a)\) taken over all field extensions \(L/k\) and all objects \(a\) in \(F(\mathcal X)\). When \(\mathcal X\) is a scheme (or more generally an algebraic space) of finite type, the essential dimension of \(\mathcal X\) recovers the usual dimension of \(\mathcal X\).
The main result of the paper is to compute the essential dimension of the stack \(\mathcal M_{g,n}\) of smooth \(n\)-pointed curves of genus \(g\) over a field \(k\) of characteristic \(0\). The authors show that \(\text{ed} (\mathcal M_{g,n})\) equals \(2\) if \((g,n) = (0,0)\) or \((1,1)\); \(0\) if \((g,n) = (0,1)\) or \((0,2)\); \(+\infty\) if \((g,n) = (1,0)\); \(5\) if \((g,n) = (2,0)\); and \(3g-3+n\) for all other pairs \((g,n)\). They furthermore show that the stack \(\overline{\mathcal M}_{g,n}\) of stable \(n\)-pointed curves of genus \(g\) has essential dimension \(\text{ed}(\overline{\mathcal M}_{g,n}) = \text{ed}(\mathcal M_{g,n})\) for \(2g-2+n > 0\). Finally, they show that the stack of hyperelliptic curves has essential dimension \(2g\) if \(g\) is odd, and \(2g+1\) if \(g\) is even.
Stacks of abelian varieties are treated in the appendix. Let \(\mathcal A_g\) denote the stack of principally polarized \(g\)-dimensional abelian varieties over a field \(k\), and let \(\mathcal B_g\) denote the stack of all \(g\)-dimensional abelian varieties over \(k\). Let \(2^a\) be the largest power of \(2\) dividing \(g\). Then the main result is that \(\text{ed}(\mathcal A_g) = g(g+1)/2 + 2^a\) if \(\text{char}(k) = 0\), or if \(\text{char}(k) = p > 0\) and \(p\nmid \#Sp_{2g}(\mathbb Z/\ell\mathbb Z)\) for some prime \(\ell > 2\); and that \(\text{ed}(\mathcal B_g) = \text{ed}(\mathcal A_g)\) if \(\text{char}(k) = 0\). For \(g\) odd this result is attributed to Miles Reid. It is not known whether the restriction on \(\text{char}(k)\) is necessary.
A key ingredient in the proofs of the main results in both the main body and the appendix is a genericity result asserting that the essential dimension of a smooth integral tame Deligne-Mumford stack is the sum of its dimension and the essential dimension of its generic gerbe. A consequence of this is that the essential dimension of any such stack is equal to the essential dimension of any nonempty open substack. essential dimension; stack; gerbe; moduli of curves; moduli of abelian varieties Patrick Brosnan, Zinovy Reichstein, and Angelo Vistoli, \emph{Essential dimension of moduli of curves and other algebraic stacks}, J. Eur. Math. Soc. (JEMS) 13 (2011), no.~4, 1079--1112, With an appendix by Najmuddin Fakhruddin. DOI 10.4171/JEMS/276; zbl 1234.14003; MR2800485; arxiv math/0701903 Generalizations (algebraic spaces, stacks), Families, moduli of curves (algebraic), Algebraic moduli of abelian varieties, classification Essential dimension of moduli of curves and other algebraic stacks | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians We prove irreducibility for the space of cyclic covers of fixed numerical type between smooth projective curves, and also for the space of cyclic covers of prime order and of fixed numerical-combinatorial type between moduli-stable projective curves. As an application we describe the irreducible components of the singular locus of the compactified moduli space of curves \(\overline{\mathfrak M_g}\), extending the work of Cornalba, who described the irreducible components of the singular locus of the moduli space of curves \(\mathfrak M_g\). Catanese, F., Irreducibility of the space of cyclic covers of algebraic curves of fixed numerical type and the irreducible components of \(S i n g(\overline{\mathfrak{M}}_g)\), (The Conference on Geometry in Honour of Shing-Tung Yau's 60-th Birthday, Advances in Geometric Analysis, Adv. Lect. Math. (ALM), vol. 21, (2012), Int. Press Somerville, MA), 281-306 Families, moduli of curves (algebraic), Coverings of curves, fundamental group, Families, moduli of curves (analytic), Moduli of Riemann surfaces, Teichmüller theory (complex-analytic aspects in several variables) Irreducibility of the space of cyclic covers of algebraic curves of fixed numerical type and the irreducible components of \(\mathrm{Sing}(\overline{\mathfrak{M}_g})\) | 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 The moduli space \(\mathcal M_g\) of Riemann surfaces of positive genus \(g\) is a quasi-projective variety of complex dimension \(3g-3\). This is the space whose points correspond bijectively to isomorphism classes of nonsingular complex projective curves of genus \(g\). Understanding the geometry or cohomology of these spaces for various \(g\) continues to be an attractive and challenging problem. Most challenging has been a conjecture of David Mumford dating from 1983 [\textit{D. Mumford}, Arithmetic and geometry, Pap. dedic. I. R. Shafarevich, Vol. II: Geometry, Prog. Math. 36, 271-328 (1983; Zbl 0554.14008)] to the effect that the rational cohomology of the \textsl{stable} moduli space of Riemann surfaces is a polynomial algebra generated by certain classes \(\kappa_i\) of dimension \(2i\) (the Mumford-Miller-Morita classes).
To state and explain the conjecture, one first considers the mapping class group \(\Gamma_g\) which consists of all isotopy classes of orientation preserving diffeomorphisms of a fixed compact oriented smooth surface \(F_g\) of positive genus \(g\). This is a discrete group that acts properly discontinuously on the space of conformal structures on \(F_g\) up to isotopy (i.e. Teichmuller space \({\mathcal T}_g\)). The quotient \({\mathcal T}_g/\Gamma_g\) is identified with \({\mathcal M}_g\) the moduli space. Because \(\mathcal T_g\) is contractible and the action of \(\Gamma_g\) has finite stabilizers, it follows that \(\mathcal M_g\) is rationally homotopy equivalent to the classifying space \(B\Gamma_g\). From the topologist point of view, and at least with rational coefficients, it is enough to study \(B\Gamma_g\).
By introducing boundary components and the associated mapping class groups \(\Gamma_{g,b}\) where \(b\) is the number of boundary circles, it becomes possible to define maps \(\Gamma_{g,b}\rightarrow \Gamma_{g+1,b}\) and \(\Gamma_{g,b}\rightarrow \Gamma_{g,b-1}\) which at the level of classifying spaces induce isomorphisms in integral cohomology in degrees less than \(g/2-1\) by fundamental work of Harer and Ivanov. The mapping telescope of the maps \(B\Gamma_{g+i,b}\rightarrow B\Gamma_{g+i+1,b}\) yield in the limit a space whose cohomology is independent of \(b\) and which is written \(B\Gamma_{\infty}\). For a given integer \(k\), \(H^k({B\Gamma}_\infty;{\mathbb Q} )\cong H^k(B\Gamma_g;{\mathbb Q})\cong H^k({\mathcal M}_g;{\mathbb Q})\) for sufficiently large \(g\). The Mumford conjecture is the statement that there is an algebra isomorphism
\[
H^*(B\Gamma_{\infty};{\mathbb Q} )\cong {\mathbb Q} [\kappa_1,\kappa_2,\ldots ]
\]
Miller and Morita proved that the polynomial algebra on the right injects into the term on the left. In the paper at hand, the authors complete the proof of this conjecture by establishing the surjection. In fact they prove a much strengthened version of this conjecture due to \textit{I. Madsen} and \textit{U. Tillmann} [Invent. Math. 145, No. 3, 509--544 (2001; Zbl 1050.55007)].
Write \(\Gamma_{g,2}\) as \(\Gamma_{g,1+1}\) denoting the fact that \(2\)-boundary components are now made into an incoming and an outgoing component. By gluing outgoing to incoming we make \(\coprod_{g\geq 0}B\Gamma_{g,1+1}\) into a monoid and its ``group completion'' is given by \({\mathbb Z}\times B\Gamma_{\infty}^+\), where \(B\Gamma_{\infty}^+\) is a space having the same homology groups as \(B\Gamma_{\infty}\) but is simply-connected (that this space exists is due to the fact that \(\Gamma_g\) is a perfect group for \(g\geq 3\)). One major theorem of \textit{U. Tillmann} states that this group completion is an infinite loop space [Invent. Math. 130, No.2, 257-275 (1997; Zbl 0891.55019)]. Madsen and Tillmann later conjectured that this infinite loop space is \(\Omega^\infty{{\mathbb C} P}^\infty_{-1}\) the bottom space of a Thom spectrum associated to the anti-tautological bundles over complex projective spaces. Their conjecture involved the construction of an infinite loop map \(\alpha_\infty : {\mathbb Z}\times B\Gamma_{\infty,2}^+\rightarrow\Omega^\infty{{\mathbb C} P}^\infty_{-1}\) and the assertion that this map is a homotopy equivalence. The main theorem of the paper under review is to establish this conjecture in its full strength. Since the rational cohomology of connected components of \(\Omega^\infty{{\mathbb C} P}^\infty_{-1}\) is isomorphic to \({\mathbb Q} [\kappa_1,\kappa_2,\ldots ]\), the Mumford conjecture becomes a consequence.
The proof provided here for the Madsen-Tillmann conjecture is a tour de force blending techniques from homotopy theory, algebraic geometry and sheaf theory. It is no easy task to even summarize it.
The first main idea is to describe the Madsen-Tillmann map \(\alpha_\infty\) as a map between classifying spaces. Introduce sheaves \(h\mathcal{V}\) and \(\mathcal{V}\) defined on the category of smooth manifolds (the dimension \(d=2\) corresponds to the Mumford conjecture) as follows : \(h\mathcal{V}(X)\) for a manifold \(X\) is the set of pairs \((\pi,\hat f)\) where \(\pi\) is a smooth submersion \(E\rightarrow X\) with \((d+1)\)-dimensional oriented fibers and \(\hat f\) is an appropriate section of the vertical or fiberwise jet bundle over \(E\) (section 1). Similarly one defines \(\mathcal{V} (X)\) (resp. \(\mathcal{V}_c(X)\)) as the set of pairs \((\pi, f)\) with \(\pi\) a submersion \(E\rightarrow X\) and \(f\) a smooth real function on \(E\) which is regular on fibers and such that the projection \((\pi,f): E\rightarrow X\times{\mathbb R}\) is proper (resp. proper and having connected fibers). This is another way of describing a bundle of smooth closed \(d\)-manifolds on \(X\times {\mathbb R}\) (see section 2). After introducing the notion of concordance of sheaves, one defines their representing spaces (or classifying spaces) denoted in our case by \(|\mathcal{V}|\), \(|\mathcal{V}_c|\) and \(|h\mathcal{V}|\). Using the Ehresmann's fibration lemma and Phillip's submersion theorem, one is able to identify \(|\mathcal{V}_c|\) and \(|h\mathcal{V}|\) with \({\mathbb Z}\times B\Gamma_{\infty}^+\) and \(\Omega^\infty{{\mathbb C} P}^\infty_{-1}\) respectively (in dimension \(d=2\)). As one expects there is a natural map \(|\mathcal{V}_c|\rightarrow |h\mathcal{V}|\) that models the Madsen-Tillmann map \(\alpha_\infty\). To show homotopy equivalence, and hence establish the conjecture, the authors proceed by showing that both spaces are homotopy fibers of compatible and equivalent fibrations. This is explained next.
Define new sheaves \(\mathcal{W}\) and \(h\mathcal{W}\) by enlarging for each smooth closed manifold \(X\) the set \(\mathcal{V}(X)\) to the set \(\mathcal{W}(X)\) which consists of pairs \((\pi,f)\) with \(\pi\) as before but with \(f: E\rightarrow{\mathbb R}\) a fiberwise Morse function rather than a fiberwise regular function. There is a similar enlargement for \(h\mathcal{V}(X)\) to \(h\mathcal{W}(X)\). There is a so-called jet prolongation map \(j : |\mathcal{W}|\longrightarrow |h\mathcal{W}|\) and one main theorem establishes that \(j\) is a homotopy equivalence (section 4). The main ingredient here is \textit{V. A. Vassiliev}'s ``first main theorem'' on complements of discriminants [Complements of discriminants of smooth maps; Topology and applications. Translations of Math. Monographs 98, A.M.S., (Providence), RI. (1992; Zbl 0762.55001)].
The next key step is to show that the spaces \(|h\mathcal{V}|\) and \(|h\mathcal{W}|\) fit in a homotopy fibration sequence of infinite loop spaces \(|h\mathcal{V}|\rightarrow |h\mathcal{W}|\rightarrow |h\mathcal{W}_{loc}|\), where the latter classifying space is derived from an equally explicit sheaf construction. This homotopy fiber sequence is constructed in section 3 by identifying each of \(h{\mathcal V}(X)\), \(h{\mathcal W}(X)\) and \(h{\mathcal W}_{loc}(X)\) on closed manifolds with a bordism construction and then identifying their classifying spaces, via the Thom-Pontryagin construction, with the corresponding infinite loop spaces of their Thom spectra. These Thom spectra are derived from the tautological bundle over the Grassmanianns which classify \((d+1)\)-dimensional oriented vector bundles whose fibers are equipped with a Morse type map and with a linear embedding in \({\mathbb R}^{n+d+1}\) (section 3.1).
Analogously there is a space \(|\mathcal{W}_{loc}|\) and maps \(|\mathcal{W}|\rightarrow |\mathcal{W}_{loc}| \rightarrow |h\mathcal{W}_{loc}|\) such that the second map of this composite is a homotopy equivalence (section 3) and the homotopy fiber of the first map (in the two dimensional case) is the space \({\mathbb Z}\times B\Gamma_{\infty}^+\). The proof of this last claim takes up the largest bulk of the paper and is technically the most demanding part. It rests on three main steps: first giving homotopy colimit decompositions for the spaces \(|{\mathcal W}|\) and \(|{\mathcal W}_{loc}|\) (this is part of the very long section 5). The stratification pieces of these decompositions classify certain smooth surface bundles. It becomes necessary however to consider classifying spaces for surface bundles with connected fibers. Passing from not necessarily connected surfaces to connected surfaces is done through a well-organized surgery procedure in a way that doesn't change the homotopy type of the classifying space (section 6). Lastly Harer theorem on the homology stability of mapping class groups is used to finish off the identification of the homotopy fiber of \(|{\mathcal W}|\rightarrow |{\mathcal W}_{loc}|\) (section 7).
Putting everything together, the homotopy equivalence \(\alpha_\infty\) becomes a consequence of the fact that \(|{\mathcal{W}}|\rightarrow |\mathcal{W}_{loc}|\) maps to \(|{h\mathcal{W}}|\rightarrow |h\mathcal{W}_{loc}|\) through commuting homotopy equivalences and so the homotopy fiber of the first map; \({\mathbb Z}\times B\Gamma_{\infty}^+\), is homotopy equivalent to the fiber of the second map; \(|h\mathcal{V}|\simeq \Omega^\infty{{\mathbb C} P}^\infty_{-1}\).
This long paper introduces a host of powerful ideas and techniques which will likely play in the future larger roles in all of algebraic topology and geometry. In section 4 and in Appendix A several pages are devoted to introducing and discussing the notion of sheaves taking values in the category of small categories, and a ``classifying'' space construction for such sheaves is given. In Appendix B, the homotopy invariance property of homotopy colimits is translated into the language of sheaves. moduli space of curves; mapping class group; classifying spaces; thom spectra. I. Madsen and M. Weiss, The stable moduli space of Riemann surfaces: Mumford's conjecture, Ann. of Math. (2) 165 (2007), no. 3, 843-941. Families, moduli of curves (algebraic), Loop spaces The stable moduli space of Riemann surfaces: Mumford'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 Let \(Y:=\{algebraic\) surfaces of degree d in \({\mathbb{P}}_ 3\}\) and \(\Sigma_ d:=\{S\in Y| \quad S\quad is\quad\) smooth and Pic(S) is not generated by the hyperplane bundle\(\}\). Previously, the author proved the explicit Noether-Lefschetz theorem [J. Differ. Geom. 20, 279-289 (1984; Zbl 0559.14009)]: For \(d\geq 3\), every component of \(\Sigma_ d\) has codimension \(\geq d-3\) in Y. Here the author gives a new and short proof of this result as a consequence of some vanishing theorem for Koszul cohomology on \({\mathbb{P}}_ n\), the proof of which is given in this paper. explicit Noether-Lefschetz theorem; vanishing theorem for Koszul cohomology M. Green, A new proof of the explicit Noether-Lefschetz theorem, J. Differential Geom. 27 (1988), 155-159. Zbl0674.14005 MR918461 Structure of families (Picard-Lefschetz, monodromy, etc.), Picard groups, Parametrization (Chow and Hilbert schemes), Étale and other Grothendieck topologies and (co)homologies A new proof of the explicit Noether-Lefschetz 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 The central results of this paper directly address the following question: given a pair of Fano toric varieties -- and a pair of anti-canonical sections -- can one provide natural conditions under which the zero loci of these sections define birational Calabi-Yau varieties? Alternatively, after a crepant partial resolution, when can we obtain a derived equivalence between such pairs of Calabi-Yau varieties?
As well as the general theory of toric varieties, the relevant aspects of which are summarised in the opening pages of the article, the central results draw on the derived equivalences obtained from variation of GIT by \textit{M. Ballard} et al. [J. Reine Angew. Math. 746, 235--303 (2019; Zbl 1432.14015)] and \textit{D. Halpern-Leistner} [J. Am. Math. Soc. 28, No. 3, 871--912 (2015; Zbl 1354.14029)], as well as the recent related work of \textit{D. Favero} and \textit{T. L. Kelly} [Adv. Math. 352, 943--980 (2019; Zbl 1444.14076)] on Berglund-Hübsch-Krawitz mirror symmetry.
The authors provide a wide array of applications for the derived and birational equivalences they obtain. In particlar, the authors describe birational and derived equivalences between mirrors constructed from anti-canonical hypersurfaces in the same toric Fano variety using a range of mirror constructions. These equivalences make use of the mirror construction of \textit{P. Clarke} [Adv. Theor. Math. Phys. 21, No. 1, 243--287 (2017; Zbl 1386.81130)], which interprets various mirror constructions as an exchange of \textit{linear data} associated to a pair of Landau-Ginzburg models via the sigma model/Landau-Ginzburg correspondence.
As well as the derived equivlance of Calabi-Yau mirrors, the authors obtain birational and derived equivalences between various \(K3\) surfaces obtained from the list of 95 weighted projective spaces whose anti-canonical linear system contain quasi-smooth \(K3\) surfaces with ADE singularities, recovering birational equivalences of \textit{M. Kobayashi} and \textit{M. Mase} [Tokyo J. Math. 35, No. 2, 461--467 (2012; Zbl 1262.14046)]. Moreover, applying their results to the case of quartic \(K3\) surfaces, the authors obtain equivalances between 52 of the 95 families in Reid's list and linear systems of quartic surfaces. Calabi-Yau varieties; toric varieties; \(K3\) surfaces; derived equivalences; Picard groups; mirror symmetry Toric varieties, Newton polyhedra, Okounkov bodies, Picard groups, Mirror symmetry (algebro-geometric aspects), Calabi-Yau manifolds (algebro-geometric aspects), \(K3\) surfaces and Enriques surfaces Equivalences of families of stacky toric Calabi-Yau 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 \(I\) be an ideal in a local ring \(A\). We want to investigate the Cohen-Macaulay properties of various Rees algebras of \(A\) with respect to \(I\), namely the multi-Rees algebras \((R_A({\mathcal I})_r))=:\bigoplus_{{(n_1,\dots,n_r)\atop n_i\geq 0}}I^{n_1}\dots I^{n_r}\), the Rees algebras of powers of \(I\) \((R_A(I^s))\) and the ordinary Rees algebra \((R_A(I))\).
In the first part of this paper we will discuss how the Cohen-Macaulay properties of \(R_A({\mathcal I}_r)\) and \(R_A(I^s)\) are related. We will describe situations in which the Cohen-Macaulayness of one of these algebras implies the same property for the other one in terms of the \(a\)-invariants of the corresponding form ring \(G_A(I)\). In particular, we will show that both implications always hold under the assumption that \(\text{depth } G_A(I)\geq \dim A-1\).
In the second part of the paper we will address the question when the Cohen-Macaulayness of \(R_A({\mathcal I}_r)\) or \(R_A(I^s)\) implies that \(R_A(I)\) is Cohen-Macaulay, too. To investigate this question, we will concentrate on the case that \(A\) is a Cohen-Macaulay ring of dimension at most three. Cohen-Macaulay properties; Rees algebras Korb, T.; Nakamura, Y.: On the Cohen--Macaulayness of multi-Rees algebras and Rees algebras of powers of ideals. J. math. Soc. Japan 50--52, 451-467 (1998) Cohen-Macaulay modules, Associated graded rings of ideals (Rees ring, form ring), analytic spread and related topics, Families, moduli of curves (algebraic) On the Cohen-Macaulayness of multi-Rees algebras and Rees algebras of powers of ideals | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is 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 solves three different and important enumerative problems in the setting of algebraic geometry, which are united by the fact that the relevant formulas which are used can be expressed as a sum of certain weights over isomorphism classes of marked trees. The technique used involves some tricks well known to physicists. The sum is written as a partition function that can be computed by evaluating a formal potential function at an appropriate critical point. Although these steps cannot be justified completely in the full generality of marked trees, the author is able to apply them, according to the mathematical rigour, in order to solve the following three problems.
The first problem is the computation of the Betti numbers of the moduli space \(\overline M_{0,n}\) of stable n-pointed curves of genus zero. These moduli spaces appear in the computation of Gromov-Witten invariants occurring in the theory of quantum cohomology. Some nice formulas are obtained for the generating function of the Poincaré polynomials (and of the Euler characteristic) which imply recursive formulas which are simpler than the formulas obtained by \textit{S. Keel} [Trans. Am. Math. Soc. 330, No. 2, 545-574 (1992; Zbl 0768.14002)].
The second problem is the computation of the Betti numbers of the configuration space \(X[n]\), which is a natural compactification constructed by \textit{W. Fulton} and \textit{R. MacPherson} [Ann. Math., II. Ser. 139, No. 1, 183-225 (1994; Zbl 0820.14037)] of the space of n pairwise distinct labeled points on a smooth compact variety \(X\).
The third and last problem comes from the work of \textit{M. Kontsevich} in the same volume as the paper under review [in: ``The moduli space of curves'', Prog. Math. 129, 335-368 (1995)] and consists in the calculation of the number \(m_d\) which gives the contribution of maps of degree \(d\) from \({\mathbb{P}}^1\) to a quintic threefold \(X\) (which is Calabi-Yau) whose image has normal sheaf \({\mathcal O}(-1)\oplus {\mathcal O}(-1)\). The author shows that \(m_d=d^{-3}\) by managing a formula of Kontsevich. This last problem has been considered and generalized in a recent preprint of \textit{Graber} and \textit{Pandharipande} (alg-geom/9708001). enumerative problems; configuration space; Calabi-Yau threefold; sum over trees; Betti numbers; moduli space of stable n-pointed curves; Gromov-Witten invariants Manin, Y.I.: Generating functions in algebraic geometry and sums over trees. In: The Moduli Space of Curves (Texel Island, 1994), Volume 129 of Progress in Mathematics, pp. 401-417. Birkhäuser, Boston (1995) Families, moduli of curves (algebraic), Enumerative problems (combinatorial problems) in algebraic geometry, Algebraic moduli problems, moduli of vector bundles Generating functions in algebraic geometry and sums over trees | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is 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 primary focus of this paper is to present constructive existence proofs of Azumaya algebras on a variety \(X\), which splits at every point of \(X\). For certain varieties \(X\), the author's techniques allow him to construct algebras which were not previously known to exist. Also, a procedure is given for computing both the Picard group and the cohomological Brauer group of any toric variety. existence proofs; Azumaya algebras; Picard group; cohomological Brauer group; toric variety Separable algebras (e.g., quaternion algebras, Azumaya algebras, etc.), Toric varieties, Newton polyhedra, Okounkov bodies, (Co)homology of commutative rings and algebras (e.g., Hochschild, André-Quillen, cyclic, dihedral, etc.), Brauer groups of schemes, Picard groups Examples of locally trivial Azumaya algebras | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians Let T be an integral domain of the form \(K+M\), where K is a field and M is a (non-zero) maximal ideal. This note is concerned with the class group Cl(R), Picard group Pic(R) and local class group G(R) for a subring of T of the form \(R=D+M\), where D is a ring contained in K. When K is the quotient field of D the inclusion maps induce an exact sequence \(0\to F(D)\to F(R)\to F(T)\) for \(F=Cl, Pic\) and G; in particular when \(F=Pic\) the last homomorphism is onto and the resulting short exact sequence splits. Im(Cl(R)\(\to Cl(T))\) is identified in a number of special cases including \(Im=Pic(T)\), when this occurs \(Cl(R)=Cl(D)\oplus Pic(T)\) and \(G(R)=G(D)\). As an application of this calculation of Cl(R), a criterion for R to be a PVMD is established. The note ends by showing how the D\(+M\) construction provides examples of rings with arbitrarily preassigned class groups. Prüfer v-multiplication domain; integral domain; Picard group; local class group; \(D+M\) construction D.F. Anderson and A. Ryckaert, The class group of \(D+M\), J. Pure Appl. Alg. 52 (1988), 199--212. Theory of modules and ideals in commutative rings, Extension theory of commutative rings, Divisibility and factorizations in commutative rings, Integral domains, Picard groups, Dedekind, Prüfer, Krull and Mori rings and their generalizations The class group of \(D+M\) | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians \textit{E. Witten} [in: Surveys in differential geometry. Vol. I: Proceedings of the conference on geometry and topology, held at Harvard University, Cambridge, MA, USA, April 27-29, 1990. Providence, RI: American Mathematical Society; Bethlehem, PA: Lehigh University. 243--310 (1991; Zbl 0757.53049)] gave a remarkable conjecture relating the intersection theory of the (Deligne-Mumford compactified) moduli space \(\overline{\mathcal{M}}_{g,k}\) of genus \(g\) Riemann surfaces with \(k\) punctures and integrable systems in KdV hieracrchies. This was later proved by \textit{M. Kontsevich} [Commun. Math. Phys. 147, No. 1, 1--23 (1992; Zbl 0756.35081)]. More precisely, the generating (partition) function \(\exp\left(\mathcal{F}\right)\) where \( \mathcal{F} = \sum\limits_{g=0}^\infty \hbar^{g-1} \sum\limits_{k=0}^\infty \frac{t_{l_1} \ldots t_{l_k}}{k!} I_g \) with intersection numbers \(I_g = \left< \tau_{l_1}, \ldots, \tau_{l_k} \right>_g = \int_{\overline{\mathcal{M}}_{g,k}} \prod_i \psi_i^{l_i}\) for \(\psi_i \in H^{2i}(\overline{\mathcal{M}}_{g,k})\) being the first Chern class of certain line-bundles, should be a tau-function of the KdV hierarchy.
A host of activities ensued in the past 2 decades, in various generalization relating the intersection theory, especially in light of Gromov-Witten invariants and other integrable hierarchies, by \textit{A. Okounkov} and \textit{R. Pandharipande} [Ann. Math. (2) 163, No. 2, 561--605 (2006; Zbl 1105.14077)], \textit{B. Dubrovin} and \textit{Y. Zhang} [Commun. Math. Phys. 198, No. 2, 311--361 (1998; Zbl 0923.58060)], \textit{E. Getzler} [in: Symplectic geometry and mirror symmetry. Proceedings of the 4th KIAS annual international conference, Seoul, South Korea, August 14--18, 2000. Singapore: World Scientific. 51--79 (2001; Zbl 1047.37046)], Fan-Jarvis-Ruan (FJRW) [\textit{H. Fan} et al., Ann. Math. (2) 178, No. 1, 1--106 (2013; Zbl 1310.32032)] et al. This correspondence can be thought of a manifestation of mirror symmetry with the geometry side playing the role of the A-model and the integrable side, the B-model.
Of particular note is the beautiful result of FJRW that the Drinfeld-Sokolov hierarchy for affine simply-laced Lie algebras ADE corresponds to geometric orbifolds of ADE-type. However, the Saito-Givental-Dubronvin-Zhang partition functions of the non-simply-laced BCGF singularities have been shown not to be tau-functions of the corresponding Drinfeld-Sokolov hierarchy. The purpose of the current paper is to nicely complete this story (cf.~Theorems 1.3 and 1.4): a correction is needed. In particular, the partition function of the \(\Gamma\)-invariant sectors of \(A,D^T,E_6\) (where the \(D^T\) is a mirror version of the D-type singularity) FJRW theory with maximal diagonal symmetry group is a tau-function of the corresponding \(B,C,F_4\) Drinfeld-Sokolov hierarchy (note the foldings of the respective Dynkin diagrams). Moreover, the partition function of the \(\mathbb{Z}_3\)-invariant sector of a \(D_4\) FJRW theory gives that of the \(G_2\) case. Delign-Mumford moduli space; integrable hierarchy; mirror symmetry Liu, S.-Q.; Ruan, Y.; Zhang, Y., BCFG Drinfeld-Sokolov hierarchies and FJRW-theory, Invent. Math., 201, 711-772, (2015) Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects), Topological field theories in quantum mechanics, Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.), Gromov-Witten invariants, quantum cohomology, Frobenius manifolds, Families, moduli of curves (algebraic) BCFG Drinfeld-Sokolov hierarchies and FJRW-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 In this article, we apply counting formulas for the number of morphisms from a curve to a toric variety to three different though related contexts (the first two are to be understood over global function fields): Manin's problem for rational points of bounded non-anticanonical height, asymptotics for algebraic points of bounded height and irreducibility of certain moduli spaces of curves, with application to the Severi problem for toric surfaces. Severi problem Toric varieties, Newton polyhedra, Okounkov bodies, Families, moduli of curves (algebraic), Heights, Varieties over global fields Algebraic points, non-anticanonical heights and the Severi problem on toric varieties | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians While there is much work and many conjectures surrounding the intersection theory of the moduli space of curves, relatively little is known about the intersection theory of the Hurwitz space \(\mathcal{H}_{k,g}\) parametrizing smooth degree \(k\), genus \(g\) covers of \(\mathbb{P}^1 \). Let \(k=3,4,5\). We prove that the rational Chow rings of \(\mathcal{H}_{k,g}\) stabilize in a suitable sense as \(g\) tends to infinity. In the case \(k=3\), we completely determine the Chow rings for all \(g\). We also prove that the rational Chow groups of the simply branched Hurwitz space \(\mathcal{H}^s_{k,g}\subset\mathcal{H}_{k,g}\) are zero in codimension up to roughly \(\frac{g}{k} \). In [\textit{S. Canning} and \textit{H. Larson}, The Chow rings of the moduli spaces of curves of genus 7, 8 and 9, preprint 2021, \url{arXiv:2104.05820}], results developed in this paper are used to prove that the Chow rings of \(\mathcal{M}_7\), \(\mathcal{M}_8\), and \(\mathcal{M}_9\) are tautological. Families, moduli of curves (algebraic), Algebraic cycles, Stacks and moduli problems, Coverings of curves, fundamental group Chow rings of low-degree Hurwitz spaces | 0 |
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