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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 This paper studies the intersection theory of moduli spaces of cyclic admissible covers. The main result in this paper gives an explicit formula for computing the first Chern class of the Hodge bundle over the space of admissible \(\mu_3\)-covers of \(n\)-pointed rational stable curves as a linear combination of boundary strata. The authors apply this formula to give a recursive formula for calculating certain families of Hodge integrals containing \(\lambda_1\). They also consider covers with a \(\mu_2\)-action for which they compute \(\lambda_2\) as a linear combination of codimension-\(2\) boundary strata.
This paper is organized as follows: Section 1 is an introduction to the subject and summarizes the main results. In Section 2 the authors recall some of the notions they need about moduli spaces of rational, pointed and stable curves. In Section 3 they introduce moduli spaces of admissible cyclic covers of rational curves. In Section 4 the authors compute an expression for the class \(\lambda_1\) on moduli spaces of \(\mu_3\)-admissible covers as a linear combination of boundary divisors. The main technique of proof consists of intersecting divisors with boundary curves. In Section 5 they derive a recursive structure among certain Hodge integrals on spaces of cyclic degree-\(3\) covers. Section 6 deals with the study of the second Chern class of the Hodge bundle over spaces of degree-\(2\) cyclic admissible covers. admissible covers; Hodge bundle; tautological classes; moduli space of curves Families, moduli of curves (algebraic) Boundary expression for Chern classes of the Hodge bundle on spaces of cyclic covers | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians We prove a conjecture due to Goncharov and Manin which states that the periods of the moduli spaces \(\mathfrak M_{0,n}\) of Riemann spheres with n marked points are multiple zeta values. We do this by introducing a differential algebra of multiple polylogarithms on \(\mathfrak M_{0,n}\) and proving that it is closed under the operation of taking primitives. The main idea is to apply a version of Stokes' formula iteratively to reduce each period integral to multiple zeta values.
We also give a geometric interpretation of the double shuffle relations, by showing that they are two extreme cases of general product formulae for periods which arise by considering natural maps between moduli spaces. moduli spaces; multiple zeta values; iterated integrals; polylogarithms; associators; associahedra F.C.S. Brown, \textit{Multiple zeta values and periods of moduli spaces} \(\mathcal{M}\)\_{}\{0,\(n\)\}(\(\mathbbR\)), \textit{Annales Sci. Ecole Norm. Sup.}\textbf{42} (2009) 371 [math/0606419] [INSPIRE]. Multiple Dirichlet series and zeta functions and multizeta values, Families, moduli of curves (algebraic), Polylogarithms and relations with \(K\)-theory Multiple zeta values and periods of moduli spaces \(\overline{\mathfrak M}_{0,n}\) | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians In the present note the authors study geometric criteria for the finite generation of the Cox rings of rational projective surfaces. Let us recall the following notions which are now specify to the case of surfaces. Let \(X\) be a smooth projective surface over an algebraically closed field \(k\). Then the Cox ring of \(X\) is defined as the \(k\)-algebra
\[
\text{Cox}(X) = \bigoplus_{(n_{1},\dots, n_{r}) \in \mathbb{Z}^{r}} H^{0}(X, \mathcal{O}(L_{1}^{n_{1}}) \otimes \cdots \otimes \mathcal{O}(L_{r}^{n_{r}})),
\]
where \(L_{1},\dots,L_{r}\) is a basis of the \(\mathbb{Z}\)-module \(\mathrm{Pic}(X)\) of classes of invertible sheaves on \(X\) modulo isomorphisms under the tensor product. Moreover, we assume that the linear and numerical equivalences on the group of Cartier divisors on \(X\) are the same (this is for instance the case of smooth rational projective surfaces). Denote by \(K_{X}\) the canonical divisor of \(X\) and by \(NS(X)\) the Neron-Severi group. The set of all effective elements in \(NS(X)\) is denoted by \(M(X)\). Since \(M(X)\) has an algebraic structure of a monoid, thus \(M(X)\) is called the effective monoid of \(X\).
The main result of the paper is the following criterion.
{ Theorem 1.}
Let \(X\) be a smooth projective rational surface defined over an algebraically closed field \(k\) of arbitrary characteristic such that the invertible sheaf associated to \(-K_{X}\) has a non-zero global section. Then the following conditions are equivalent: {\parindent=6mm \begin{itemize} \item[a)] \(\text{Cox}(X)\) is finitely generated, \item [b)] \(M(X)\) is finitely generated, \item [c)] \(X\) has only a finite number of \((-1)\)-curves and only a finite number of \((-2)\)-curves.
\end{itemize}} The second result of the paper provides another criterion.
{Theorem 2.}
Let \(X\) be a smooth projective rational surface defined over an algebraically closed filed \(k\) of arbitrary characteristic such that the invertible sheaf associated to the divisor \(-rK_{X}\) has at least two linearly independent sections for a certain positive integer \(r\). The following conditions are equivalent: {\parindent=6mm \begin{itemize} \item[a)] \(\text{Cox}(X)\) is finitely generated, \item [b)] the set of smooth projective rational curves of self-intersection \(-1\) on \(X\) is finite.
\end{itemize}} Cox rings; rational surfaces; effective monoids De La Rosa Navarro, B.L., Frías Medina, J.B., Lahyane, M., Moreno Mejía, I., Osuna Castro, O.: A geometric criterion for the finite generation of the Cox ring of projective surfaces. Rev. Mat. Iberoam. 31(4), 1131-1140 (2015) Divisors, linear systems, invertible sheaves, Picard groups, Riemann-Roch theorems A geometric criterion for the finite generation of the Cox rings of projective surfaces | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians Following a claim by \textit{A. Néron} [Proc. internat. Congr. Math. 1954 Amsterdam 3, 481-488 (1956; Zbl 0074.15901)] the authors construct an infinite family of curves of genus \(g\geq 2\) over \(\mathbb{Q}\) with Mordell-Weil rank \(r\) of its Jacobian variety \(\geq 3g+6\). Actually Néron claimed that \(r\) could be \(\geq 3g+7\) but here it is shown that Néron's method does not give such curves. In the meantime the second author succeeded to construct such curves with \(r\geq 3g+7\) modifying Néron's method [\textit{Y. Umezu}, Comment. Math. Univ. St. Pauli 48, No. 2, 169-179 (1999; see the following review Zbl 0958.14014)] and the first author gave families of such curves with \(r\geq 4g+7\), \(g\geq 2\) [\textit{T. Shioda}, Am. J. Math. 120, No.3, 551-566 (1998; Zbl 0919.14015)] using a different method. genus; Mordell-Weil rank; Jacobian variety Families, moduli of curves (algebraic), Jacobians, Prym varieties, Pencils, nets, webs in algebraic geometry, Representations of orders, lattices, algebras over commutative rings On Néron's construction of curves with high rank. I | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians We study the Grothendieck conjecture for the moduli spaces of hyperbolic curves of genus one. A consequence of the main results is that the isomorphism class of a certain moduli space of hyperbolic curves of genus one over a sub-\(p\)-adic field is completely determined by the isomorphism class of the étale fundamental group of the moduli space over the absolute Galois group of the sub-\(p\)-adic field. We also prove related results in absolute anabelian geometry. anabelian geometry; Grothendieck conjecture; moduli space; hyperbolic curve; configuration space Families, moduli of curves (algebraic), Universal profinite groups (relationship to moduli spaces, projective and moduli towers, Galois theory), Fundamental groups and their automorphisms (group-theoretic aspects), Coverings of curves, fundamental group The Grothendieck conjecture for the moduli spaces of hyperbolic curves of genus one | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians The Prym moduli space $\mathcal{R}_g$ parametrizes isomorphism classes of pairs $[(C,\alpha)]$, for $C$ a smooth complex projective genus $g$ curve and $\alpha$ a two-torsion line bundle on $C$. \par Let $\mathcal{A}_g$ denote the moduli space of principally polarized abelian varieties of dimension $g$. The Prym map $\mathrm{Pr}: \mathcal{R}_g \rightarrow \mathcal{A}_{g-1}$ sends points $[(C,\alpha)] \in \mathcal{R}_g$ to their principally polarized Prym variety $P(C,\alpha)$. \par Let $\mathcal{R}^0_g \subseteq \mathcal{R}_g$ be the open set where the Prym map is an immersion with respect to the orbifold metric on $\mathcal{A}_{g-1}$. This metric is induced by the symmetric metric on the Siegel space. \par In the present article, the authors study geodesic submanifolds of $\mathcal{A}_{g-1}$. For example, they establish the following result. \par Theorem. Assume that $[(C,\alpha)] \in \mathcal{R}^0_g$ where $C$ is a $k$-gonal curve of genus $g$ with $g \geq 9$ and $k \geq 3$. Let $Y$ be a germ of a totally geodesic submanifold of $\mathcal{A}_{g-1}$ which is contained in the Prym locus and passes through the Prym variety $P(C,\alpha)$. Then $\dim Y \leq 2g + k - 1$. \par Using this theorem, the authors prove: \par Theorem. If $g \geq 9$ and if $Y$ is a germ of a totally geodesic submanifold of $\mathcal{A}_{g-1}$ which is contained in the Prym locus and intersects $\text{Pr}(\mathcal{R}^0_g)$, then $\dim Y \leq \frac{5}{2} g + \frac{1}{2}$. \par The techniques used to establish these results are partly based on those of \textit{E. Colombo} et al. [Int. J. Math. 26, No. 1, Article ID 1550005, 21 p. (2015; Zbl 1312.14076)]. Prym locus; Shimura subvarieties; geodesic submanifolds Families, moduli of curves (algebraic), Subvarieties of abelian varieties A bound on the dimension of a totally geodesic submanifold in the Prym locus | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians The aim of this paper is to study the relationship between stable vector bundles of rank two on projective three space and their determinant surface determined by two sections of the bundle. A general surface has Picard number \(\rho (S) = 1\). A smooth surface \(S\) is not general because its Picard number is at least two by the following result of the author:
A smooth surface \(S \subset \mathbb{P}^3\) occurs as a determinant of a rank two vector bundle \({\mathcal E}\) on \(\mathbb{P}^3\) if and only if \(S\) has a surjective morphism onto \(\mathbb{P}^1\).
In the following the author estimates \(\rho (S)\). There is an estimate of \(\rho (S)\) from below in terms of the behaviour of \({\mathcal E}\) under the restriction to lines and planes. In particular, defining jumping planes there is a sufficient condition for \(S\) to have Picard number \(\rho (S) \geq 3\). stable vector bundles of rank two; determinant surface; Picard number Picard groups, Determinantal varieties, Hypersurfaces and algebraic geometry Determinant surfaces of rank 2 bundles on \(\mathbb{P}^ 3\) | 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 We give an explicit formula of the normalized Mumford form which expresses the second tautological line bundle by the Hodge line bundle defined on the moduli space of algebraic curves of any genus. This formula is represented as an infinite product which is a higher genus version of the Ramanujan delta function under the trivialization by normalized abelian differentials and Eichler integrals of their products. Furthermore, this formula gives a universal expression of the normalized Mumford form as a computable power series with integral coefficients by the moduli parameters of algebraic curves. Therefore, one can describe the behavior of this form and hence of the Polyakov string measure around the Deligne-Mumford boundary. normalized Mumford form; moduli space of algebraic curves; Ramanujan delta function; Polyakov string measure Families, moduli of curves (algebraic), Families, moduli of curves (analytic), Moduli of Riemann surfaces, Teichmüller theory (complex-analytic aspects in several variables), Arithmetic ground fields for curves, Arithmetic varieties and schemes; Arakelov theory; heights, Local ground fields in algebraic geometry, Theta functions and curves; Schottky problem, String and superstring theories; other extended objects (e.g., branes) in quantum field theory An explicit formula of the normalized Mumford form | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is 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 0518.00005.]
In this paper the author intends to begin the study of the Chow ring for the moduli space \({\mathcal M}_ g\) of curves of genus g and for its compactification \(\bar {\mathcal M}_ g\), the moduli space of stable curves. The point is that, though \({\mathcal M}_ g\) itself is far more complicated than the Grassmann varieties (the former is not unirational for large g), its Chow ring seems to have a similar simple structure as that of the latter in lower degrees.
In part one, as preliminary, an intersection product is defined in the Chow ring A(\(\bar {\mathcal M}_ g)\) of \(\bar {\mathcal M}_ g\) by using two facts that it has only the quotient singularity and is globally the quotient of a Cohen-Macaulay variety, i.e. the moduli space of curves with level structure. Both the Grothendieck and Baum-Fulton-MacPherson forms of the Riemann-Roch theorem give the necessary tool. - In part II a sequence of what he calls ``tautological'' classes \(\kappa_ i\in A^ i(\bar {\mathcal M}_ g)\) is defined, and auxiliary classes \(\lambda_ i\) as follows: on \(\bar {\mathcal M}_ g\) there are a canonical family of curves \(\pi:\bar {\mathcal C}_ g\to \bar {\mathcal M}_ g\) whose fibre is C/Aut(C) for [C]\(\in \bar {\mathcal M}_ g\), and a sheaf \(\omega_{\bar {\mathcal C}_ g/\bar {\mathcal M}_ g}\) of relative 1-forms which is a Q-sheaf (i.e. quotient of an invertible sheaf by a finite action). With the results in part I one can define the intersection on Chow classes in \(A^.(\bar {\mathcal C}_ g)\) or in \(A^.(\bar {\mathcal M}_ g)\), hence we set \(K_{\bar {\mathcal C}_ g/\bar {\mathcal M}_ g}=c_ 1(\omega_{{\mathcal C}_{\bar gg}/\bar {\mathcal M}_ g})\in A^ 1(\bar {\mathcal C}_ g)\) and \(\kappa_ i=(\pi_*K^{i+1}_{\bar {\mathcal C}_ g/\bar {\mathcal M}_ g})\in A^ i(\bar {\mathcal M}_ g).\) Moreover for \(E=\pi_*(\omega_{\bar {\mathcal C}_ g/\bar {\mathcal M}_ g})\), set \(\lambda_ i=c_ i(E)\). - The author then studies relations between them in \(A^.({\mathcal M}_ g)\) and expresses certain subvarieties such as the hyperelliptic locus with them. One of the important results is that all classes \(\kappa_ i\), \(\lambda_ i\) are polynomials in \(\kappa_ 1,...,\kappa_{g-1}\) in \(A^.({\mathcal M}_ g)\). - In this direction \textit{S. Morita} has found a number of new interesting relations between them but in \(H^.({\mathcal M}_ g)\) with different, i.e. topological method [see Proc. Japan Acad., Ser. A 60, 373-376 (1984)]. Its geometric counterpart in \(A^.({\mathcal M}_ g)\) is unknown up to now. - In part III, as an example, the author works out \(A^.(\bar {\mathcal M}_ 2)\) completely. moduli space of curves; Chow ring; moduli space of stable curves; intersection product; Riemann-Roch theorem Mumford, D.: Towards an Enumerative Geometry of the Moduli Space of Curves, Arithmetic and Geometry, Vol. II, Progress in Mathematics, vol.~36, pp.~271-328. Birkhäuser Boston, Boston (1983) Families, moduli of curves (algebraic), Enumerative problems (combinatorial problems) in algebraic geometry, Parametrization (Chow and Hilbert schemes), Algebraic moduli problems, moduli of vector bundles, Fine and coarse moduli spaces, Riemann-Roch theorems Towards an enumerative geometry of the moduli space of curves | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians Let \(C(D,r)\) be the Chow variety parametrizing nondegenerate irreducible curves of degree \(d\) in projective \(r\)-space. The authors show that the dimension of \(C(d,r)\) for \(r=3\) is given by a function \(\delta(d,3)\). Furthermore, they show that the component with the maximal dimension corresponding to those curves lies on a quadric with balanced bidegrees. The idea of the proof is to show that if the normal bundle of a curve has more than \(\delta(d,3)\) sections then the genus of the curve is larger than \((d^ 2/4)-2d+6\). Then using the classical Halphen bound on the genus of space curves, one sees that the curve must be contained in a quadric surface.
More generally the authors show that the dimension of \(C(d,r)\) is given by a function \(\delta(d,r)\) for \(d>4r^ 2-4r+3\). Again the components with the maximal dimension correspond to curves contained in a two dimensional rational normal scroll. dimension of Chow variety; Halphen bound Eisenbud, D.; Harris, J., The dimension of the Chow variety of curves, Compos. Math., 83, 3, 291-310, (1992) Parametrization (Chow and Hilbert schemes), Plane and space curves, Families, moduli of curves (algebraic), Projective techniques in algebraic geometry The dimension of the Chow variety of curves | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians Let \(X\) be a smooth projective variety of dimension \(n\) over an algebraically closed field \(k\). Let \({\mathcal C}_d(X)\) be the \textit{Chow variety} parametrizing \(d\)-dimensional cycles on \(X\) (see, for example, the book of [\textit{J. Kollár}, Rational curves on algebraic varieties. Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. 32. Berlin: Springer-Verlag (1995; Zbl 0877.14012)] for its construction). For \(a+b = n-1\), let \({\mathcal I}\subset {\mathcal C}_a(X)\times {\mathcal C}_b(X)\) be the incidence variety parametrizing the pairs \((A,B)\) with \(A\cap B \neq \emptyset\). When \(k = {\mathbb C}\), B. Mazur constructed, in 1993, using intersection theory operations on the universal cycles, a Weil divisor on \({\mathcal C}_a(X)\times {\mathcal C}_b(X)\) supported on \(\mathcal I\) and posed the problem whether \(\mathcal I\) is the support of a Cartier divisor, satisfying some additional properties. \textit{B. Wang} [Compos. Math. 115, No. 3, 303--327 (1999; Zbl 0982.14017)] showed that the Weil divisor \((n-1)!\, {\mathcal I}\) is Cartier.
In the paper under review, the author proposes a new approach to Mazur's question. Let \({\mathcal H}_d(X)\) be the \textit{Hilbert scheme} parametrizing \(d\)-dimensional subschemes of \(X\). Let \({\mathcal U}_a\), \({\mathcal U}_b\) be the closed subschemes of \(X\times {\mathcal H}_a(X)\times {\mathcal H}_b(X)\) obtained by pulling back the universal families over \({\mathcal H}_a(X)\) and \({\mathcal H}_b(X)\). Using the \textit{determinant functor} constructed by \textit{F. Knudsen} and \textit{D. Mumford} [Math. Scand. 39, No. 1, 19--55 (1976; Zbl 0343.14008)], one gets a line bundle \({\mathcal L} := \text{det}\, \text{R}\, pr_{23\ast}({\mathcal O}_{{\mathcal U}_a}\otimes^{\text{L}} {\mathcal O}_{{\mathcal U}_b})\) on \({\mathcal H}_a(X)\times {\mathcal H}_b(X)\). Let \(U\subset {\mathcal H}_a(X)\times {\mathcal H}_b(X)\) be the open subset over which the fibers of \({\mathcal U}_a\) and \({\mathcal U}_b\) are disjoint. One expects that, for \(a+b = n-1\), \(U\) is dense. Since \(\text{R}\, pr_{23\ast}({\mathcal O}_{{\mathcal U}_a}\otimes^{\text{L}} {\mathcal O}_{{\mathcal U}_b})\) is acyclic on \(U\), the ``Div'' construction of Knudsen and Mumford shows that \(\mathcal L\) is the invertible sheaf associated to a canonically defined Cartier divisor on \({\mathcal H}_a(X)\times {\mathcal H}_b(X)\).
One faces, now, the problem of showing that \(\mathcal L\) descends to a line bundle on \({\mathcal C}_a(X)\times {\mathcal C}_b(X)\) (via the product of the Hilbert-Chow morphisms). In order to solve this problem, the author studies the morphism of Picard groups induced by a seminormal proper hypercovering of a seminormal scheme, using some results of \textit{L. Barbieri-Viale} and \textit{V. Srinivas} [``Albanese and Picard 1-motives'', Mém. Soc. Math. Fr., Nouv. Sér. 87 (2001; Zbl 1085.14011)].
Then, by analysing the Hilbert-Chow morphism for 0-cycles, using the results of [\textit{B. Iversen}, Linear determinants with applications to the Picard scheme of a family of algebraic curves.Lecture Notes in Mathematics. 174. Berlin-Heidelberg-New York: Springer-Verlag. (1970; Zbl 0205.50802)], and for codimension 1 cycles, using the results of Knudsen and Mumford, the author shows that, in the case \(a=0\), \(b = n-1\), \(\mathcal L\) descends from \({\mathcal H}_0(X)\times {\mathcal H}_{n-1}(X)\) to \({\mathcal C}_0(X)\times {\mathcal C}_{n-1}(X)\). Chow variety; incidence divisor; Hilbert-Chow morphism; determinant functor; seminormal variety; simplicial Picard functor Parametrization (Chow and Hilbert schemes), Picard groups, Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20], Varieties defined by ring conditions (factorial, Cohen-Macaulay, seminormal) The Hilbert-Chow morphism and the incidence divisor: zero-cycles and divisors | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians Let \(\overline{M}_{g,n}\) be the non-singular moduli stack of a genus \(g\), \(n\)-pointed Deligne-Mumford stable curve \(C\). For each marking \(i\) there is an associated cotangent bundle \(\mathbb L_i\rightarrow\overline {M}_{g,n}\) with fiber \(T^*_{C,p_i}\) over the moduli point \([C,p_1,\ldots,p_n]\). Write \(\psi_i\) for the first Chern class \(c_1(\mathbb L_i)\in H^*(\overline{M}_{g,n},\mathbb Q)\). For a curve \(C\) let \(\omega_C\) denote its dualizing sheaf. Then the Hodge bundle \(\mathbb E\rightarrow\overline{M}_{g,n}\) is the rank \(g\) vector bundle with fiber \(H^0(C,\omega_C)\) over \([C,p_1,\ldots,p_n]\). Let \(\lambda_j=c_j(\mathbb E)\). A Hodge integral over \(\overline {M}_{g,n}\) is defined to be an integral of products of the \(\psi\) and \(\lambda\) classes.
Let \(X\) be a non-singular projective variety over \(\mathbb C\). Write \(\overline{M}=\overline{M}_{g,n}(X,\beta)\) for the moduli stack of stable maps to \(X\) representing the class \(\beta\in H_2(X,\mathbb Z)\). Let \([\overline{M}]^{\text{vir}}\in A_*(\overline{M})\) denote the virtual class (in the expected dimension). As a first result the following theorem (reconstruction theorem) is proven:
Theorem 1: The set of Hodge integrals over moduli stacks of maps to \(X\) may be uniquely reconstructed from the set of descendent integrals of the form
\[
\int_{[\overline{M}_{g,n}(X,\beta)]^{\text{vir}}}\prod_{i=1}^n\psi_i^{a_i}\cup e_i^*(\gamma_i)\cup\prod_{j=1}^g\lambda_j^{b_j}.
\]
The proof of this result relies on an interpretation of Mumford's calculation of Grothendieck-Riemann-Roch in Gromov-Witten theory.
The main result of the paper can be formulated as follows:
Theorem 2: Let \(F(t,k)\in{\mathbb Q}[k][[t]]\) be defined by
\[
F(t,k)=1+\sum_{g\geq 1}\sum_{i=0}^gt^{2g}k^i\int_{\overline{M}_{g,1}}\psi_1^{2g-2+i}\lambda_{g-i},
\]
then \(F(t,k)=\left({t/2\over\sin(t/2)}\right)^{k+1}.\)
Let \(C(g,d)=\int_{[\overline{M}_{g,0}(\mathbb P^1,d)]^{\text{vir}}}c_{\text{top}}(R^1\pi_*\mu^*N)\) denote the contribution to the genus \(g\) Gromov-Witten invariant of a Calabi-Yau \(3\)-fold of multiple covers of a fixed rational curve with normal bundle \(N=\mathcal O(-1)\oplus\mathcal O(-1)\). One knows that \(C(0,d)=1/d^3\) and \(C(1,d)=1/12d\). Here the general case is calculated:
Theorem 3: For \(g\geq 2\) one has
\[
C(g,d)=|\chi(M_g)|\cdot{d^{2g-3}\over(2g-3)!},
\]
where \(\chi(M_g)=B_{2g}/2g(2g-2)\) is the Harer-Zagier formula for the orbifold Euler characteristic of \(M_g\), and where \(B_{2g}\) is the \(2g\)-th Bernoulli number.
Theorem 4: For \(g\geq 2\) one has
\[
\int_{\overline{M}_g}\lambda^3_{g-1}={|B_{2g}|\over 2g}{|B_{2g-2}|\over 2g-2}{1\over(2g-2)!}.
\]
Several methods to obtain relations between Hodge integrals are discussed in some detail: (i) via virtual localization, (ii) via classical curve theory, first via the canonical system, second via Weierstraß loci.
\noindent In the introduction the authors end with an interesting combinatorial conjecture relating Gromov-Witten theory to the intrinsic geometry of \(M_g\) via Hodge integrals. Let \(\mathcal R^*(M_g)\) be the ring of tautological Chow classes in \(M_g\). This ring is conjectured to be a Gorenstein ring with socle in degree \(g-2\). The top intersection pairings in \(\mathcal R^*(M_g)\) are determined by the Hodge integrals \(\int_{\overline{M}_{g,n}}\psi_1^{k_1}\ldots\psi_n^{k_n} \lambda_g\lambda_{g-1}\).
It was conjectured by \textit{C. Faber} [in: Moduli of Curves and Abelian Varieties. The Dutch Intercity Seminar on Moduli, Aspects Math. E 33, 109-129 (1999; Zbl 0978.14029)] that
\[
\int_{\overline{M}_{g,n}}\psi_1^{k_1}\ldots\psi_n^{k_n} \lambda_g\lambda_{g-1}={{(2g+n-3)!(2g-1)!!}\over{(2g-1)!\prod_{i=1}^n (2k_i-1)!!}}\int_{\overline{M}_{g,1}}\psi_1^{g-1}\lambda_g\lambda_{g-1},
\]
where \(g\geq 2\) and \(k_i>0\). The conjecture has been shown to be implied by the so-called degree \(0\) Virasoro conjecture applied to \({\mathbb P}^2\) [cf. \textit{E. Getzler} and \textit{R. Pandharipande}, Nucl. Phys. B 530, No. 3, 701-714 (1998; Zbl 0957.14038)]. moduli stack; Hodge integral; Gromov-Witten theory; Virasoro conjecture; Deligne-Mumford stable curves C. Faber and R. Pandharipande, \textit{Hodge integrals and Gromov-Witten theory}, math.AG/9810173. Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects), Families, moduli of curves (algebraic) Hodge integrals and Gromov-Witten theory | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians Let \(S\) be a minimal analytic surface and \(C\) be a curve (i.e. a reduced, irreducible effective divisor on \(S)\). The `arithmetic genus' of \(C\) is the number \(\pi(C)=1+C(C+K_ S)/2\), where \(K_ S\) is the canonical class of \(S\). Fix an integer \(g\) and look to all the algebraic families of curves with arithmetic genus \(g\). Namikawa showed that, modulo the automorphisms of \(S\), there are only finitely many such families, when the Kodaira dimension of \(S\) is different from 1. Using a result of Miyaoka and Umezu, the author shows here that also for surfaces of Kodaira dimension 1, the number of algebraic families of curves with fixed arithmetic genus is finite \((\bmod Aut(S))\). curves on surfaces; minimal analytic surface; number of algebraic families of curves with fixed arithmetic genus Families, moduli of curves (algebraic), Enumerative problems (combinatorial problems) in algebraic geometry, Algebraic cycles Finiteness of numbers of curves on a minimal surface with \(\kappa=1\) | 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 \(k\) be an algebraically closed field of characteristic 0 and \(R\) a reduced irreducible curve singularity over \(k\), i.e. \(R \supset k\) is a complete local domain of Krull dimension 1 having \(k\) as residue field. Any torsion free rank one module \(M\) can be seen as a submodule of the normalization \(R'=k[[t]]\) of \(R\) such that \(R \subseteq M \subseteq R'\). The aim of this paper is to construct coarse moduli spaces parametrizing the isomorphism classes of such modules corresponding to some fixed invariants as e.g. \(\delta(M)= \dim_ kR'/M\), \(\Gamma(M)=v(M)\), where \(v\) is the valuation of \(R'\). The idea is to show that the above isomorphism classes can be seen as orbits of the multiplicative group \(R'{}^*\) of invertible elements of \(R'\) (in fact of the Jordan group \(J \cong R'{}^*/k^*)\) acting on some Grassmannians. Then using the authors' results from Proc. Lond. Math. Soc., III. Ser. 67, No. 1, 75-105 (1993) it is enough to see that there exist geometric quotients for a certain stratification given by some invariants. The paper contains an algorithm for computation of these moduli spaces and many useful nice examples. irreducible curve singularity; coarse moduli spaces; Grassmannians Greuel, G.-M., Pfister, G.: Moduli spaces for torsion free modules on curve singularities I. J. of Algebraic Geometry2, 81--135 (1993) Families, moduli of curves (algebraic), Singularities of curves, local rings, Grassmannians, Schubert varieties, flag manifolds, Singularities in algebraic geometry Moduli spaces for torsion free modules on curve singularities. I | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians \textit{L. Illusie} [in: Barsotti symposium in algebraic geometry. Memorial meeting in honor of Iacopo Barsotti, in Abano Terme, Italy, June 24-27, 1991. San Diego, CA: Academic Press. 183--203 (1994; Zbl 0832.14015)] has suggested that one should think of the classifying group of \(M_X^{gp}\)-torsors on a logarithmically smooth curve \(X\) over a standard logarithmic point as a logarithmic analogue of the Picard group of \(X\). This logarithmic Picard group arises naturally as a quotient of the algebraic Picard group by lifts of the chip firing relations of the associated dual graph. We connect this perspective to \textit{M. Baker} and \textit{S. Norine}'s theory [Adv. Math. 215, No. 2, 766--788 (2007; Zbl 1124.05049)] of ranks of divisors on a finite graph, and to \textit{O. Amini} and \textit{M. Baker}'s [Math. Ann. 362, No. 1--2, 55--106 (2015; Zbl 1355.14007)] metrized complexes of curves. Moreover, we propose a definition of a combinatorial rank for line bundles on \(X\) and prove that an analogue of the Riemann-Roch formula holds for our combinatorial rank. Our proof proceeds by carefully describing the relationship between the logarithmic Picard group on a logarithmic curve and the Picard group of the associated metrized complex. This approach suggests a natural categorical framework for metrized complexes, namely the category of logarithmic curves. Families, moduli of curves (algebraic), Logarithmic algebraic geometry, log schemes, Combinatorial aspects of tropical varieties, Geometric aspects of tropical varieties Logarithmic Picard groups, chip firing, and the combinatorial rank | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians Let \(C\) be an integral curve with one unibranch singularity of embedding dimension greater than two. We study decomposition of the compactified Picard variety \((\text{Pic}^ 0C)^ =\) into irreducible components for a unibranch singularity. We give a criterion for determining the beginning of a decomposition, defined by the number of generators of a module locally at the singularity. We discuss some particular cases where it is possible to give more precise descriptions. In the case of curves characterised by ``maximal ideal of \(M\) of the singularity equal to the conductor of \(C\)'' the basic decomposition given in section two actually gives components of \((\text{Pic}^ 0C)^ =\). We deal with the obvious first extension of this, namely, \(\text{rk} (M/C) = 1\). In this case the basic decomposition of \((\text{Pic}^ 0C)^ =\) given in section 1 does not give irreducible components but is only a step towards giving the irreducible components, which we do for the case of Gorenstein curves with \(\text{rk} (M/C) = 1\). compactified Jacobian; decomposition of Picard variety; unibranch singularity; Gorenstein curves Jacobians, Prym varieties, Singularities of curves, local rings, Picard groups, Varieties defined by ring conditions (factorial, Cohen-Macaulay, seminormal) Compactified Jacobian of some unibranch curves | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians [For the entire collection see Zbl 0614.00006.]
The purpose of the paper is to give information on a certain smooth compactification of the space of all morphisms of a given degree from \({\mathbb{P}}^ 1\) to a Grassmann variety. This scheme is the Grothendieck Quot scheme of quotients of a trivial vector bundle on \({\mathbb{P}}^ 1\). We compute the additve and the multiplicative structure of its Chow ring and identify the ample cone and the corresponding projective embeddings. families of rational curves; smooth compactification; Grassmann variety; Quot scheme; Chow ring Strømme, S A, On parametrized rational curves in Grassmann varieties., Lecture Notes in Math, 1266, 251-272, (1987) Grassmannians, Schubert varieties, flag manifolds, Families, moduli of curves (algebraic), Parametrization (Chow and Hilbert schemes), Homogeneous spaces and generalizations, Algebraic moduli problems, moduli of vector bundles On parametrized rational curves in Grassmann 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 Consider the set \({\mathcal S} = \{ (\tilde S,\tilde L ) | \tilde S\) is a smooth projective surface and \(\tilde L \in \hbox{Pic}(\tilde S)\) is very ample\(\}\). An adjunction process on the pair \((\tilde S, \tilde L )\) in \(\mathcal S\) is the construction of a new pair \((S,L)\) in the following way: first contract all the \(-1\) rational curves \(C\) on \(\tilde S\) with \(\tilde L \cdot C = 1\), getting a new surface \(S\) and a reduction morphism \(\pi : \tilde S \rightarrow S\); then consider on \(S\) the line bundle \(H = K_ S \otimes L\), where \(L = \pi_ * \tilde L\). The pair \((S,L)\) is called the reduction of \((\tilde S,\tilde L)\). It is known that if \((\tilde S,\tilde L)\) is not in a certain subset \(\mathcal E\) of \(\mathcal S\) then \((S,H)\) is again in \(\mathcal S\).
Therefore there is a map \({\mathcal RA} : {\mathcal S} - {\mathcal E} \rightarrow {\mathcal S}\) which associates to a pair \((\tilde S,\tilde L)\) the pair \((S,H)\). It is not hard to check that \(\mathcal RA\) is not a surjection. This paper considers the question of characterizing the pairs in \(\mathcal S\) which are in the image of \(\mathcal RA\). Answers are given in the following cases: \smallskip \noindent (a) surfaces in \({\mathbf P}^ 4\) (i.e.\ \(h^ 0 (H) \leq 5\));
\noindent (b) surfaces of degree \(\leq 9\) (i.e.\ \(H^ 2 \leq 9\));
\noindent (c) surfaces for which \(K_ S^{\otimes -1}\) is nef. \smallskip The work relies heavily on adjunction theory and related classification results for surfaces of small sectional genus. It also uses Castelnuovo's bound for the genus of a curve in projective space. Picard group; adjunction process; surfaces of small sectional genus; Castelnuovo's bound for the genus of a curve Special surfaces, Divisors, linear systems, invertible sheaves, Picard groups, Projective techniques in algebraic geometry On projective surfaces arising from an adjunction process | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is 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 moduli space \(\mathcal{M}_{g,1}^N\) of pointed algebraic curves of genus \(g\) with a prescribed semigroup \(N\) whose complement w.r.t. \(\mathbb{N}_0:=\{0,1,2, \cdots\}\) is the gap sequence at the marked point, when the underlying base field is of positive characteristic. In zero characteristic case, there is a nice description of such a moduli space in terms of the negative part of miniversal deformation space of the monomial curve of the semigroup corresponding to the given Weierstrass gap sequence, due to the well-known work of \textit{H. C. Pinkham} [Astérisque 20, 1--131 (1974; Zbl 0304.14006)].
However in positive characteristic case, the isomorphism established by a theorem of Pinkham may not hold in general. In this paper the author establishes criteria for validity of the theorem of Pinkham up to genus 4 in positive characteristic case. Specifically it is proved that Pinkham's statement hold for any numerical semigroup \(N\) of genus \(g\leq 4\) if the characteristic of the base field does not divide any exponent of the generating monomials which defines the negative miniversal deformation of the monomial curve \(X\) w.r.t \(N\), unless \(N=\{5,6,7, 8, \cdots\}\) and \(g=4\). In the case \(N=\{5,6,7, 8, \cdots\}\) (\& \(g=4\)) - which is the semigroup of non-gap sequence at non Weierstrass points - together with additional assumption that the characteristic \(p\) is not five, it is proved that the same statement also holds. This paper has a nice review on the work of Pinkham which is easily accessible to the reader.
For part I, II, see [\textit{T. Nakano} and \textit{T. Mori}, ibid. 27, No. 1, 239--253 (2004; Zbl 1077.14037); [\textit{T. Nakano}, ibid. 31, No. 1, 147--160 (2008; Zbl 1145.14025)]. pointed algebraic curves; moduli space; monomial curves; Weierstrass point Families, moduli of curves (algebraic), Riemann surfaces; Weierstrass points; gap sequences On the moduli space of pointed algebraic curves of low genus. III: Positive characteristic. | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians Let \(H_{d,g}\) denote the Hilbert scheme of locally Cohen-Macaulay curves in \(\mathbb{P}^3\). For any \(d>4\) and \(g\leq {d-3\choose 2}\), \(H_{d,g}\) has two well-understood irreducible families: There is a component \(E\subset H_{d,g}\) corresponding to extremal curves [see \textit{M. Martin-Deschamps} and \textit{D. Perrin}, Ann. Sci. Éc. Norm. Supér., IV. Sér. 29, No. 6, 757-785 (1996; Zbl 0892.14005), these are the curves with maximal Rao function] and \(S\), the family of subextremal curves [see \textit{S. Nollet}, Manuscr. Math. 94, No. 3, 303-317 (1997; Zbl 0918.14014), these have the next largest Rao function]. In this short note we show that \(S\cap E\neq \emptyset\) in \(H_{d,g}\) by constructing an explicit specialization (proposition 1). Our construction also works for ACM curves of genus \(g= {d-3\choose 2}+1\) (remark 2) and hence \(H_{d,g}\) is connected for \(g> {d-3\choose 2}\) (corollary 3). Hilbert scheme Nollet, S., A remark on connectedness in Hilbert scheme, Communications in Algebra, 28, 5745-5747, (2000) Parametrization (Chow and Hilbert schemes), Families, moduli of curves (algebraic) A remark on connectedness in Hilbert schemes | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians We express Hurwitz numbers of polynomials of arbitrary topological type in terms of intersection numbers on the moduli space of curves of genus zero with marked points. Hurwitz numbers of polynomials; moduli space of curves; intersection numbers Shadrin, S.: Polynomial Hurwitz numbers and intersections on M\={}0,k. Funct. anal. Appl. 37, 78-80 (2003) Rational and birational maps, Families, moduli of curves (algebraic), Polynomial rings and ideals; rings of integer-valued polynomials Polynomial Hurwitz numbers and intersections on \(\overline{\mathcal{M}}_{0,k}\) | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is 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 thesis determines the singularities of the coarse moduli space \(\overline S_g\) over \(\mathbb{C}\) of spin curves. The precise description of the singularities yields the result that pluricanonical forms on the smooth locus of \(\overline S_g\) lift holomorphically to a desingularisation. The moduli space \(\overline S_g\) constructed by \textit{M. Cornalba} [Moduli of curves and theta-characteristics. Teaneck, NJ: World Scientific Publishing Co. 560--589 (1989; Zbl 0800.14011)] compactifies the coarse moduli space \(S_9\) of smooth spin curves. These are pairs \((C,L)\) of a smooth curve of (arithmetic) genus \(g\geq 2\) and a theta characteristic \(L\) on \(C\), i.e. a line bundle \(L\) on \(C\) such that \(L^{\otimes 2}\) is isomorphic to the canonical bundle \(\omega_C\). This compactification is compatible with the Deligne-Mumford compactification \(\overline M_g\) of the coarse moduli space \(M_g\) of smooth curves of genus \(g\) via stable curves [\textit{P. Deligne} and \textit{D. Mumford}, Publ. Math., Inst. Hautes Étud. Sci. 36, 75--109 (1969; Zbl 0181.48803) and Matematika, Moskva 16, No. 3, 13--53 (1972; Zbl 0233.14008)]. In particular there exists a natural morphism \(\pi:\overline S_g\to\overline M_g\) which sends the moduli point of a spin curve to the moduli point of the underlying curve. \(\pi\) is a finite map of degree \(2^{2g}\).
This thesis focuses on the local (analytic) structure of the moduli space \(\overline S_g\). As in the case of \(\overline M_g\) an analytic neighbourhood of the moduli point of a spin curve in \(\overline S_g\) is isomorphic to the quotient \(V/G\) of a \(3g-3\)-dimensional vector space \(V\) with respect to a finite group \(G\). This group is essentially the automorphism group of the spin curve under consideration. A careful analysis of the occurring quotients gives a description of the locus of canonical singularities of \(\overline S_g\) with the help of the Reid-Tai criterion. This locus has codimension 2 in \(\overline S_g\). Moreover, the smooth locus \(\overline S^{\text{reg}}_g\subset\overline S_g\) is determined. The morphism \(\pi\) plays an important role in these calculations, since it establishes a connection between the well-understood singularities of \(\overline M_g\) by \textit{J. Harris} and \textit{D. Mumford} [Invent. Math. 67, 23--86 (1982; Zbl 0506.14016)] and those of \(\overline S_g\). In order to understand this connection the ramification of the finite map \(\pi\) is described.
These local results are used to prove that all pluricanonical forms on \(\overline S^{\text{reg}}_g\), i.e. sections in \(\Gamma(\overline S^{\text{reg}}_g,{\mathcal O}_{\overline S_g}(kK_{\overline S_g}))\), extend holomorphically to a desingularisation \(\widetilde S_g\) of \(\overline S_g\). An important ingredient is the analogous result for \(\overline M_g\) by Harris and Mumford [loc. cit.]. moduli space; spin curve; singularities; pluricanonical forms Research exposition (monographs, survey articles) pertaining to algebraic geometry, Families, moduli of curves (algebraic) Moduli of 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 The Mordell conjecture states a curve \(C\) of genus \(\ge 2\) defined over a number field \(k\) has only finitely many points in \(C(k)\). The conjecture was first proved by \textit{G. Faltings} [Invent. Math. 73, 349--366 (1983; Zbl 0588.14026)], and later again by \textit{P. Vojta} [Ann. Math. (2) 133, No. 3, 509--548 (1991; Zbl 0774.14019)] using the diophantine approximation. To this day, no effective height upper bound for points in \(C(k)\) is known, which will help us to find all the points in \(C(k)\). However, estimates of \(\#C(k)\) have been obtained by \textit{E. Bombieri} [Ann. Sc. Norm. Super. Pisa, Cl. Sci., IV. Ser. 17, No. 4, 615--640 (1990; Zbl 0722.14010)], \textit{T. de Diego} [J. Number Theory 67, No. 1, 85--114 (1997; Zbl 0896.11025)], and \textit{G. Rémond} [Invent. Math. 142, No. 3, 513--545 (2000; Zbl 0972.11054)], and they involve the Faltings height of the Jacobian variety \(\mathrm{Jac}(C)\).
\textit{J. H. Silverman} [J. Lond. Math. Soc., II. Ser. 47, No. 3, 385--394 (1993; Zbl 0811.11046)] and \textit{S. David} and \textit{P. Philippon} [IMRP, Int. Math. Res. Pap. 2007, Article ID rpm006, 113 p. (2007; Zbl 1163.11049)] introduced examples of families of curves \(C/k\) for which the upper bound of \(\#C(k)\) is independent of the Faltings height of the Jacobian variety \(\mathrm{Jac}(C)\). It is known as Mazur's conjecture that there is a constant \(c\) depending on the genus \(g\) and the number field \(k\) such that \(\#C(k) \le c^{1 + \mathrm{rank}(\mathrm{Jac}(C)(k))}\), and Caporaso et al. proposed the stronger uniformity conjecture where it is independent of \(\mathrm{rank}(\mathrm{Jac}(C)(k))\). The authors of the paper under review prove a uniformity result of this quality for a one-dimensional family of curves.
Let \(S\) be a smooth, geometrically irreducible curve defined over \(k\) embedded in some projective space, and let \(h : S(\overline k) \to \mathbb{R}\) be the pull-back of the absolute logarithmic Weil height. Let \(\mathcal C\) be a geometrically irreducible, quasi-projective variety defined over \(k\) with a smooth morphism \(\pi : \mathcal C \to S\) with all fibers being smooth, geometrically irreducible, projective curves of genus \(g\ge 2\). The authors prove that there exists a constant \(c\ge 1\) depending on \(\mathcal C\), \(\pi\), and the choice of embedding of \(S\) into projective space such that given \(s\in S(\overline k)\) with \(h(s)\ge c\), \(\#\mathcal C_s(\overline k) \cap \Gamma \le c^{1+ \mathrm{rank}(\Gamma)}\) where \(\mathcal C_s=\pi^{-1}(s)\) which is embedded in \(\mathrm{Jac}(\mathcal C_s)\) based at a \(\overline k\)-point of \(\mathcal C_s\) and \(\Gamma\) is a finite rank subgroup of \(\mathrm{Jac}(\mathcal C_s)(\overline k)\). For example, if \(\Gamma = \mathrm{Jac}(\mathcal C_s)(L)\) where \(L\) is a finite extension of \(k\), then it implies \(\#\mathcal C_s( L)\le c^{1+ \mathrm{rank}(\mathrm{Jac}(\mathcal C_s)(L))}\). uniformity conjecture Rational points, Families, moduli of curves (algebraic) Uniform bound for the number of rational points on a pencil of curves | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians The author studies a special case of the so-called enlarged Witten conjecture originating in \textit{E. Witten}'s work on two-dimensional quantum gravity [in: Proc. Conf., Cambridge/MA (USA) 1990, Surv. Differ. Geom., Suppl. J. Diff. Geom. 1, 243--310 (1991; Zbl 0757.53049); in: L. R. Goldberg (ed.) et al.; Topological methods in modern mathematics, Proceedings of a symposium in honor of John Milnor's sixtieth birthday, held in the State University of New York at Stony Brook, USA, June 14-June 21, 1991, Houston, TX: Publish Perish, Inc., 235--269 (1993; Zbl 0812.14017); and in: Nucl. Phys., B 371, 191--245 (1992)].
This conjecture states that certain intersection numbers of Mumford-Morita-Miller classes on \(\overline{M_{g,n}}\), a compactification of the moduli space of genus-\(g\) algebraic curves with it marked points, are given by an asymptotic expansion of a specific string solution of the integrable Korteweg-de Vries (KdV) hierarchy. Since this was proved by \textit{M. Kontsevich} [Commun. Math. Phys. 147, 1--23 (1992; Zbl 0756.35081)] for the case of 2-KdV using methods of differential topology and a specific matrix model, the general problem for \(n\)-KdV with arbitrary genus is still under investigation. In the article under review, the author presents calculations for intersection numbers in genus 3 for the special case of the 3-KdV (Boussinesq) hierarchy. The consideration is given in the framework of the axiomatic approach suggested by \textit{T. J. Jarvis}, \textit{T. Kimura} and \textit{A. Vaintrob} [Compos. Math 126, 157--212 (2001; Zbl 1015.14028)] and is in general based on the recursion relations for the auxiliary intersection numbers obtained recently by the present author [Int. Math. Res. Not. 2003, 2051--2094 (2003; Zbl 1070.14030)]. By expressing the correlators in terms ot these auxiliary intersection numbers the author establishes a relation between the correlators of genus 3 and genus 0 and thus proves Witten's conjecture for genus 3 in the case of the Boussinesq hierarchy. intersection numbers; moduli space of curves; Witten conjecture; Boussinesq hierarchy Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.), Relationships between algebraic curves and integrable systems, Families, moduli of curves (algebraic), Relationships between algebraic curves and physics, Relations of infinite-dimensional Hamiltonian and Lagrangian dynamical systems with algebraic geometry, complex analysis, and special functions Intersections in genus 3 and the Boussinesq hierarchy | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is 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 tangent space to the period space for Riemann surfaces of genus g at a curve C is naturally isomorphic to the second symmetric product of the dual of the vector space of holomorphic differentials on C. If C is Galois, then its group of automorphisms acts on this tangent space. The object of this and future papers of the author is to study the relationship between subspaces described representation-theoretically and the geometric properties of deformations of C in directions lying in these subspaces. The present paper deals with the case where C is a cyclic cover of \({\mathbb{P}}_ 1\). The results are rather too technical to be conveniently reproduced here. deformations of curve; cyclic cover of projective line; period space for Riemann surfaces; holomorphic differentials; group of automorphisms Coverings of curves, fundamental group, Families, moduli of curves (algebraic), Differentials on Riemann surfaces, Fuchsian groups and automorphic functions (aspects of compact Riemann surfaces and uniformization), Algebraic functions and function fields in algebraic geometry, Moduli of Riemann surfaces, Teichmüller theory (complex-analytic aspects in several variables) The geometry of the period mapping on cyclic covers of \({\mathbb{P}}_ 1\) | 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 This paper continues work of the author and Hartshorne and follows in response to Hartshorne's problem: Characterize those Chern classes which are possible for reflexive sheaves of rank 2 on \({\mathbb{P}}^ 3\). In particular, the only main result says:
Theorem (2.3). For Chern classes \(c_ 1, c_ 2, c_ 3\) satisfying \(c_ 1c_ 2\equiv c_ 3 (mod 2)\), \(0\leq c_ 3\leq 4c_ 2-c^ 2_ 1\) and furthermore (a) if \(4c_ 2-c^ 2_ 1\) is 7 or 15 then \(c_ 3\neq 0\); (b) if \(c_ 1\) is even and \(c_ 2\) odd, then \(c_ 3\leq 4c_ 2- c^ 2_ 1-6\); there is a ``suitable'' rank 2 sheaf of Chern classes \(c_ 1, c_ 2, c_ 3\) admitting semi-natural cohomology.
Here a cohomology, for a sheaf \({\mathcal E}\) of rank 2 on \({\mathbb{P}}^ 3\), is semi-natural if, for \(t\geq -2-c_ 1/2\), three of the cohomology groups \(H^ i({\mathcal E}(t))\) vanish, \(0\leq i\leq 3\). The sheaves of theorem (2.3) arise as a consequence of the existence of certain curves in V(\({\mathcal O}_{{\mathbb{P}}^ 3}(-a))\) satisfying conditions relating genus, degree, the number of lines possessed and the number of points on a line meeting the curve. Chern classes; reflexive sheaves of rank 2; semi-natural cohomology Hirschowitz, Existence de faisceaux réflexifs de rang deux sur P3 à bonne cohomologie, Inst. Hautes Études Sci. Publ. Math. 66 pp 105-- (1988) Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20], Families, moduli of curves (algebraic), Characteristic classes and numbers in differential topology, Geometric invariant theory Existence de faisceaux réflexifs de rang deux sur \({\mathbb{P}}^ 3\) à bonne cohomologie. (Existence of reflexive sheaves of rank two on \({\mathbb{P}}^ 3\) with good homology) | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is 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 moduli space of curves of genus \(g, \mathcal{M}_{g}\) let \(\mathcal{GP}_g\) be the locus of curves that do not satisfy the Gieseker--Petri theorem. In the genus seven case we show that \(\mathcal{GP}_7\) is a divisor in \(\mathcal{M}_7\). A. Castorena., Curves of genus seven that do not satisfy the Gieseker-Petri theorem, Bolletino U.M.I,(8)8-B(2005), 697--706. Families, moduli of curves (algebraic) Curves of genus seven that do not satisfy the Gieseker--Petri theorem | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians Let \({\mathfrak M}_g\) be the coarse moduli space of smooth curves of genus \(g\) over the complex field \(\mathbb{C}\). Let \({\mathfrak T}_g\subseteq {\mathfrak M}_g\) be the locus parametrizing isomorphism classes \([C]\) of tetragonal curves \(C\), i.e. curves \(C\) carrying at least one \(g^1_4\) but no \(g^1_d\) with \(d\leq 3\). -- It has been shown by \textit{E. Arbarello} and \textit{M. Cornalba} [Math. Ann. 256, 341-362 (1981; Zbl 0454.14023)] that \({\mathfrak T}_g\) is irreducible and unirational, its dimension is \(2g+3\) if \(g\geq 7\) and that the general tetragonal curve \(C\) has exactly one \(g^1_4\). If \(7\leq g\leq 9\) there exist tetragonal curves \(C\) carrying more than one \(g^1_4\) without being bielliptic, i.e. a double cover of an elliptic curve. If \(g\geq 10\) each tetragonal curve \(C\) carries exactly one \(g^1_4\) unless it is bielliptic, in which case it necessarily has infinitely many \(g^1_4\)'s [see e.g. \textit{A. Del Centina} and \textit{A. Gimigliano} in: The Curves Seminar at Queens, Vol. IX, Queen's Pap. Pure Appl. Math. 95 (1993; Zbl 0838.14023); lemma 2.9].
Let \(7\leq g\leq 9\) and denote by \({\mathfrak T}_g^{(n)}\) and \({\mathfrak M}_g^{be}\) the locus in \({\mathfrak T}_g\) of isomorphism classes \([C]\in {\mathfrak T}_g\) of curves carrying exactly \(n\) linear series \(g^1_4\)'s and that of bielliptic curves respectively. We have that \({\mathfrak T}_7={\mathfrak T}_7^{(1)}\cup{\mathfrak I}_7^{(2)} \cup{\mathfrak T}_7^{(3)} \cup{\mathfrak M}_7^{be}\), \({\mathfrak T}_8= {\mathfrak T}_8^{(2)} \cup{\mathfrak M}_8^{be}\) and \({\mathfrak T}_9= {\mathfrak T}_9^{(1)} \cup{\mathfrak T}_9^{(2)} \cup{\mathfrak M}^{be}_g\). The main results proved in sections 2, 3 and 4 are summarized in the following
Theorem. The loci \({\mathfrak T}_7^{(3)}\), \({\mathfrak T}_7^{(2)} \subseteq {\mathfrak M}_7\) and \({\mathfrak T}_8^{(2)} \subseteq{\mathfrak M}_8\) are rational, irreducible subvarieties of dimensions 16, 15 and 17, respectively.
For \(g=9\) the problem of the rationality of \({\mathfrak T}_9^{(2)}\) remains open.
In section 5 we focus our attention on particular subloci of \({\mathfrak T}_7\), namely the locus \({\mathfrak M}^3_7\) of points representing curves \(C\) carrying a theta-characteristic \({\mathfrak L}\) satisfying \(h^0(C,{\mathfrak L})=3\), and its intersection \(({\mathfrak M}^3_7)':= {\mathfrak M}^3_7 \cap{\mathfrak T}_7^{(2)} \subseteq {\mathfrak M}_7\). rationality of coarse moduli space of smooth curves; tetragonal curve G. Casnati and A. Del Centina, On certain spaces associated to tetragonal curves of genus 7 and 8, Commutative algebra and algebraic geometry (Ferrara), Lecture Notes in Pure and Appl. Math., vol. 206, Dekker, New York, 1999, pp. 35 -- 45. Special divisors on curves (gonality, Brill-Noether theory), Rational and unirational varieties, Families, moduli of curves (algebraic) On certain spaces associated to tetragonal curves of genus 7 and 8 | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is 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 E be a complex differentiable vector bundle over a compact oriented manifold M and f: \(E\to E^*\) a symmetric or anti-symmetric bundle map. Such a map is called general if the \((r+1)\times(r+1)\) minors of a matrix representing \(f_ x\), \(x\in M\) generate the ideal of the r-th degeneracy locus \(M_ r=\{x\in M\); rank f\({}_ x\leq r\}\) in the local ring \({\mathcal C}^{\infty}_{M,x}\) for almost all \(x\in M\) and if moreover \(M_ r\) has the expected codimension \((^{n-r+1}\!\!_ 2)\), \(n=\dim M\). The authors give formulae for the cohomology classes of \(M_ r\) for general f, analogous to Porteous' formulae for general bundle maps \(E\to F\) between bundles on M. By a ''squaring principle'' these results are extended for twisted bundle maps f: \(E\to E^*\otimes L\), where L is a line bundle.
Two applications are given. As a first application the degrees of various determinantal varieties are computed. The second application concerns families \(\pi\) : \(X\to B\) of genus g curves over a smooth compact base B (i.e. X and B are projective complex manifolds, \(\pi\) is proper and the fibres \(X_ t=\pi^{-1}(t)\) are curves of genus g, possibly singular). The authors obtain inequalities involving the Chern classes of B and those of the Hodge bundle \(R^ 1\pi {\mathcal O}_ x\) (assumed to be non- trivial) in two cases (i) dim B\(=1\) (ii) dim B\(=2\) and \(\pi\) has no singular fibres. In the first case the inequality roughly says that the more non-trivial the bundle \(R^ 1\pi {\mathcal O}_ X\), either the larger the genus of B or the greater the number of singular fibres. degeneracy locus of symmetric bundle maps; determinantal variety; variation of weight one Hodge structure; families of genus g curves Joe Harris & Loring W. Tu, ``On symmetric and skew-symmetric determinantal varieties'', Topology23 (1984) no. 1, p. 71-84 Sphere bundles and vector bundles in algebraic topology, Determinantal varieties, Characteristic classes and numbers in differential topology, Transcendental methods, Hodge theory (algebro-geometric aspects), Families, moduli of curves (algebraic) On symmetric and skew-symmetric determinantal varieties | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians This is the author's PhD-thesis. It consists of an introduction with motivation inspired by an exact sequence and a diagram relating the Betti cohomology of the modular curve \(Y_ 1(p)\), its compactification \(X_ 1(p)\) and the cusps, and a survey of results presented in the appropriate framework of modular units and Hecke operators. From the outset, the author's conviction that the Betty cohomology is but one realization of the so-called Anderson motive leads to an interpretation in terms of 1- motives. These turn out to be particularly suitable to attack three specific problems related to the aforementioned sequence and diagram. The introduction also gives a useful `Leitfaden' which enables the reader to turn directly to chapter 4, which contains a clear exposé on the coherence of the various constituent parts and the statements of the main results.
Chapter 1 gives a thorough discussion of 1-motives and their realizations, including the crystalline part. Special attention is given to so-called Kummer 1-motives, especially over a totally real number field. They turn out to be important for the eigenspace decomposition under the Hecke algebra of the various realizations of the main object under consideration, the Anderson extension. -- In chapter 2 Anderson's 1-motive is introduced, first over Spec\((\mathbb{Q})\) and afterwards over an arbitrary locally noetherian scheme. This 1-motive is at the basis of the construction of an Eisenstein section, providing Anderson's extension. At several places the author works over a locally noetherian scheme, but the final results are most often stated over \(\mathbb{Q}\). -- Chapter 3 is rather technical, but nonetheless of fundamental importance because it gives the comparison between Anderson's motive, the Betti realization of which is the cohomology mentioned above, and the realizations of Anderson's 1- motive. -- Chapter 5 contains a fairly detailed discussion of the construction of Hecke operators on Anderson's 1-motive and Anderson's extension, making use of the theory of modular curves à la Deligne- Rapoport, to show that the 1-motivic Eisenstein section induces the classical one in Betti cohomology. The final chapter treats Siegel functions to obtain modular units. The results of these last two chapters are used to formulate the main results of the thesis in chapter 4. This interesting work shows once more the intriguing nature of modular curves and related subjects coming from various disciplines.
Let \(p\geq 5\) be a prime number and let \(Y=Y_ 1(p)\) be the fine moduli scheme over \(\mathbb{Q}\) for the moduli problem of `elliptic curves with a point of exact order \(p\)', with smooth projective model \(X=X_ 1(p)\) over \(\mathbb{Q}\) and cusp divisor \(E=X\backslash Y\). Then \(E\) decomposes as a disjoint union of subschemes \(E_ 0\) and \(E_ \infty\) (in what follows, also written \(E_ W\) and \(E_ Z\), respectively). With respect to the compactified (coarse) moduli scheme \(X_ 0(p)\) (with cusps 0 and \(\infty\)) for the moduli problem of `elliptic curves with \(p\)-isogeny' and the canonical map \(\pi_{10}:X_ 1(p)\to X_ 0(p)\), one has \(E_ 0=\pi^{-1}_{10}(0)\) and \(E_ \infty=\pi^{-1}_{10}(\infty)\). Outside the geometric points of \(X_ 0(p)\) with only automorphisms \(\pm 1\), \(\pi_{10}\) is a Galois covering with group \(\mathbb{F}^*_ p/\pm 1\). For \(\ell\in\mathbb{F}^*_ p/\pm 1\) the corresponding automorphism of \(X\) is given by \(\tau_ \ell:(E,P)\mapsto(E,\ell\cdot P)\), where \(P\) is a \(p\)-division point of the elliptic curve \(E\). Also, in the context of the (coarse) moduli schemes \(Y(p)\) and \(X(p)\) for the moduli problem of `elliptic curves with full level \(p\) structure' and the corresponding map \(\pi_ 1:X(p)\to X_ 1(p)\), \(E_ 0\) is the cusp locus where \(\pi_ 1\) is unramified and \(E_ \infty\) is the one where \(\pi_ 1\) is ramified with index \(p\).
Define open subschemes \(j_ 1':W=W_ 1(p)\hookrightarrow X\) and \(j_ 2':Z=Z_ 1(p)\hookrightarrow X\) of \(X\) as follows: \(W=Y\cup E_ \infty\) and \(Z=Y\cup E_ 0\), thus \(X=Y\cup E_ 0\cup E_ \infty=W\cup E_ W=Z\cup E_ Z\). Write \(\text{Div}^ 0_ W\) for the free \(\mathbb{Z}\)-module of degree zero divisors on \(X\) with support in \(E_ W\), and let \(J_ Z=J_{Z_ 1(p)}\) denote the generalized jacobian of \(Z\), defined over \(\mathbb{Q}\) and representing line bundles on \(X\) trivialized along \(E_ Z\). One has a \(\text{Gal}(\overline\mathbb{Q}/\mathbb{Q})\)-invariant map \(u:\text{Div}^ 0_ W\to J_ Z(\overline\mathbb{Q})\), which associates to the divisor \(D\) its line bundle \({\mathcal O}(D)\) with canonical trivialization along \(E_ Z\). These data lead to the definition of the Anderson 1-motive: \(M_ A=[\text{Div}^ 0_ W@>u>>J_ Z]\).
Let \(R_ Z\) be the kernel of the canonical map \(\varphi:J_ Z\to J_ X=\text{Jac}(X)\), and write
\[
K_ 1=\text{Ker}\{u_ 1=\varphi\circ u:\text{Div}^ 0_ W\to J_ X\},
\]
then the Drinfeld-Manin theorem implies that the 1-motive \([K_ 1@>0>>J_ X]\) has finite index in the 1-motive \(M_ 1=[\text{Div}^ 0_ W@>u_ 1>>J_ X]\). It splits canonically into the direct sum of \([K_ 1\to 0]\) and \([0\to J_ X]\). This implies the existence of the so-called Eisenstein section \(s_ 1:[K_ 1\to 0]\to[\text{Div}^ 0_ W@>u_ 1>>J_ X]\), giving a partial splitting of the sequence
\[
[0\to J_ X]\to[\text{Div}^ 0_ W@>u_ 1>>J_ X]\to[\text{Div}^ 0_ W\to 0].
\]
Similarly there is an Eisenstein section \(s_ 2:[K_ 2\to 0]\to[\text{Div}^ 0_ Y@>u_ 2>>J_ X]\), where \(K_ 2=\text{Ker}(u_ 2)\) with \(u_ 2\) defining the 1-motive \(M_ 2=[\text{Div}^ 0_ Y@>u_ 2>>J_ X]\). The Eisenstein section \(s_ 1\) can be used to define the 1-motivic Anderson extension \(M_{AE}=[K_ 1@>u_{AE}>>R_ Z]\) of \([K_ 1\to 0]\) with \([0\to R_ Z]\). One is interested in an explicit description of \(M_{AE}\). Using the theory of modular units, more specifically results from the theory of Siegel functions, this goal is achieved in the following sense: (i) For any prime number \(\ell\) (including \(\ell=p)\) the \(\ell\)-adic realization of \(M_{AE}\) considered as \(\text{Gal}(\overline\mathbb{Q}/\mathbb{Q}(\zeta_ p)^ +)\)-module, where \(\mathbb{Q}(\zeta_ p)^ +\subset\mathbb{Q}(\zeta_ p)\) is the maximal totally real subfield, is unramified outside \(\ell\); (ii) \(u_{AE}\) is injective, i.e. \(M_{AE}\) is as `non-trivial as possible'.
One can construct Hecke operators \(T_ \ell\), for all primes \(\ell \neq p\), acting as endomorphisms of order \({p-1\over 2}\), on \(M_ 1\), \(M_ 2\), \(M_ A\) and \(M_{AE}\). They induce actions on the realizations of \(M_ 2\) which can be identified with the action of the classical Hecke operators on the motivic cohomology \(H^ 1_{\mathcal M}(X,j'_{2!}Rj_{1*}\mathbb{Z})\) (1), where \(j_ 1:Y\hookrightarrow Z\) is the embedding. The automorphisms \(\tau_ \ell\) induce \(\tau_{\ell*}\) with inverse \(\tau^*_ \ell\) acting on \(\text{Div}_ Y\), \(\text{Div}_ W\) and \(\text{Div}_ Z\) such that the action of \(T_ \ell\) on \(\text{Div}^ 0_ W\) and \(\text{Div}^ 0_ Z\) is given by \(1+\ell\cdot\tau_{\ell*}\) and \(\ell+\tau^*_ \ell\), respectively. On \(\text{Div}^ 0_ Y\) the \(T_ \ell\) act by restriction of \((1+\ell\cdot\tau_{\ell*})\oplus(\ell+\tau^*_ \ell)\) to \(\text{Div}_ W\oplus\text{Div}_ Z\). For \(M_{AE}\) one obtains: For primes \(\ell\neq p\) the Hecke operators \(T_ \ell\) are given by \(T_ \ell=1+\ell\cdot\tau_{\ell*}\).
According to the Drinfeld-Manin theorem \(K_ 2=\text{Ker}(u_ 2)\) has finite index in \(\text{Div}^ 0_ Y\) and one may ask for the Galois module structure of the quotient \({\mathcal C}^ 0_ Y=\text{Div}^ 0_ Y/K_ 2\), the \(E\)-divisor class group. Analogously, one has the \(E_ W\)- and \(E_ Z\)-divisor class groups \({\mathcal C}^ 0_ W\) and \({\mathcal C}^ 0_ Z\). A formula for the order \(|{\mathcal C}^ 0_ Y|\) is derived: \(|{\mathcal C}^ 0_ Y|={A_ Y\over B_ Y}\cdot|{\mathcal C}^ 0_ W|\cdot|{\mathcal C}^ 0_ Z|\), where \(A_ Y\) divides \(\left(\text{numerator of } {p-1\over 2}\right)\cdot{p-1\over 2}\), and where \(|{\mathcal C}^ 0_ Z|=|{\mathcal C}^ 0_ W|\) can be expressed in terms of generalized Bernoulli numbers by a result of Kubert and Lang. The \(B_ Y=\left|{\text{HDiv}^ 0_ Y\over\text{HDiv}_ W\oplus\text{HDiv}_ Z}\right|\), with HDiv denoting the principal divisors, remains unkown. For the \(p\)-part \({\mathcal C}_ Y^{0(p)}\) of \({\mathcal C}^ 0_ Y\) one obtains: \({\mathcal C}_ Y^{0(p)}={\mathcal C}_ W^{0(p)}\oplus{\mathcal C}_ Z^{0(p)}\) as Galois module. Hecke operator; modular curve; Kummer 1-motive; Eisenstein section; Anderson extension; Anderson 1-motive; modular units; divisor class group; generalized Anderson extension C. BRINKMANN , Die Andersonextension und 1-Motive , Bonner Mathematische Schriften. 223. Bonn : Univ. Bonn, Math.-Naturwiss. Fak. ( 1991 ). MR 94a:11087 | Zbl 0749.14001 Generalizations (algebraic spaces, stacks), Families, moduli of curves (algebraic), Jacobians, Prym varieties, Applications of methods of algebraic \(K\)-theory in algebraic geometry The Anderson extension and 1- motives | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians It is well known that the variety of irreducible plane projective algebraic curves of a given degree with a given number of nodes is non-singular [\textit{F. Severi}, 1921], irreducible [\textit{J. Harris}, 1985] and that locally this variety is the transversal intersection of germs of equisingular strata corresponding to individual singular points [\textit{Severi}] (a variety with the latter property will henceforth be referred to as \(T\)-variety, or variety with the property \(T)\). The curves are defined over an algebraically closed field of characteristic zero. -- For more complicated singularities (beginning with ordinary cusps) there are examples of reducible equisingular strata (shortly ESS) [\textit{O. Zariski}, 1935], non-smooth strata [\textit{I. Luengo}, 1987], or strata without property \(T\) [\textit{J. Wahl}, 1974 and the author, 1994].
On the other hand, various sufficient conditions for ESS to have the properties listed above were found [\textit{Zariski}, \textit{Greuel}-\textit{Karras}, \textit{Ran}, the author, \textit{Vassiliev}]. The author does not formulate them, but only emphasizes that they guarantee the smoothness, \(T\)-property and irreducibility of ESS when the number of all singularities, or singularities different from nodes, is bounded above by a linear function of the degree \(d\) of the curve, whereas this number may be of order \(d^2\).
In a previous paper the author [Bull. Soc. Math. Fr. 122, No. 2, 235-253 (1994; Zbl 0854.14015)] proved the smoothness and irreducibility of families of curves with nodes and cusps when the total Milnor number was bounded above by a quadratic function of the curve degree. The goal of the paper under review is to develop the method suggested in the latter paper and to get similar results for curves with arbitrary singularities. equisingular strata; families of curves [Sh4] Shustin, E.: Geometry of equisingular families of plane algebraic curves. J. Algebr. Geom.5, 209--234 (1996). Families, moduli of curves (algebraic), Singularities of curves, local rings Geometry of equisingular families of plane 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 Let \(Y\) be a smooth curve of positive genus \(g = g(Y)\), and \(U_Y = U_Y(n,d)\) be the moduli space of semi-stable vector bundles of rank \(n\) and degree \(d\) on \(Y\). One approach to the study of \(U_Y\) is to specialize \(Y\) to a curve \(X_o\) with one ordinary double point, and then to introduce an appropriate moduli object \(U'_{X_o}\) with good specialization properties from \(U_Y\) to \(U'_{X_o}\) when \(Y\) specializes to \(X_o\). Since \(X_o\) is of genus \(g-1\) then one would expect to get this way a machinery for studying \(U_Y\) by induction on \(g(Y)\). Such a strategy (a construction of a special moduli object \(U'_{X_o} = G_{X_o}\) in the rank 2 case) has been used by \textit{D. Gieseker} [J. Differ. Geom. 19, 173-206 (1984; Zbl 0557.14008)] to prove a conjecture of Newstead and Ramanan that the Chern classes \(c_i\) of \(U_Y(2,1)\) vanish for \(i \geq 2g-1\), and lately generalized by the author and Nagaraj for arbitrary rang [\textit{D. S. Nagaraj} and \textit{C. S. Seshadri}, Proc. Indian Acad. Sci., Math. Sci. 109, No. 2, 165-201 (1999; Zbl 0957.14021)].
One of the main goals of these lectures is to give a construction (see \S 4 and \S 5) of such good moduli object \(U'_{X_o} = G(n,d)\) for any rank \(n \geq 2\) with \((n,d) = 1\), which generalizes Gieseker's \(G_{X_o}\) for \(n = 2\). In particular, in \S 6 is given a concrete realization of \(G(2,1)\) together with an argument how this model of \(U'_{X_o}\) (for \(n = 2\) and \(d = 1\)) is used to re-prove the conjecture of Newstead and Ramanan. degenerations; moduli space; vector bundles on curves C. S. Seshadri, Degenerations of the moduli spaces of vector bundles on curves, School on Algebraic Geometry (Trieste, 1999) ICTP Lect. Notes, vol. 1, Abdus Salam Int. Cent. Theoret. Phys., Trieste, 2000, pp. 205 -- 265. Vector bundles on curves and their moduli, Families, moduli of curves (algebraic), Formal methods and deformations in algebraic geometry Degenerations of the 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 This survey article contains the lecture notes for the author's mini-course taught at the Graduate Summer School of the IAS/Park City Mathematics Institute in July 2011. As the author points out, the main goal of his five lectures is to explain why the absolute Galois group \(\mathrm{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})\) and its action on the profinite completion of the fundamental group of a \(n\)-pointed Riemann surface of genus \(g\) appear quite naturally in the study of the corresponding mapping class group, and therefore also in the geometric description of the moduli space \(M_{g,n}\) of such surfaces.
This approach is based on Grothendieck's theory of the algebraic fundamental group of an algebraic variety, and consequently this abstract framework is briefly introduced in Lecture 1. In this context, the algebraic fundamental group of an algebraic variety \(K\) defined over a subfield \(K\) of \(\mathbb{C}\) appears as the profinite completion of the classic topological fundamental group, and its connection with the Galois group \(\mathrm{Gal}(\overline K/K)\) is explained via Grothendieck's seminal work developed in the famous seminar notes ``SGA 1'' [Lecture Notes in Mathematics. 224. Berlin-Heidelberg-New York: Springer-Verlag. XXII, 447 p. (1971; Zbl 0234.14002)]. In the special case of the Teichmüller description of the moduli space \(M_{g,n}(\mathbb{C})\) it is shown that the Galois group \(\mathrm{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})\) naturally appears in a short exact sequence containing the algebraic fundamental groups of \(M_{g,n}(\mathbb{C})\) and \(M_{g,n}(\mathbb{Q})\), on the one hand, and in an exact sequence containing the so-called arithmetic mapping class group \(\mathrm{AMCG}_{g,n}\) and the profinite completion of the ordinary mapping class group \(\mathrm{MCG}_{g,n}\) alternatively.
Lecture 2 briefly recalls the monodromy representation on topological fundamental groups, thereby pointing out that the above Galois representation of \(\mathrm{Gal}(\mathbb{Q}/\mathbb{Q})\) may be viewed as an analogue of the latter in the arithmetic/algebraic case.
Lecture 3 provides a sketchy description of the arithmetic mapping class groups \(\mathrm{AWCG}_{g,n}\), with the main reference being the related work of \textit{T. Oda} [Étale homotopy type of the moduli spaces of algebraic curves. Geometric Galois actions 1. Cambridge: Cambridge University Press. Lond. Math. Soc. Lect. Note Ser. 242, 85--95 (1997; Zbl 0902.14019)].
Lecture 4 points to some more recent results concerning the properties of various Galois representations of \(\mathrm{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})\) with regard to outer fundamental groups of \((g,n)\)-surfaces, or to the profinite completion of the mapping class group \(\mathrm{MCG}_{g,n}\), respectively. One of the main references, in this context, is the author's own work [\textit{M. Matsumoto} and \textit{A. Tamagawa}, Am. J. Math. 122, No. 5, 1017--1026 (2000; Zbl 0993.12002)].
Finally, Lecture 5 discusses the question of how these Galois actions of \(\mathrm{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})\) can be used to obtain a bound for the size of the mapping class group \(\mathrm{MCG}_{g,n}\) in the group of automorphisms of the Lie algebraization of the topological fundamental group of a \((g,n)\)-surface by giving Galois obstructions to the surjectivity of the so-called higher Johnson homomorpbisms. In this context, the geometry of the mixed Hodge structure on the topological fundamental group of a \((g,n)\)-surface plays a crucial role, just as the recent progress concerning related conjectures of Deligne-Ihara and of T. Oda does. The paper ends with an appendix, the purpose of which is to describe algebraic fundamental groups in terms of étale coverings and fiber functors à la A. Grothendieck. algebraic fundamental group; arithmetic mapping class group; Galois representations; Johnson homomorphisms; graded Lie algebras M. Matsumoto, Introduction to arithmetic mapping class groups, Moduli spaces of Riemann surfaces, IAS/Park City Math. Ser. 20, American Mathematical Society, Providence (2013), 319-356. Homotopy theory and fundamental groups in algebraic geometry, Families, moduli of curves (algebraic), Coverings of curves, fundamental group, Coverings in algebraic geometry, Families, moduli of curves (analytic), Moduli of Riemann surfaces, Teichmüller theory (complex-analytic aspects in several variables) Introduction to arithmetic 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 A generalized Fermat curve of type \((p,n)\) is a closed Riemann surface \(S\) admitting a group \(H\cong \mathbb{Z}_p^n\) of conformal automorphisms with \(S/H\) being the Riemann sphere with exactly \(n+1\) cone points, each one of order \(p\). If \((p-1)(n-1)\geq 3\), then \(S\) is known to be non-hyperelliptic and generically not quasiplatonic. Let us denote by \(\operatorname{Aut}_H(S)\) the normalizer of \(H\) in \(\operatorname{Aut}(S)\). If \(p\) is a prime, and either (i) \(n=4\) or (ii) \(n\) is even and \(\operatorname{Aut}_H(S)/H\) is not a non-trivial cyclic group or (iii) \(n\) is odd and \(\operatorname{Aut}_H (S)/H\) is not a cyclic group, then we prove that \(S\) can be defined over its field of moduli. Moreover, if \(n\in \{3,4\}\), then we also compute the field of moduli of \(S\). algebraic curves; Riemann surfaces; field of moduli; field of definition Hidalgo, R. A.; Reyes-Carocca, S.; Valdés, M. E.: Field of moduli and generalized Fermat curves, Rev. colomb. Mat. 47, No. 2, 205-221 (2013) Automorphisms of curves, Families, moduli of curves (algebraic), Special algebraic curves and curves of low genus, Compact Riemann surfaces and uniformization Field of moduli and generalized Fermat 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 genus \(g\) hyperelliptic curves with reduced automorphism group \(A _{5}\) and give equations \(y ^{2} = f(x)\) for such curves in both cases where \(f(x)\) is a decomposable polynomial in \(x ^{2}\) or \(x ^{5}\). For any fixed genus the locus of such curves is a rational variety. We show that for every point in this locus the field of moduli is a field of definition. Moreover, there exists a rational model \(y ^{2} = F(x)\) or \(y ^{2} = x F(x)\) of the curve over its field of moduli where \(F(x)\) can be chosen to be decomposable in \(x ^{2}\) or \(x ^{5}\). While similar equations have been given by \textit{E. Bujalance, F. J. Cirre, J. M. Gamboa} and \textit{G. Gromadzki} [Symmetry types of hyperelliptic Riemann surfaces. Mém. Soc. Math. Fr,. Nouv. Sér. 86, (2001; Zbl 1078.14044)] over \({\mathbb R}\), this is the first time that these equations are given over the field of moduli of the curve. Automorphisms of curves, Arithmetic ground fields for curves, Families, moduli of curves (algebraic) Hyperelliptic curves with reduced automorphism group \(A_{5}\) | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians We give an overview of known and new techniques on how one can obtain explicit equations for candidates of good towers of function fields. The techniques are founded in modular theory (both the classical modular theory and the Drinfeld modular theory).
In the classical modular setup, optimal towers can be obtained, while in the Drinfeld modular setup, good towers over any nonprime field may be found. We illustrate the theory with several examples, thus explaining some known towers as well as giving new examples of good explicitly defined towers of function fields. towers of function fields; Drinfeld modules; curves with many points Algebraic functions and function fields in algebraic geometry, Families, moduli of curves (algebraic), Arithmetic theory of algebraic function fields, Computational aspects of algebraic curves Good towers of function fields | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians In this article, the author studies the equivariant cohomology of the moduli space of semistable rank two vector bundles of even degree over a compact Riemann surface. For the odd degree case, the moduli space is a smooth projective variety, and this case has been extensively studied. For the even degree case considered here, the moduli space is a singular projective variety. The author proves an analogue to the Mumford conjecture, which concerns finding a complete set of relations for generators of the cohomology. He also gives a structure theorem which completely determines the ring structure of the equivariant cohomology in this case. equivariant cohomology ring; moduli spaces; intersection cohomology; Riemann surface; Mumford conjecture Vector bundles on curves and their moduli, Riemann surfaces; Weierstrass points; gap sequences, Algebraic moduli problems, moduli of vector bundles, Families, moduli of curves (algebraic), Other algebro-geometric (co)homologies (e.g., intersection, equivariant, Lawson, Deligne (co)homologies), Equivariant homology and cohomology in algebraic topology The equivariant cohomology ring of the moduli space of vector bundles over a Riemann surface | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians We show explicitly that the perturbative \(\text{SU}(N)\) Chern-Simons theory arises naturally from two Penner models, with opposite coupling constants. As a result computations in the perturbative Chern-Simons theory are carried out using the Penner model, and it turns out to be simpler and transparent. It is also shown that the connected correlators of the puncture operator in the Penner model are related to the connected correlators of the operator that gives the Wilson loop operator in the conjugacy class. Families, moduli of curves (algebraic), Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects), Moduli of Riemann surfaces, Teichmüller theory (complex-analytic aspects in several variables), Topological properties in algebraic geometry, String and superstring theories; other extended objects (e.g., branes) in quantum field theory, String and superstring theories in gravitational theory, Applications of manifolds of mappings to the sciences Perturbative Chern-Simons theory from the Penner model | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians \textit{K. Kato} proved a formula which compares the dimension of the space of vanishing cycles in a finite morphism between formal germs of curves over a complete discrete valuation ring [Duke Math. J. 55, 629--659 (1987; Zbl 0665.14005)]. Kato's formula is explicit only in the case where the morphism in question is generically separable on the level of special fibres.
In the paper under review, using formal patching techniques à la Harbater, the author proves an analogous explicit formula in the case of a Galois cover of degree \(p\) between formal germs of curves over a complete discrete valuation ring of unequal characteristic \((0,p)\). This formula includes the case when one has inseparability on the level of special fibres and has applications in the study of semi-stable reduction of Galois covers of curves. finite morphism; Galois cover; formal germs of curves; complete discrete valuation ring Saïdi, M.: Wild ramification and a vanishing cycles formula. J. algebra 273, No. 1, 108-128 (2004) Families, moduli, classification: algebraic theory, Families, moduli of curves (algebraic), Singularities in algebraic geometry Wild ramification and a vanishing cycles formula. | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is 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 past few decades, moduli spaces of curves have become increasingly
prominent and important in mathematics. In fact, the study of moduli spaces
lies at the centre of a rich confluence of rather disparate areas such as geometry, combinatorics and string theory. Starting from baby principles, I will describe exactly what a moduli space is and motivate the study of its
intersection theory. This scenic tour will guide us towards a discussion of
Kontsevich's combinatorial formula, including a description of a new proof. Families, moduli of curves (algebraic), (Equivariant) Chow groups and rings; motives A tourist's guide to intersection theory on 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 In the paper under review the authors use localization techniques to express the ratios of certain tautological classes in terms of simple Hurwitz numbers.
Let \(R^*(\mathcal M_g)\) be the tautological ring of \(\mathcal M_g\). A conjecture due to \textit{C. Faber} predicts that \(R^*(\mathcal M_g)\) satisfies Poincaré duality with socle in degree \(g-2\) [see Faber, Carel (ed.) et al., Moduli of curves and abelian varieties. The Dutch intercity seminar on moduli. Braunschweig: Vieweg. Aspects Math. E33, 109--129 (1999; Zbl 0978.14029)] and, in fact, by results of \textit{C. Faber} in [loc. cit.] and of \textit{E. Looijenga} in [Invent. Math. 121, No. 2, 411--419 (1995; Zbl 0851.14017)], its degree \(g-2\) part is known to be one dimensional. A family of degree \(g-2\) tautological classes can be obtained by considering the \((2g-1)\)-dimensional moduli space \(\mathcal A_{dd}^g\) of connected genus \(g\) degree \(d\) covers of \(\mathbb P^1\) fully ramified over \(0\) and \(\infty\) and simply ramified over the other (\(2g\)) branch points, together with its natural (source) map \(\mathcal A_{dd}^g\to\mathcal M_g\) (for \(d=2\) this corresponds to the hyperelliptic locus of \(\mathcal M_g\)). This map can be extended to a map \(\overline{\mathcal A}_{dd}^g\to\overline{\mathcal M}_g\), where \(\overline{\mathcal A}_{dd}^g\) is the smooth and proper stack of admissible covers constructed by \textit{D. Abramovich, A. Corti} and \textit{A. Vistoli} in [Commun. Algebra 31, No. 8, 3547--3618 (2003; Zbl 1077.14034)], allowing to evaluate the classes \([\mathcal A_{dd}^g]\) in \(R^*(\mathcal M_g)\) by integrating with suitable elements of the Chow ring of \(\overline{\mathcal M}_g\).
Simple Hurwitz numbers are combinatorial objects counting ramified covers of \(\mathbb P^1\) with prescribed ramification data over \(0\) and simple ramification over the other (fixed) branch points. Even if, in general, Hurwitz numbers are hard to compute, in the case that we have full ramification over \(0\), there is a simple way of write them down in generating function form due to computations of \textit{B. Shapiro, M. Shapiro} and \textit{A. Vainshtein} in [Khovanskij, A. (ed.) et al., Topics in singularity theory. V. I. Arnold's 60th anniversary collection. Providence, RI: American Mathematical Society. Transl., Ser. 2, Am. Math. Soc. 180(34), 219--227 (1997; Zbl 0883.05072)].
By localizing on spaces of admissible covers, the authors get a relation between generating functions for Hurwitz-Hodge integrals [see \textit{J. Bryan, T. Graber} and \textit{R. Pandharipande}, J. Algebr. Geom. 17, No. 1, 1--28 (2008; Zbl 1129.14075)] and for simple Hurwitz numbers fully ramified over \(0\). Combining this formula with computations due to the second author in [Algebra Number Theory 1, No. 1, 35--66 (2007; Zbl 1166.14036)], the ratios of several tautological classes can be expressed in a purely combinatorial way. In particular, this master relation gives back a formula due to \textit{J. Bryan} and \textit{R. Pandharipande} in [J. Am. Math. Soc. 21, No. 1, 101--136 (2008; Zbl 1126.14062)] computing all proportionalities of \([\mathcal A_{dd}^g]\) in \(R^*(\mathcal M_g)\). Hurwitz numbers; tautological classes; admissible covers; Gromov-Witten theory Bertram A., Cavalieri R., Todorov G.: Evaluating tautological classes using only Hurwitz numbers. Trans. Amer. Math. Soc. 360(11), 6103--6111 (2008) Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects), Families, moduli of curves (algebraic) Evaluating tautological classes using only Hurwitz numbers | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians We run Mori's program for the moduli space of stable pointed rational curves with divisor \(K+\sum a_i\psi _i\). We prove that the birational model for the pair is either the Hassett space of weighted pointed stable rational curves for the same weights or the GIT quotient of the product of projective lines with the linearization given by the same weights. Carr, S.: A polygonal presentation of \({Pic(\overline{\mathfrak{M}}_{0,n})}\) , (2009). arXiv:0911.2649 Algebraic moduli problems, moduli of vector bundles, Minimal model program (Mori theory, extremal rays), Families, moduli of curves (algebraic) Log canonical models for the moduli space of stable pointed rational curves | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians Let \(k\) be a field of positive characteristic, and let \(X\) be a \(K3\) surface over \(k\). A finite group action \(G\) on \(X\) is called \textit{symplectic} if every element of \(G\) fixes the nowhere vanishing \(2\)-form on \(X\). The paper under review looks at symplectic group actions on \(K3\) surfaces defined over a field \(k\) of positive characteristic, and discuss their influences on the quantities that exist exclusively in positive characteristic, e.g., the height of formal Brauer groups, Artin invariants of \(K3\) surfaces. The main results established here are that the height of the formal Brauer group and the Artin invariant of a \(K3\) surface are invariant under symplectic group actions. Special attention is given to the \(K3\) surfaces, which are one-parameter deformations of \(K3\) surfaces in weighted projective \(3\)-spaces. A typical example of such a surface is a deformation of a weighted diagonal \(K3\) surface, and is defined by the equation of the form \(x_0^{d_0}+x_1^{d_1}+x_2^{d_2}+x_3^{d_3}-\lambda x_0x_1x_2x_3=0\) with a parameter \(\lambda\) in a weighted projective \(3\)-space. Picard numbers, the height of formal Brauer groups, and Artin invariants are computed for these \(K3\) surfaces. Moreover, the Tate conjecture is proved for (some) of these \(K3\) surfaces over a finite field \(k\). Also a formula counting the number of rational points over a finite field of \(q\)-elements on these \(K3\) surfaces is given. formal Brauer groups; Artin invariants; mirror symmetry \(K3\) surfaces and Enriques surfaces, Varieties over finite and local fields, Brauer groups of schemes, Group actions on varieties or schemes (quotients), Picard groups \(K3\) surfaces with symplectic group actions | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is 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 tautological ring of the moduli space of stable \(n\)-pointed curves of genus 2 with rational tails. The algebra is described in terms of explicit generators and relations. It is proved that this algebra is Gorenstein. Families, moduli of curves (algebraic), (Equivariant) Chow groups and rings; motives The tautological ring of the moduli space \(M_{2,n}^{rt}\) | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is 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 stable curve. Fix an integer \(r>0\). Here we give a bijection between the set of all disconnecting nodes of \(X\) and the depth 1 sheaves with pure rank \(r\) on \(X\) satisfying certain properties (among them \(F\) spanned, \(\deg(F)=r\), \(h^0(X,F)\geq 2r\) and \(F\) ``maximally non-locally free''). Let \(X_1\) and \(X_2\) be the closures in \(X\) of \(X\setminus\{P\}\), If \(F\) is \(\omega_X\)-semistable, then \(p_a(X)= 2\cdot p_a(X_1)= 2\cdot p_a(X_2)\). The converse is true if \(X_1\) and \(X_2\) are irreducible. stable curve; reducible curve; Brill-Noether theory; Brill-Noether theory for vector bundles Vector bundles on curves and their moduli, Families, moduli of curves (algebraic), Special divisors on curves (gonality, Brill-Noether theory) Brill-Noether theory of rank \(r\) sheaves on stable curves: an extremal case | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is 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 construct an open substack \(\mathcal{U}\subset\mathcal{M}_{g,1}\) with the complement of codimension \(\ge 2\) and a morphism from \(\mathcal{U}\) to a weighted projective stack, which sends the Weierstrass locus \(\mathcal{W}\cap\mathcal{U}\) to a point, and maps \(\mathcal{M}_{g,1}\setminus\mathcal{W}\) isomorphically to its image. The construction uses alternative birational models of \(\mathcal{M}_{g,1}\) and \(\mathcal{M}_{g,2}\) studied earlier by the author. Families, moduli of curves (algebraic), Stacks and moduli problems, Singularities of curves, local rings, Riemann surfaces; Weierstrass points; gap sequences Contracting the Weierstrass locus to a point | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians Let \(n>1\) be an integer such that \(X_0(n)\) has genus \(0\), and let \(K\) be a field of characteristic \(0\) or relatively prime to \(6n\). In this paper, we explicitly classify the isogeny graphs of all rational elliptic curves that admit a nontrivial isogeny over \(\mathbb{Q}\). We achieve this by introducing \(56\) parameterized families of elliptic curves \(\mathcal{C}_{n, i}(t,d)\) defined over \(K(t,d)\), which have the following two properties for a fixed \(n\): the elliptic curves \(\mathcal{C}_{n, i}(t,d)\) are isogenous over \(K(t,d)\), and there are integers \(k_1\) and \(k_2\) such that the \(j\)-invariants of \(\mathcal{C}_{n, k_1}(t,d)\) and \(\mathcal{C}_{n, k_2}(t,d)\) are given by the Fricke parameterizations. As a consequence, we show that if \(E\) is an elliptic curve over a number field \(K\) with isogeny class degree divisible by \(n\in\{4,6,9\}\), then there is a quadratic twist of \(E\) that is semistable at all primes \(\mathfrak{p}\) of \(K\) such that \(\mathfrak{p}\nmid n\). elliptic curves; isogenies; additive reduction; parameterized families of elliptic curves Elliptic curves over global fields, Isogeny, Families, moduli of curves (algebraic), Elliptic curves Explicit classification of isogeny graphs of rational elliptic curves | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians Let \({\mathcal M}_ g\) be the moduli space of stable curves of genus \(g\). The author first finds \(a={[(g^ 2-1)/4]}+3g-3\) generators for the Chow group \(A^ i_ \mathbb{Q}(\overline{{\mathcal M}}_ g-{\mathcal M}_ g)\) and uses this fact to show that \(a\) is an upper bound on the rank of the homology group \(H_{2(3g-3)-4}(\overline {\mathcal M}_ g-{\mathcal M}_ g,\mathbb{Q})\). The dual basis for \(H_ 4(\overline{\mathcal M}_ g)\) is seen to be algebraic. tautological class; intersection product; moduli space of stable curves; generators for the Chow group; rank of the homology group Edidin, D, The codimension-two homology of the moduli space of stable curves is algebraic, Duke Math. J., 67, 241-272, (1992) Families, moduli of curves (algebraic), Parametrization (Chow and Hilbert schemes), Étale and other Grothendieck topologies and (co)homologies The codimension-two homology of the moduli space of stable curves is algebraic | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is 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.]
Let \(C\) be a smooth connected complex curve of genus \(g\), and for all integers \(r\) and \(d\), let \(W^ r_ d(C)\) be the subscheme of \(Pic^ d(C)\) parametrizing isomorphism classes of line bundles of degree \(d\) with at least \(r\) independent sections. When \(C\) is general, the latter has dimension \(\rho = g - (r + 1)(g - d + r)\); when \(\rho \geq 2\), results of Fulton and Lazarsfeld imply that the Albanese variety of \(W^ r_ d(C)\) is isomorphic to \(\text{Pic}^ d(C)\). When \(\rho = 1\), the scheme \(W^ r_ d(C)\) is a smooth curve; it generates \(\text{Pic}^ d(C)\) hence induces a surjection \(\text{Jac}(W^ r_ d(C)) \to \text{Pic}^ d(C)\). The kernel \(K^ r_ d(C)\) is defined over the field \(\mathcal K\) of rational functions of the moduli space of pointed curves of genus \(g\), and is in general nontrivial. It is proved that if \(C\) is general of genus \(g\geq 3\) and \(\rho = 1\), the kernel \(K^ r_ d(C)\) is connected, and that the ring of rationally determined endomorphisms (i.e. defined over \(\mathcal K\)) of the abelian variety \(K^ 1_ d(C)\) is isomorphic to \(\mathbb{Z}\). An example of Pirola shows that the latter does not hold for \(r\geq 2\). The authors believe that the ring of all endomorphisms of \(K^ 1_ d(C)\) should still be isomorphic to \(\mathbb{Z}\), but they could not prove it. This theorem implies the extension to the case \(\rho = r = 1\) of a result of \textit{C. Ciliberto} [Duke Math. J. 55, 909-917 (1987; Zbl 0657.14013)] who proved it for \(\rho \geq 2\): the group of rationally determined line bundles on the curves of the universal family over any component of the Hurwitz scheme of coverings of \(\mathbb{P}^ 1\) of degree \(d\) and genus \(g \geq 3\) is generated by the relative canonical bundle and the bundle \({\mathcal O}_{\mathbb{P}^ 1}(1)\). The question remains whether this statement still holds for \(\rho = 1\) and \(r\geq 2\), or for \(\rho = 0\). The proof of the first result proceeds by degenerating the curve to a reducible tree-like curve, and by studying the limit of \(K^ 1_ d(C)\) using limit linear series. limit linear series; higher Picard group; higher Jacobian; complex curve; isomorphism classes of line bundles of degree \(d\); Albanese variety; moduli space of pointed curves Ciro Ciliberto, Joe Harris, and Montserrat Teixidor i Bigas, On the endomorphisms of \?\?\?(\?\textonesuperior _{\?}(\?)) when \?=1 and \? has general moduli, Classification of irregular varieties (Trento, 1990) Lecture Notes in Math., vol. 1515, Springer, Berlin, 1992, pp. 41 -- 67. Jacobians, Prym varieties, Families, moduli of curves (algebraic), Picard schemes, higher Jacobians, Divisors, linear systems, invertible sheaves On the endomorphisms of \(\text{Jac}(W_ d^ 1(C))\) when \(\rho=1\) and \(C\) has general 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 The main object of the paper under review is the elementary obstruction \(ob(X)\) for the existence of a rational point (or, more generally, of a zero-cycle of degree 1), on a smooth, geometrically integral variety \(X\) defined over an arbitrary field \(k\), as well as its various analogues. The elementary obstruction \(ob(X)\), first introduced and studied by \textit{J.-L.~Colliot-Thélène and J.-J.~Sansuc} [Duke Math. J. 54, 375--492 (1987; Zbl 0659.14028)] and revisited by \textit{M.~Borovoi, J.-L.~Colliot-Thélène and A.~N.~Skorobogatov} [Duke Math. J. 141, 321--364 (2008; Zbl 1135.14013)], is defined as the class in \({\text{Ext}}^1_{\mathfrak g}(\bar k(X)^*/\bar k^*, \bar k^*)\) of the extension of \(\mathfrak g\)-modules
\[
1\to \bar k^*\to \bar k(X)^* \to \bar k(X)^*/\bar k^*\to 1
\]
(here \(\mathfrak g\) stands for the Galois group \({\text{Gal}}(\bar k/k)\) of a separable closure of \(k\)).
The author compares the order of \(ob(X)\) with the following invariants of \(X\): the index \(I(X)\), the greatest common divisor of the degrees of the closed points on \(X\); the period \(P(X)\), the order of the class of the Albanese torsor \({\text{Alb}}^1_{X/k}\) in the Galois cohomology group \(H^1(k, {\text{Alb}}^0_{X/k})\) (in the appendix to the paper the author proves that \({\text{Alb}}^1_{X/k}\) and \({\text{Alb}}^0_{X/k}\) exist for any geometrically integral variety \(X\) over an arbitrary field \(k\)); the generic period \(P_{\text{gen}}(X)\), the supremum of \(P(U)\) where \(U\) ranges over all dense open subsets of \(X\). The first result of the paper states that 1) \(P(X) | P_{\text{gen}}(X) | I(X)\); 2) the order of \(ob(X)\) divides \(P_{\text{gen}}(X)\); 3) these invariants satisfy no further divisibility relations.
This result has interesting applications in the cases where \(k\) is a \(p\)-adic field, a real closed field, a number field, or a field of dimension \(\leq 1\). Some of these results are related to the above cited paper by Borovoi et al. and answer some questions raised there. Among the topics discussed here one can note the behaviour of the elementary obstruction under extensions of the ground field.
Finally, motivated by the fact that \(ob(X)=0\) if and only if the Yoneda equivalence class of a certain 2-extension \(e(X)\) of \({\text{Pic}}(X\otimes _k\bar k)\) by \(\bar k^*\) (viewed as Galois modules) is trivial, the author considers a generalization of the elementary obstruction related to an analogous 2-extension \(E(X)\) of the relative Picard functor \({\text{Pic}}_{X/k}\) by the multiplicative group \({\mathbb G}_{\text{m}}\). In particular, he shows that the Yoneda equivalence class of \(E(X)\) is trivial if and only if \(P_{\text{gen}}(X)=1\). Some of results obtained in the course of the proof are interesting by their own (in particular, an explicit Poincaré sheaf on any abelian variety is exhibited). zero-cycle; elementary obstruction; Albanese torsor; 1-motive Wittenberg, O.\!, On Albanese torsors and the elementary obstruction, Math. Ann., 340, 4, 805-838, (2008) Rational points, Zeta functions and related questions in algebraic geometry (e.g., Birch-Swinnerton-Dyer conjecture), Global ground fields in algebraic geometry, Picard groups, Algebraic cycles, Picard schemes, higher Jacobians On Albanese torsors and the elementary obstruction | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is 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 studies the space of all Riemann surfaces of genus \(g>1\). The main result is the establishing of the explicit analytic isomorphism between the set of all equivalence classes of principally polarized Jacobi varieties and some quasiprojective variety. Teichmüller theory; Riemann surfaces; principally polarized Jacobi varieties Families, moduli of curves (algebraic), Moduli of Riemann surfaces, Teichmüller theory (complex-analytic aspects in several variables) Ein algebraischer Modulraum für die kompakten Riemannschen Flächen von Geschlecht 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 \emph{genus} of a lattice polygon \(P\) is the number of interior vertices. A lattice polygon \(P\) is called \emph{hyperelliptic} if its interior lattice points are collinear. Let \(\mathbb{M}_P\) denote the moduli space of skeletons of smooth tropical plane curves with Newton polygon \(P\). The authors classify the possible values for \(\dim \mathbb{M}_P\). More precisely, the possible values of \(\dim \mathbb{M}_P\) for non-hyperelliptic \(P\) and \(g \geq 3\) are \(\{g, \ldots, 2g+1\}\) minus the impossible combinations \((g, \dim \mathbb{M}_P) \in \{ (3,7), (4, 4), (7, 7) \}\) plus the exceptional case \((7, 16)\). With hyperelliptic \(P\) the possible values for \(\dim \mathbb{M}_P\) and \(g \geq 2\) are \(\{g, \ldots, 2g-1\}\).
For the non-hyperelliptic case note that the upper bounds on \(\dim \mathbb{M}_P\) have been established in [\textit{S. Brodsky} et al., Res. Math. Sci. 2, Paper No. 4, 31 p. (2015; Zbl 1349.14043)]. The bulk of the work in the present article is to establish the lower bound, especially for the exceptional cases. In order to show that every intermediate value for \(\dim \mathbb{M}_P\) is attained, the authors give explicit polygons. Furthermore, the authors show their theorem to be false when only \emph{maximal} lattice polygons are allowed.
For the hyperelliptic case, the authors use the complete classification of hyperelliptic lattice polygons. They derive a formula for the dimension of the moduli space of smooth tropical plane curves dual to a given unimodular triangulation. Based on this formula, \(\dim \mathbb{M}_P\) is computed for any hyperelliptic polygon of genus \(g \geq 2\). tropical curves; lattice polygons; moduli spaces Foundations of tropical geometry and relations with algebra, Lattice polytopes in convex geometry (including relations with commutative algebra and algebraic geometry), Families, moduli of curves (algebraic) Moduli dimensions of lattice polygons | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is 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 \({\mathcal X}\) be a separated Deligne-Mumford stack, and \(\overline X\) a compactification of it coarse scheme. The author defines the abelian monodromy extension (AME) property for the couple \(({\mathcal X}, \overline X)\): if \(S\) is a very small analytic neighborhood of a point \(p\), and \(U:=S\setminus \{p\}\), any morphism \(U\rightarrow {\mathcal X}\) extends to a regular map \(S\rightarrow \overline X\) whenever the image of the induced map \(\pi_1(U)\rightarrow \pi_1({\mathcal X})\) is virtually abelian (i.e. if it contains an abelian subgroup of finite index). If the AME property holds, \(\overline X\) is said to be an AME compactification of \(\mathcal X\). If such a compactification exists, it is proved that there is a unique maximal one. The author shows that, over the complex numbers,
- the standard D-M compactification of the moduli space of curves \(\mathcal M_g\) is its unique maximal AME compactification;
- the Baily-Borel compactification \(\mathcal A_g^{BB}\) of the moduli space of abelian varieties \(\mathcal A_g\) is its unique maximal projective AME compactification.
Let \(V\) be an open subavariety of an irreducible normal variety \(Y\).
From the above results, the author can derive a necessary and sufficient condition for extending a family of smooth curves over \(V\) to a family of stable curves over \(Y\), and a sufficient condition for extending a family abelian varieties over \(V\) to a family \(Y\rightarrow \mathcal A_g^{BB}\). moduli space of curves; moduli space of abelian variety; monodromy Cautis, S, The abelian monodromy extension property for families of curves, Math. Ann., 344, 717-747, (2009) Families, moduli of curves (algebraic), Structure of families (Picard-Lefschetz, monodromy, etc.), Compactifications; symmetric and spherical varieties The abelian monodromy extension property for families 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 These are notes that accompany a short course given at the ``School on algebraic geometry 1999'' at the ICTP, Trieste. The author provides a brief but very lucid survey on the various approaches to the (coarse) moduli spaces of smooth complex projective curves, their compactifications via stable curves, and their geometry. In this vein, the emphasis is on outlining the basic notions, construction methods and multifarious techniques for moduli spaces, explaining and illustrating some of the central (classical) results, and sketching a few more recent developments in the enumerative geometry of moduli spaces of curves. This survey article consists of seven sections covering the following specific topics:
1. Riemann surfaces; 2. Riemann's moduli count; 3. Orbifolds and the Teichmüller approach; 4. Grothendieck's viewpoint (including Kodaira-Spencer maps, deformation theory, the orbifold structure of the moduli spaces \(M_g\), and stable curves); 5. The approach via geometric invariant theory (à la Mumford-Gieseker); 6. Moduli of pointed stable curves; 7. Tautological classes in the cohomology of moduli spaces (including Witten classes, Mumford classes, the tautological algebra, and C. Faber's conjectures on the structure of the tautological algebra of the moduli space of (stable) genus-\(g\) curves).
According to their intended survey character, these notes provide a beautiful, very first introduction to the moduli theory of algebraic curves for non-specialists in the field, including interested theoretical physicists and researchers in other fields of pure mathematics.
As to the more transcendental aspects (moduli of Riemann surfaces), the reader should also consult the survey article ``Moduli of Riemann surfaces, transcendental aspects'' by \textit{R. Hain}, published in the same volume of proceedings, pp. 293-353 (2000; see the following review Zbl 0995.14007), which treats the analytic Teichmüller approach in greater detail, on the one hand, and gives a complementary account on the enumerative geometry of moduli spaces via their Picard group, on the other hand. Riemann surfaces; orbifolds; coarse moduli spaces; tautological classes; enumerative geometry; moduli spaces of curves; geometric invariant theory; pointed stable curves Families, moduli of curves (algebraic), Classification theory of Riemann surfaces, Geometric invariant theory, Families, moduli of curves (analytic), Fine and coarse moduli spaces A minicourse on moduli 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 L'objet de cet article est l'étude de l'action du groupe de Galois absolu sur les groupes fondamentaux des espaces de modules des courbes pointées non ordonnées \(\mathcal{M}_{g,\left\{n\right\}}\). Cette étude se fait dans l'esprit de [\textit{A. Grothendieck}, Lond. Math. Soc. Lect. Note Ser. 242, 5--48; English translation: 243--283 (1997; Zbl 0901.14001)].
Après une première section introductive, la seconde section de l'article est constituée de rappels sur les espaces de modules de courbes. En particulier, sur les lieux \(\mathcal{M}_{g,\left\{n\right\}}\left[G\right]\) paramétrisant les courbes ayant le groupe fini cyclique \(G\) comme groupe d'automorphisme.
Afin d'étuder les composantes irréductibles de \(\mathcal{M}_{g,\left\{n\right\}}\left[G\right]\), les donnés de branchement et d'Hurwitz sont décrites. Ces dernières ont été introduites dans [\textit{M. Cornalba}, Ann. Mat. Pura Appl. (4) 149, 135--151 (1987; Zbl 0649.14013)] dans le cas complexe et sont ici étendues à tout corps complet.
Grâce aux donnés d'Hurwitz, les composantes irréductibles de \(\mathcal{M}_{g,\left\{n\right\}}\left[G\right]/\text{Aut}(G)\) sont décrites dans la quatrième section.
Enfin, l'action du groupe de Galois absolu sur ces composantes irréductibles est décrite dans la cinquième section. Le résultat principal énoncé ici est que, sous certaines contraintes géométriques, le groupe de Galois absolu agit sur les sous groupes finis cycliques du groupe fondamental de \(\mathcal{M}_{g,\left\{n\right\}}\) par conjugaison. Ce résultat permet de plus de montrer qu'en genre deux, l'action du groupe de Galois absolu sur la torsion profinie de \(\pi_{1}^{\text{geom}}\left(\mathcal{M}_{2,\left\{n\right\}}\right)\) est aussi donnée par conjugaison. algebraic fundamental group; stack inertia; special loci; good groups; absolute Galois group; moduli space of algebraic curves; branched covering Benjamin Collas & Sylvain Maugeais, ''Composantes irréductibles de lieux spéciaux d ?espaces de modules de courbes, action galoisienne en genre quelconque'', Ann. Inst. Fourier 65 (2015) no. 1, p. 245-276 Galois theory, Families, moduli of curves (algebraic), Coverings of curves, fundamental group, Special algebraic curves and curves of low genus Irreducible components of special loci in moduli spaces of curves, Galois action in general 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 There are open questions about which Newton polygons and Ekedahl-Oort types occur for Jacobians of smooth curves of genus \(g\) in positive characteristic \(p\). In this chapter, I survey the current state of knowledge about these questions. I include a new result, joint with Karemaker, which verifies, for each odd prime \(p\), that there exist supersingulat curves of genus \(g\) defined \(\overline{\mathbb{F}}_p\) for infinitely many new values of \(g\). I sketch a proof of Faber and van der Geer's theorem about the geometry of the \(p\)-rank stratification of the moduli space of curves. The chapter ends with a new theorem, in which I prove that questions about the geometry of the Newton polygon and Ekedahl-Oort strata can be reduced to the case of \(p\)-rank 0. Algebraic moduli of abelian varieties, classification, Families, moduli of curves (algebraic), Toric varieties, Newton polyhedra, Okounkov bodies, Jacobians, Prym varieties Current results on Newton polygons of curves | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians In this note, the author proves that for a very ample \((2k-1)\)-spanned line bundle \(L\) on a smooth projective surface, the so-called Severi variety \(\tilde{V}_k(L)\) parameterizing all irreducible \(k\)-nodal curves \(D\in |L|\) is irreducible. Followed from the spannedness \(h^0(L)\geq 3k\), so that \(\dim \tilde{V}_k(L)=h^0(L)-k-1\) if \(h^0(L)>3k\) and \(\tilde{V}_k(L)=\emptyset\) otherwise. This result improves the same statement in [\textit{M. Kemeny}, Bull. London Math. Soc. 43, No. 1, 159--174 (2013; Zbl 1032.14005)] proved for \((3k-1)\)-very ample line bundles and is along the result in [\textit{C. Ciliberto} and \textit{Th. Dedieu}, Proc. Am. Math. Soc. 147, No. 10, 4233--4244 (2019; Zbl 1423.14192)].
Reviewer's remark: In the definition of \(k\)-spanned (\(k\)-very ample) line bundles, the rôle of \(k\) is missing. \(K3\) surface; Severi variety; nodal curve; Hilbert scheme of nodal curves Families, moduli of curves (algebraic), Surfaces and higher-dimensional varieties, Divisors, linear systems, invertible sheaves On the irreducibility of the Severi variety of nodal curves in a smooth surface | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians Here it is proved that the monodromy of the Riemann surfaces of genus \(g\) acts as the full symmetric group on the Weierstraß points of a general curve. The proof uses a degeneration to stable reducible curves with \(Pic^ 0\) compact introduced and developed by the authors [Invent. Math. 85, 337-371 (1986; Zbl 0598.14003)]. An example: if the curve X is union of a smooth curve Y of genus \(g-1\) and an elliptic curve E glued at a point \(p\in Y\cap E,\quad p\quad not\) a Weierstraß point of Y, the Weierstraß points on X are the ramification points on Y of the complete linear series \(| K_ Y+2p|\) and the \(g^ 2-1\) points of \(E\setminus \{p\}\) which differ from p by g-torsion. Part of the monodromy is constructed by fixing a (reducible) curve and varying its canonical series. reducible curve; canonical divisor; Weierstraß points of a general curve; ramification points Eisenbud, D. andHarris, J., The monodromy of Weierstrass points,Invent. Math. 90 (1987), 333--341. Riemann surfaces; Weierstrass points; gap sequences, Families, moduli of curves (algebraic), Classification theory of Riemann surfaces, Coverings of curves, fundamental group The monodromy of 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 In the famous article [Compos. Math. 110, No. 1, 65--126 (1998; Zbl 0894.18005)] \textit{E. Getzler} and \textit{M. Kapranov} studied operadic type structures related to the moduli space of algebraic curve. Intuitively, algebraic curves with marked points can be glued along the marked points generating operations on the moduli spaces. However, when considering arbitrary genuses of curves, the classical operadic picture, in which operations are labeled by trees, is replaced by operation labeled instead by graphs. They call this operadic structure modular. Moreover, moduli of curves with marked points do have typically (for instance when considering genus \(0\) curves) an extra cyclic symmetry obtained by permuting the punctures. In the same paper, they show how to construct a Feynman transform on the category of dg-modular operads and how to compute its Euler characteristic in terms of the Wick's theorem, hence highlighting the relation of this operad with mathematical physics.
In this paper the authors present a generalization of these results for curves with marked points \(k-\log\) canonically embedded, meaning admitting a projective embedding by a complete linear system. The study of log canonical models for curves has been central in the study of moduli spaces of curves and for its relationships to the Minimal Model program. There are three results presented: first, they show that for \(k\geq 5\) the moduli spaces of \(k-\log\) canonically embedded curves assemble together in a modular operad in Deligne-Mumford stacks. Second, they show that for \(k\geq 1\) the moduli spaces of \(k-\log\) canonically embedded curves of genus \(0\) assemble together in a cyclic operad in schemes. Third, they show that for \(k\geq 2\) the moduli spaces of \(k-\log\) canonically embedded curves assemble together in a stable cyclic operad in Deligne-Mumford stacks. In order to prove these results, they construct morphisms on these moduli spaces corresponding to the gluing of two embedded curves and to the gluing of two points together on the same embedded curve.
The proofs of these statements appear correct. Would be interesting, as a follow up work, to understand weather the construction of Getzler and Kapranov of the Feynman transform could be generalized to this setting. modular operad; log-canonical Hilbert scheme Families, moduli of curves (algebraic), , Parametrization (Chow and Hilbert schemes) Modular operads of embedded curves | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians A Hurwitz space \(H\) is the space of (ramified) coverings of compact Riemann surfaces with their genera, covering degree and ramification fixed. The author constructs a combinatorial model of a compactification \(\overline {H}\) of \(H\) and uses it to give a cellular decomposition of \(\overline{H}\). This generalizes the cellular decomposition of the moduli space \(M_{g,n}\) of punctured Riemann surfaces (Harer-Kontsevich theorem) [\textit{M. Kontsevich}, Commun. Math. Phys. 147, 1--23 (1992; Zbl 0756.35081)]. The author also gives an analogy between Hurwitz spaces of cyclic Galois coverings and moduli spaces of spin curves. ramified coverings; Galois coverings; compactification; moduli spaces of spin curves Coverings of curves, fundamental group, Coverings in algebraic geometry, Planar graphs; geometric and topological aspects of graph theory, Moduli of Riemann surfaces, Teichmüller theory (complex-analytic aspects in several variables), Families, moduli of curves (algebraic) Cellular decomposition of compactified Hurwitz spaces | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians For a stable curve \(C\), the dual graph is the weighted graph \(\Gamma\), such that: the vertices of \(\Gamma\) correspond to irreducible components of \(C\), the edges of \(\Gamma\) correspond to nodes joining the two components corresponding to the two endpoints and the weights on the vertices of \(\Gamma\) are the genera of the normalizations of the corresponding components of \(C\). Let \(\Gamma\) be now a fixed abstract weighted graph. Then \(\mathfrak{M}_g(\Gamma)\) is the substack of the stack \(\overline{\mathfrak{M}}_g\) of stable curves consisting of curves with dual graphs \(\Gamma\). This is a locally closed substack.
The current paper analizes the above substacks, concentrating on their normalizations. Though a general method of understanding normalizations of moduli functors (if possible at all) would be very much desired, the authors here follow a different path. An explicit normalization is constructed, given by the special setup of stable curves: every stable curve can be constructed by gluing the normalizations of its components. In the case of weighted graphs \(\Gamma\) that yield 1 dimensional strata of \(\overline{\mathfrak{M}}_g\), the normalization is even more explicitly described as a gerbe over an orbifold. moduli of stable curve; boundary strata of moduli of curves Stacks and moduli problems, Families, moduli of curves (algebraic), Automorphisms of curves Normalization of the 1-stratum of the moduli space of stable curves | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians We define the \(p\)-density of a finite subset \(D\subset \mathbb N^{r}\), and show that it gives a sharp lower bound for the \(p\)-adic valuations of the reciprocal roots and poles of zeta functions and \(L\)-functions associated to exponential sums over finite fields of characteristic \(p\). When \(r=1\), the \(p\)-density of the set \(D\) is the first slope of the generic Newton polygon of the family of Artin-Schreier curves associated to polynomials with their exponents in \(D\). character sums; \(L\)-functions and zeta functions; Newton polygons; Chevalley-Warning theorem Blache, Régis: P-density, exponential sums and Artin-Schreier curves Exponential sums, Families, moduli of curves (algebraic) Valuation of exponential sums and the generic first slope for Artin-Schreier curves | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians The main result of this paper provides a formula for the so-called covering gonality $\mathrm{Cov.gon}(X)$ of a general hypersurface $X\subset \mathbb{P}^N$ of large degree in arbitrary dimension. The covering gonality is a measure for the failure of a variety to be uniruled, and is defined as the least gonality of an irreducible curve through a general point on a variety. The main tool used in proofs is the cone of tangent lines having a certain intersection multiplicity with the hypersurface at a point. A curve of gonality $\mathrm{Cov.gon}(X)$ through a general point $p\in X$ is then described as a plane curve, obtained by intersection of $X$ and the linear span of $p$ and a line $\ell$ inside the cone with $p\notin \ell$. Accordantly, the projection from the singular point $ p$ of the curve gives the gonal map. gonality; hypersurfaces; covering gonality; families of curves; cone of tangent lines Hypersurfaces and algebraic geometry, Families, moduli of curves (algebraic), Special divisors on curves (gonality, Brill-Noether theory), Rationality questions in algebraic geometry Gonality of curves on general hypersurfaces | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians The simplest, and for many the most familiar singularities, are those presented by plane curves. Their local topology was first investigated in the late 1920's, and they have provided an excellent proving ground for theories and theorems ever since. This paper addresses three central problems concerning plane curve singularities. Suppose we are given a natural number \(d\) and a finite list of types of plane curve singularities. We can then ask:
(1) Is there an (irreducible) curve of degree \(d\) in the complex projective plane \(\mathbb{C}\mathbb{P}^2\) whose singularities are precisely those in the given list?
(2) Is the family of such curves smooth, and of the expected dimension?
(3) Is this set connected?
These questions have a long history, appearing in the works of Plücker, Severi, Segre and Zariski. They are part of a more general problem: that of moving from local information to global results. The authors have made spectacular recent progress, which is surveyed in this paper. They extend their discussion to curves on more general surfaces. Combined with recent ideas of Viro their results also have application in real algebraic geometry. equisingular families of plane curves; curves on surfaces; real algebraic geometry; plane curve singularities Gert-Martin Greuel and Eugenii Shustin, Geometry of equisingular families of curves, Singularity theory (Liverpool, 1996) London Math. Soc. Lecture Note Ser., vol. 263, Cambridge Univ. Press, Cambridge, 1999, pp. xvi, 79 -- 108. Singularities of curves, local rings, Global theory and resolution of singularities (algebro-geometric aspects), Families, moduli of curves (algebraic), Plane and space curves, Topology of real algebraic varieties Geometry of equisingular families 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 \(\mathcal T_g\) be the moduli space of trigonal curves of genus \(g\), i.e., the moduli space parametrizing irreducible smooth projective curves which admit a degree \(3\) morphism to \(\mathbb P^1\). For \(g\geq 5\), \(\mathcal T_g\) can be regarded as a sub locus of the moduli space \(\mathcal M_g\) of smooth curves of genus \(g\) as all such trigonal curves have a unique \(g^1_3\).
In the paper under review, the author studies the problem of giving a birational classification for \(\mathcal T_g\). More precisely, the author shows that if \(g=2b+1\) and \(b\geq 2\), then \(\mathcal T_g\) is rational.
The proof consists of a geometrical classical argument relating \(\mathcal T_g\) with Hirzebruch surfaces \(\mathbb F_N=\mathbb P(\mathcal O_{\mathbb P^1}\oplus \mathcal O_{\mathbb P^1}(N))\). In detail, for odd genus, \(\mathcal T_g\) is birationally equivalent to the rational quotient \(|L_{3,b}|/\text{ Aut}(\mathbb F_1)\), where \(L_{3,b}=\mathcal O_{F_1}(3(\Sigma+F)+bF)\) is a line bundle determined by the fiber \(F\) of the natural projection \(\pi:\mathbb F_1\to\mathbb P^1\) and by a section \(\Sigma\) of \(\pi\) with self-intersection \(-1\). Therefore, the result follows by showing that \(|L_{3,b}|/\text{ Aut}(\mathbb F_1)\) is rational for \(b\geq 2\), which the author does.
The fact that \(\mathcal T_g\) is rational for \(g=4n+2\) with \(n\geq 2\) was already known due to work of \textit{N. I. Shepherd-Barron} [Proc. Symp. Pure Math. 46, 165--171 (1987; Zbl 0669.14015)]. The remaining case of curves of genus divisible by four has been successively studied by the author in [``The Rationality of the Moduli Spaces of Trigonal Curves'', Int. Math. Res. Not. IMRN No. 14, 5456--5472 (2015)]. trigonal curves; rationality S. Ma, The rationality of the moduli spaces of trigonal curves of odd genus. J. Reine. Angew. Math. 683 (2013), 181-187. arXiv:1012.0983. Families, moduli of curves (algebraic), Rationality questions in algebraic geometry The rationality of the moduli spaces of trigonal curves of odd 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 This paper studies the realizability problem for effective tropical canonical divisors in equicharacteristic zero. More precisely, given a stable tropical curve \(\Gamma\) and a divisor \(D\) in the canonical linear system on \(\Gamma\), the authors provide a combinatorial condition to decide when there exists a smooth curve \(X\) over a non-Archimedean field whose stable reduction has \(\Gamma\) as its dual tropical curve together with an effective canonical divisor \(K_X\) that specializes to \(D\). The proof of the result relies on the description of the incidence variety compactification of the strata of abelian differentials with prescribed orders of zeros by \textit{M. Bainbridge} et al. [Duke Math. J. 167, No. 12, 2347--2416 (2018; Zbl 1403.14058)], where the dual graph \(\Gamma\) is equipped with an enhanced level structure and certain residue conditions have to be satisfied. The authors translate these conditions into tropical geometry and solve the corresponding combinatorial problem. In particular, the result implies that the canonical divisor \(K_\Gamma\) on \(\Gamma\) is in general not realizable. The authors also develop a moduli-theoretic framework to understand specialization of divisors from algebraic to tropical curves as a natural toroidal tropicalization map in the sense of \textit{D. Abramovich} et al. [Ann. Sci. Éc. Norm. Supér. (4) 48, No. 4, 765--809 (2015; Zbl 1410.14049)]. tropical geometry; moduli spaces; Hodge bundle; abelian differentials; canonical divisors; flat surfaces; Berkovich spaces Families, moduli of curves (analytic), Families, moduli of curves (algebraic), Moduli of Riemann surfaces, Teichmüller theory (complex-analytic aspects in several variables) Realizability of tropical canonical divisors | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is 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 \(d\)-dimensional compact Kähler manifold with canonical bundle \(\omega_X\) and assume that \(b_1(X)>0\). Consider the cohomology modules \(P_X=\bigoplus_{i=0}^d H^i(X,{\mathcal{O}}_X)\) and \(Q_X=\bigoplus_{i=0}^d H^i(X,\omega_X)\), viewed as graded modules over the exterior algebra \(E=\bigwedge^*H^1(X,{\mathcal{O}}_X)\) via the cup product. The authors follow the idea of the Bernstein-Gelfand-Gelfand (BGG) correspondence between regularity properties of graded modules over exterior algebras and exactness properties of linear complexes of modules over symmetric algebras to study the structure of the canonical module \(Q_X\). They use the notion of \(m\)-regularity which is analogous to Castelnuovo-Mumford regularity for graded modules over a polynomial ring. For \(P_X\) and \(Q_X\), the BGG corresponence is realized by complexes which are defined as follows. Let \({\mathbf P}\) denote the projective space of one-dimensional subspaces of \(H^1(X,{\mathcal {O}}_X)\). The cup product with \(0\not=v\in H^1(X,{\mathcal {O}}_X)\) leads to a complex \({ \underline{\mathbf L}}_X\) of vector bundles on \({\mathbf P}\):
\[
0\rightarrow {\mathcal {O}}_{\mathbf P}(-d)\otimes H^0(X,{\mathcal O}_X)\rightarrow{\mathcal {O}}_{\mathbf P}(-d+1)\otimes H^1(X,{\mathcal O}_X)\rightarrow\dots
\]
\[
\rightarrow{\mathcal {O}}_{\mathbf P}(-1)\otimes H^{d-1}(X,{\mathcal O}_X)\rightarrow{\mathcal {O}}_{\mathbf P}\otimes H^d(X,{\mathcal O}_X)\rightarrow 0.
\]
Let \(S=\)Sym\((H^1(X,{\mathcal O}_X)^\vee)\) denote the symmetric algebra on the vector space \(H^1(X,{\mathcal O}_X)^\vee\). The global sections in \(\underline{\mathbf L}_X\) define a linear complex \({\mathbf L}_X\) of graded \(S\)-modules,
\[
0\rightarrow S\otimes_\mathbb C H^0(X,{\mathcal O}_X)\rightarrow S\otimes_\mathbb C H^1(X,{\mathcal O}_X)\rightarrow\dots
\]
\[
\rightarrow S\otimes_\mathbb C H^d(X,{\mathcal{O}}_X)\rightarrow 0,
\]
which reflects the regularity properties of \(Q_X\) in the sense of the BGG correspondence. Let alb\(_X\) be the Albanese map of \(X\) and \(k:=\dim X - \dim\) alb\(_X(X)\). The main technical result of the paper (Theorem A) shows that the exactness properties of \({\underline{\mathbf L}}_X\) and \({\mathbf L}_X\) are determined by the fiber-dimension of the Albanese map: Both complexes are exact in the first \(d-k\) terms from the left, but \({\mathbf L}_X\) has non-trivial homology at the next term to the right. If \(X\) allows no irregular fibration, i.e., no holomorphic map \(X\rightarrow Y\) onto a normal variety \(Y\) with connected positive-dimensional fibers and \(\dim Y=\dim\) alb\(_{\hat Y}(\hat Y)\) for a smooth model \(\hat Y\) of \(Y\), then the cokernel \({\mathcal F}\) of the map \({\mathcal {O}}_{\mathbf P}(-1)\otimes H^{d-1}(X,{\mathcal{O}}_X)\rightarrow{\mathcal {O}}_{\mathbf P}\otimes H^d(X,{\mathcal{O}}_X)\) is a vector bundle on \({\mathbf P}\) with rk\(({\mathcal F})=\chi(\omega_X)\), and \({\underline{\mathbf L}}_X\) is a resolution of \({\mathcal F}\). Based on results of \textit{D. Eisenbud, G. Fløystad} and \textit{F.-O. Schreyer} [Trans. Am. Math. Soc. 355, No.~11, 4397--4426 (2003; Zbl 1063.14021)] the authors deduce from Theorem A that the fiber-dimension of alb\(_X\) is completely determined by the algebraic structure of \(Q_X\) in the way that the \(E\)-module \(Q_X\) is \(k\)-regular but not \((k-1)\) regular. In particular, \(k=0\) if and only if \(Q_X\) is generated by elements of degree \(0\) and has a linear free resolution (Theorem B). Another direction of applications of Theorem A leads to inequalities for numerical invariants, in particular for the Hodge numbers of \(X\), mainly when \(X\) admits no irregular fibrations. Several examples are discussed and the applications are related to former results of several authors, among them \textit{F. Catanese} [Invent. Math. 104, No.~2, 263--289; Appendix 289 (1991; Zbl 0743.32025)], \textit{A. Causin} and \textit{G. P. Pirola} [Manuscr. Math. 121, No.~2, 157--168 (2006; Zbl 1107.32006)], and \textit{G. Pareschi} and \textit{M. Popa} [Duke Math. J. 150, No.~2, 269--285 (2009; Zbl 1206.14067)]. Finally, the authors give a description of the geometrical meaning of \({\mathcal F}\) in terms of paracanonical divisors. Hodge numbers; Albanese map; BGG correspondence; irregular fibration; paracanonical divisor; Schur polynomial R. Lazarsfeld and M. Popa, Derviation complex, BGG correspondence, and numerical inequalities for compact Kähler manifolds , Invent. 182 (2010), 605-633. Kähler manifolds, Compact Kähler manifolds: generalizations, classification, Picard groups, Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20], Compact complex \(n\)-folds Derivative complex, BGG correspondence, and numerical inequalities for compact Kähler manifolds | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians [For the entire collection see Zbl 0695.00011.]
Given any s points \(P_ 1,...,P_ s\) in \({\mathbb{P}}^ 2\) and s positive integers \(m_ 1,...,m_ s\), let \(\Sigma_ n\) be the linear system of plane curves of degree \(n\) through \(P_ i\) with multiplicity at least \(m_ i (1\leq i\leq s)\). We give numerical bounds for the regularity of \(\Sigma_ n\) in the following cases: (a) the points \(P_ i\) are nonsingular points of an integral curve of degree \(d;\) (b) the \(P_ i\) are in general position; (c) the \(P_ i\) are in uniform position; (d) the \(P_ i\) are generic points of \({\mathbb{P}}^ 2\). We also study the sharpness of such bounds. regularity of linear systems of plane curves; fat points Varieties defined by ring conditions (factorial, Cohen-Macaulay, seminormal), Divisors, linear systems, invertible sheaves, Families, moduli of curves (algebraic) Linear systems of plane curves through fixed ``fat'' points of \({\mathbb{P}}^ 2\) | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians Francis Brown introduced a partial compactification \(M_{0,n}^\delta\) of the moduli space \(M_{0,n}\). We prove that the gravity cooperad, given by the degree-shifted cohomologies of the spaces \(M_{0,n}\), is cofree as a nonsymmetric anticyclic cooperad; moreover, the cogenerators are given by the cohomology groups of \(M_{0,n}^\delta\). As part of the proof we construct an explicit diagrammatically defined basis of \(H^\bullet(M_{0,n})\) which is compatible with cooperadic cocomposition, and such that a subset forms a basis of \(H^\bullet(M_{0,n}^\delta)\). We show that our results are equivalent to the claim that \(H^k(M_{0,n}^\delta)\) has a pure Hodge structure of weight \(2k\) for all \(k\), and we conclude our paper by giving an independent and completely different proof of this fact. The latter proof uses a new and explicit iterative construction of \(M_{0,n}^\delta\) from \(\mathbb{A}^{n-3}\) by blow-ups and removing divisors, analogous to Kapranov's and Keel's constructions of \(\overline{M}_{0,n}\) from \(\mathbb{P}^{n-3}\) and \((\mathbb{P}^1)^{n-3}\), respectively. moduli of curves; mixed Hodge theory; operads; multiple zeta values; Koszul duality for operads [AP17]almpetersen J.~Alm and D.~Petersen, \emph Brown's dihedral moduli space and freedom of the gravity operad, Annales scientifiques de l'ENS 50, fascicule 5 (2017), 1080-1122. Families, moduli of curves (algebraic), , Polylogarithms and relations with \(K\)-theory, Loop space machines and operads in algebraic topology, de Rham cohomology and algebraic geometry, Multiple Dirichlet series and zeta functions and multizeta values Brown's dihedral moduli space and freedom of the gravity operad | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is 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 smooth genus \(g\) curves with an \(l\)-torsion line bundle was given a nice modular compactification \(\overline R_{g,l}\) by \textit{D. Abramovich} et al. [Commun. Algebra 31, No. 8, 3547--3618 (2003; Zbl 1077.14034)]. The paper under review studies the singularities of the coarse space of this compactification: it describes precisely the singular locus, as well as the sublocus of noncanonical singularities.
A main motivation is the study of the Kodaira dimension of moduli spaces of curves. The Kodaira dimension is intrinsically a property of the coarse space (it is defined to be the Kodaira dimension of any desingularization of the coarse space). One would like to know whether or not it is enough to construct pluricanonical forms that are defined away from the locus of curves with automorphisms (which coincides with the singular locus of the coarse space), which in particular makes it important to determine the locus of noncanonical singularities of the coarse space. In their original study of the Kodaira dimension of \(\overline M_g\), Harris and Mumford determined completely the non-canonical singularities of \(\overline M_g\) (for \(g\geq 4\) they come from elliptic tails of \(j\)-invariant 0) and proved that all pluricanonical forms defined on the whole smooth locus extend across them.
The authors prove that the moduli space of stable level \(l\) curves only has two kinds of non-canonical singularities: a non-canonical singular point corresponds either to what they call a T-curve or a J-curve. The T-curves are the level \(l\) versions of the noncanonical singularities encountered by Harris and Mumford, and Harris-Mumford's argument works mutatis mutandis to show that pluricanonical forms extend across them. They also prove that J-curves occur if and only if \(l = 5\) or \(l \geq 7\), so that for small values of \(l\) they will at least not pose any problem. These results are applied to the study of the Kodaira dimension of \(\overline R_{g,2}\) by the second author and \textit{K. Ludwig} [J. Eur. Math. Soc. (JEMS) 12, No. 3, 755--795 (2010; Zbl 1193.14043)] and \(\overline R_{g,3}\) by \textit{A. Chiodo} et al. [Invent. Math. 194, No. 1, 73--118 (2013; Zbl 1284.14006)]. moduli of curves; admissible covers; twisted curves Chiodo, A., Farkas, G.: Singularities of the moduli space of level curves, preprint, arXiv:1205.0201 Families, moduli of curves (algebraic) Singularities of the moduli space of level 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 It is shown that there are precisely two birationally inequivalent non- singular curves of degree 7 and genus 4. Both can occur only in \(P^ 3\). One is a (2,5) curve \(C_ 1\) on a quadric. The other is obtained as the projective model \(C_ 2\) of a linear series \(g^ 3_ 7\) on a binodal quintic \(\Gamma\) cut out by cubics through the two double points and four simple points on \(\Gamma\). The existence of \(C_ 2\) was observed by \textit{A. Seidenberg} [Rend. Semin. Mat. Fis. Milano 51, 173-178 (1981; Zbl 0519.14024)]. It had been remarked by \textit{L. Gruson} and \textit{C. Peskine} [Ann. Sci. Éc. Norm. Supér., IV. Sér. 15, 401-418 (1982; Zbl 0517.14007)] that a connected curve of degree d and genus g belongs to a quadric if and only if \(\rho =(d-2)^ 2-4g\) is a square. The author observes that every non-singular curve on a quadric has \(\rho\) a square since for an \((\alpha,\beta)\) curve, \(\rho =(\alpha +\beta -2)^ 2- 4(\alpha -1)(\beta -1)=(\alpha -\beta)^ 2\). Further, the example \(C_ 2\) has \(\rho\) a square but does not lie on a quadric.
(In the article \(\alpha -\beta +2\) is printed instead of \(\alpha +\beta - 2.)\) non-singular curves of degree 7 and genus 4 Families, moduli of curves (algebraic), Special algebraic curves and curves of low genus, Projective techniques in algebraic geometry On non singular curves of degree 7 and genus 4 | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians The authors study the cones of ample and numerically effective divisors of moduli spaces of stable maps to Grassmannians \(\overline{M}_{0,m}(G(k,n),d)\). The cones of NEF divisors are described in Theorem 1.1. Let the \(2\leq k<k+2\leq n\) (that is, \(G(k,n)\) is not a projective space; such cases are studied in the previous papers) and let \(m+d\geq 3\). Then there is a injective map
\[
\nu\colon \text{Pic}_{\mathbb Q}(\overline{M}_{0,m+d}/S_d)\to \text{Pic}_{\mathbb Q}(\overline{M}_{0,m}(G(k,n),d))
\]
(where the symmetric group \(S_d\) permutes the last \(d\) marked points). Let \(H_{\sigma_2}\) and \(H_{\sigma_{1,1}}\) be the divisors of maps whose image intersects fixed Schubert varieties \(\Sigma_2\) and \(\Sigma_{1,1}\) respectively. Let \(T\) be the tangency divisor and let \(L_1,\ldots,L_m\) be the pullbacks of the ample generator of \(\text{Pic}(G(k,n))\) under the evaluation maps. Then the cone of NEF divisors of \(\overline{M}_{0,m}(G(k,n),d)\) is the product of cones spanned by the image under \(\nu\) of the NEF cone of \(\overline{M}_{0,m+d}/S_d\) and \(H_{\sigma_{2}}\), \(H_{\sigma_{1,1}}\), \(L_1,\ldots,L_m\). Theorem 5.1 generalizes this theorem to the products of flag varieties. The cones of effective cones are harder to understand. The authors prove that the effective cones of \(\overline{M}_{0,0}(G(k,n),d)\) stabilize as long as \(n\geq k+d\). To study the cone of \(\overline{M}_{0,0}(G(k,k+d),d)\) authors define the particular divisors \(D_{\text{unb}}\) and \(D_{\text{deg}}\) (surprisingly, the construction depends on whether \(k\) divides \(d\)). Theorem 1.2 states that the effective cone of \(\overline{M}_{0,0}(G(k,k+d),d)\) is a simplicial cone generated by \(D_{\text{unb}}\), \(D_{\text{deg}}\) and the boundary divisors. NEF cone; effective cone; Grassmannian; space of stable maps Coskun, Izzet; Starr, Jason, Divisors on the space of maps to Grassmannians, Int. Math. Res. Not., 2006, (2006) Families, moduli of curves (algebraic), Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects), Grassmannians, Schubert varieties, flag manifolds Divisors on the space of maps to Grassmannians | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians Let \(\pi: X \to S\) be a proper, flat, finite presentation, geometrically reduced, integral morphism of fine logarithmic algebraic spaces. Let \(Y\) be a logarithmic stack over \(S\), not necessarily algebraic. The main result of the paper shows that the morphism \(\text{Hom}_{\text{LogSch}/S}(X, Y) \to \text{Hom}_{\text{LogSch}/S}(\underline{X}, \underline{Y})\) is representable by logarithmic algebraic spaces locally of finite presentation over \(S\). As a corollary, the author obtains the representability of the stack of stable logarithmic maps from logarithmic curves to a fixed target without restriction on the logarithmic structure of the target.
This improves on several previous results by \textit{Q. Chen} [Ann. Math. (2) 180, No. 2, 455--521 (2014; Zbl 1311.14028)], \textit{D. Abramovich} and \textit{Q. Chen} [Asian J. Math. 18, No. 3, 465--488 (2014; Zbl 1321.14025)], and \textit{M. Gross} and \textit{B. Siebert} [J. Am. Math. Soc. 26, No. 2, 451--510 (2013; Zbl 1281.14044)]. logarithmic geometry; moduli Wise, J., \textit{moduli of morphisms of logarithmic schemes}, Algebra Number Theory, 10, 695-735, (2016) Families, moduli of curves (algebraic), Stacks and moduli problems, Generalizations (algebraic spaces, stacks) Moduli of morphisms of logarithmic schemes | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians Let \(X\) be a smooth genus \(g\) curve, \(P\in X\) and \(R\in\text{Pic} (X)\). Here we study the following problem. Under what assumptions there is \(D\in|R|\) and a divisor \(A\) on \(X\) such that Supp\((A)\subset\text{Supp}(D)\) and \(P\notin\text{Supp}(D)\)? Special divisors on curves (gonality, Brill-Noether theory), Picard groups, Vector bundles on surfaces and higher-dimensional varieties, and their moduli Points of smooth curves linearly equivalent to divisors with prescribed support | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians The author generalizes Freitag's result ''the coordinate ring of \({\mathcal D}/\Gamma\), with the Siegel space \({\mathcal D}\) and the modular group \(\Gamma\), is factorial if the degree is greater than 2'' to the case where \({\mathcal D}\) is a connected domain with Pic(\({\mathcal D})=0\) and \(\Gamma\) is a discontinuous group of holomorphic automorphisms under some conditions.
Then the result is applicable to the case D/\(\Gamma\) is the moduli space \({\mathcal M}_ g\) of curves of genus g and the author proves Mumford's conjecture on the second Betti number of \({\mathcal M}_ g\) for \(g=3,4\). Further, as an application of his method, the author shows that the closure in \({\mathcal M}_ 4\) of the moduli space of non-hyperelliptic curves of genus 4 is a subvariety defined by a principal ideal. factorial ring of automorphic forms; Satake compactification; Picard group; theta constant; Schottky invariant; Mumford's conjecture; second Betti number; moduli space of non-hyperelliptic curves S. Tsuyumine: Factorial property of a ring of automorphic forms. Trans. Amer. Math. Soc. (to appear). JSTOR: Theta series; Weil representation; theta correspondences, Families, moduli of curves (algebraic) Factorial property of a ring of automorphic forms | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians [For the entire collection see Zbl 0547.00007.]
Let \(M_ g\) (resp. \(A_ g)\) be the moduli space of complete smooth curves of genus g (resp. principally polarized abelian varieties of dimension g). Here we work only over the field of complex numbers. There is a canonical holomorphic map \(i: M_ g\to A_ g\) by associating a curve C with its Jacobian variety J(C), which is known to be injective (Torelli theorem). There arises naturally a problem to characterize the image of i, which is now called ''the Schottky problem'' (in a wide sense). Here the author gives a compact but very well organized review on recent development on this subject. According to \textit{D. Mumford}'s earlier work ''Curves and their Jacobians'' (1975; Zbl 0316.14010) the author classifies 4 approaches and gives recent results in each direction: [1] algebraic equations (to find the relations of theta constants characterizing the closure of Im i in \(A_ g)\); [2] d trisecants (to use special properties of the Kummer surface of the Jacobian); [3] geometry of the moduli space (to use a special embedding \(A_ g(2,4)\) in \({\mathbb{P}}^ N\) by theta constants); [4] rings of differential operators (to characterize the image as the set of points where a certain theta function with the corresponding modulus satisfies the so-called K-P equation (abbr. to Kadomtsev-Petviashvili-Novikov's conjecture). The author himself gives a main contribution to the case [3] with van Geemen. The most definite result is Shiota's solution for the Novikov's conjecture by using the theory of K-P hierarchy due to M. Mulase. Now almost all papers in the references have been published: e.g. \textit{B. van Geemen} [Invent. Math. 78, 329-349 (1984; Zbl 0568.14015)]; \textit{T. Shiota} [Invent. Math. 83, 333-382 (1986)] and \textit{G. E. Welters} [Ann. Math., II. Ser. 120, 497- 504 (1984; Zbl 0574.14027)]. moduli space of complete smooth curves; principally polarized abelian varieties; Torelli theorem; Schottky problem; Kummer surface of the Jacobian; theta constants; K-P equation Jacobians, Prym varieties, Families, moduli of curves (analytic), Theta functions and abelian varieties, Transcendental methods, Hodge theory (algebro-geometric aspects), Algebraic moduli problems, moduli of vector bundles, Partial differential equations of mathematical physics and other areas of application, Families, moduli of curves (algebraic) The Schottky problem | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians Given a subspace arrangement \(G^*\) contained in \(\mathbb{C}^n\) it can be described as a set of subspaces \(G\) in \((\mathbb{C}^n)^*\) dual to the ones in \(G^*\). We call \(M(G)\) the complement of \(G^*\) in \(\mathbb{C}^n\). A model for \(M(G)\) is a smooth irreducible variety \(Y_G\) having a proper map \(\pi : Y_G \rightarrow \mathbb{C}^n\) which is an isomorphism on the preimage of \(M(G)\) and such that the complement of this preimage is a divisor with normal crossings. \textit{C. De Concini} and \textit{C. Procesi} constructed one such model [Sel. Math., New Ser. 1, 459-494 (1995; Zbl 0842.14038)]. The main purpose of this paper is to study the cohomology rings of that model. arrangements of subspaces; cohomology bases; Poincaré polynomial Giovanni Gaiffi, Blowups and cohomology bases for De Concini-Procesi models of subspace arrangements, Selecta Math. (N.S.) 3 (1997), no. 3, 315 -- 333. Arrangements of points, flats, hyperplanes (aspects of discrete geometry), Families, moduli of curves (algebraic) Blowups and cohomology bases for De Concini-Procesi models of subspace arrangements | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians Some technical results on the deformations of varieties of general type and on permanence of semi-log-canonical singularities are proved. These results are applied to show that the connected component of the moduli space of stable surfaces containing the moduli point of a product of stable curves is the product of the moduli spaces of the curves, assuming the curves have different genera. An application of this result shows that even after compactifying the moduli space and fixing numerical invariants, the moduli spaces are still very disconnected. semi-log-canonical singularities; components of moduli schemes Michael A. van Opstall, Moduli of products of curves, Arch. Math. (Basel) 84 (2005), no. 2, 148 -- 154. Families, moduli, classification: algebraic theory, Families, moduli of curves (algebraic), Formal methods and deformations in algebraic geometry, Singularities in algebraic geometry Moduli of products of curves | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians The explicit computation of families of covers of the Riemann sphere is a difficult computational problem. For families of genus \(0\) covers, the author has developed several sophisticated methods for such a computation, see \textit{J.-M. Couveignes} [Lond. Math. Soc. Lect. Note Ser. 256, 38-65 (1999; Zbl 1016.14012)].
The main ingredients are the study of degenerate covers and patching arguments of \textit{D. Harbater} [Aspects of Galois theory, Lect. Notes Math. 1240 (1987; Zbl 0627.12015)]. A striking application of these methods was the computation of a polynomial with Galois group isomorphic to the Mathieu group \(M_{24}\) by \textit{L. Granboulan} [Exp. Math. 5, 3-14 (1996; Zbl 0871.12006)].
In the paper under review, the author generalizes his methods to the higher genus case. Using the theory of Jacobians, the new problems which arise are reduced to the combinatorial study of the intersection graph of singular fibers. This graph is studied using an associated \(CW\)-complex, called the ``Kirchhoff complex''.
As applications of his methods, the author treats the case of modular curves (in order to produce modular units) and explicitly computes a family of genus one covers. Hurwitz space; patching; Jacobian; intersection graph; Kirchhoff complex Couveignes, J. -M.: Boundary of Hurwitz spaces and explicit patching. J. symbolic comput. 30, 739-759 (2000) Inverse Galois theory, Coverings of curves, fundamental group, Arithmetic algebraic geometry (Diophantine geometry), Families, moduli of curves (algebraic), Universal profinite groups (relationship to moduli spaces, projective and moduli towers, Galois theory), Computational aspects of algebraic curves Boundary of Hurwitz spaces and explicit patching | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians [For the entire collection see Zbl 0711.00011.]
Pour tout entier \(n\), notons \(G_ n\) le polynôme \(G_ n(T)=\prod^{[n/2]}_{k=1}(T-2\cos 2k\pi/n)\), où \([x]\) est la partie entière de \(x\). Disons qu'une courbe \(C\) de genre \([n/2]\), définie sur un corps \(k\), est à multiplications réelles par \(G_ n\) s'il existe une correspondance \({\mathcal C}\) sur \(C\) telle que \(G_ n\) soit le polynôme caractéristique de l'endomorphisme induit par \({\mathcal C}\) sur les différentielles de premiére espèce de \(C\). Dans cet article, nous construisons, pour tout entier \(n\geq 4\), une famille de dimension 2, définie sur \(\mathbb{C}\), de courbes hyperelliptiques de genre \([n/2]\) à multiplications réelles par \(G_ n\). hyperelliptic curves with real multiplication J.-F. Mestre, Familles de courbes hyperelliptiques à multiplications réelles, Arithmetic algebraic geometry (Texel, 1989) Progr. Math., vol. 89, Birkhäuser Boston, Boston, MA, 1991, pp. 193 -- 208 (French). Elliptic curves, Families, moduli of curves (algebraic), Complex multiplication and abelian varieties Families de courbes hyperelliptiques à multiplications réelles. (Families of hyperelliptic curves with real multiplication) | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is 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 0563.00006.]
This is a survey note about some recent questions and results concerning the Weierstrass points (W-points), on families of curves. Let C be a smooth complete curve of genus g and \(\alpha =\alpha (C,p)\) the Schubert index of the point \(p\in C\), where \(\alpha =(\alpha_ 0,\alpha_ 1,...,\alpha_{g-1})\). The point p is a Weierstrass one if \(\alpha\) \(\neq (0,...,0)\). Let \(M_ g\) be the moduli spaces of curves and \(C_ g\) the moduli spaces of pointed curves. There is a stratification of \(C_ g\) by W-points such that for each index \(\alpha\) one has \(C_{\alpha}=\{(C,p),\alpha (C,p)=\alpha \}.\) The authors discuss some old and new problems on the subject as for instant: the determination of \(\alpha\) so that \(C_{\alpha}=\emptyset\), the codimension of \(C_{\alpha}\), the geometric relationship between \(C_{\alpha}\), the locus of the W-points in \(C_ g\), and so on. Many of these questions are still open. Weierstrass points; Schubert index; moduli spaces of curves Eisenbud, D., Harris, J.: Recent progress in the study of Weierstrass points. Conf. Proceedings, Rome 1984. Lect. Notes Math. (to appear) Families, moduli of curves (analytic), Riemann surfaces; Weierstrass points; gap sequences, History of algebraic geometry, Families, moduli of curves (algebraic), History of mathematics in the 20th century Recent progress in the study of 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 \(X/k\) be a smooth projective surface over an algebraically closed field \(k\). The Picard number \(\rho (X)\) is the rank of the Néron-Severi group \(NS (X)\) of \(X/k\), and for any subfield \(k_ 0 \subset k\) over which \(X\) is defined, \(\rho (X/k_ 0)\) is the rank of the subgroup of \(NS (X)\) generated by the classes of \(k_ 0\)-rational divisors. It is well known that \(\rho (X) \leq 20\) for complex K3-surfaces. In the note under review it is proved that \(\rho (X/ \mathbb{Q}) \leq 19\) for any K3- surface \(X/ \mathbb{Q}\), and an example \(X\) is given with \(\rho (X/ \mathbb{Q}) = 19\). The proof uses reduction mod \(p\) and the geometry of K3-surfaces over finite fields. Picard number; Néron-Severi group; K3-surfaces Picard groups, \(K3\) surfaces and Enriques surfaces, Elliptic curves, Arithmetic theory of algebraic function fields On the rank of elliptic curves over \(\mathbb{Q}(t)\) arising from 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 We study the moduli space \(\mathcal{AS}_g\) of Artin-Schreier curves of genus \(g\) over an algebraically closed field \(k\) of positive characteristic \(p\). The moduli space is partitioned into strata, which are irreducible. Each stratum parameterizes Artin-Schreier curves whose ramification divisors have the same coefficients. We construct deformations of these curves to study the relations between those strata. As an application, when \(p = 3\), we prove that \(\mathcal{AS}_g\) is connected for all \(g\). When \(p > 3\), it turns out that \(\mathcal{AS}_g\) is connected for a sufficiently large value of \(g\). In the course of our work, we answer a question of \textit{R. Pries} and \textit{H. J. Zhu} [Ann. Inst. Fourier 62, No. 2, 707--726 (2012; Zbl 1281.11062)] about how a combinatorial graph determines the geometry of \(\mathcal{AS}_g\). Artin-Schreier theory; moduli space; characteristic \(p\) ramification Families, moduli of curves (algebraic), Positive characteristic ground fields in algebraic geometry, Coverings of curves, fundamental group Connectedness of the moduli space of Artin-Schreier curves of fixed 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 One considers a relatively minimal bielliptic fibration \(\pi : S\to B\) of genus \(g\geq 6\) (\(S\) surface, \(B\) curve). Let \(\omega:=\omega _{S/B}\) be the relative canonical bundle and let \(\Delta (\pi)\) be the degree of \(\pi _\ast (\omega)\). One shows that the slope \(\lambda (\pi) := \omega ^2 /\Delta (\pi)\) of \(\pi\) is at least \(4\) and it is \(4\) iff \(S\) is the minimal desingularization of a double cover of a smooth elliptic surface \(V\), such that the branch locus of the double cover has only negligible singularities and the fibres of \(\tau : V\to B\) are smooth and isomorphic. bielliptic fibration; elliptic curve; branch divisor ------, On the slope of bielliptic fibrations, Proc. Amer. Math. Soc. 129 (2001), 1899--1906. JSTOR: Families, moduli of curves (algebraic), Elliptic surfaces, elliptic or Calabi-Yau fibrations, Surfaces of general type On the slope of bielliptic fibrations | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is 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 \(d\) and \(g\) be integers, and consider the Hilbert scheme \(H(d,g)\) parametrizing smooth, irreducible projective space curves of degree \(d\) and genus \(g\). The author proves that if \(H(d,g)\) is nonempty, then it contains a generically smooth component of the ``expected'' dimension. Moreover, the cohomological properties of a general curve in this component is studied. Hilbert scheme of smooth connected space curves; degree; genus Kleppe, J.O.: On the existence of nice components in the Hilbert Scheme \(\text{H}(d,g)\) of Smooth Connected Space Curves. Boll. U.M.I (7) 8-B, 305-326 (1994) Families, moduli of curves (algebraic), Plane and space curves, Parametrization (Chow and Hilbert schemes), Vector bundles on curves and their moduli On the existence of nice components in the Hilbert scheme \(H(d,g)\) of smooth connected 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 \(V\) denote an open irreducible subset of the Hilbert-scheme of \(\mathbb P^r\) parametrizing irreducible and smooth (if \(r\geq 3)\) resp. nodal (if \(r=2)\) curves of genus \(g\) and \(\pi\colon V\to \mathcal M_ g\) the natural map into the moduli space of curves of genus \(g\). \(V\) is called to have the expected number of moduli if \(\dim \pi(V)=\min(3 g-3, 3g - 3+\rho(g,r,n))\), where \(\rho(g,r,n)\) is the Brill-Noether number. It is the aim of the paper to construct such families of curves. For plane curves the result is quite complete. It is shown that for all \(g\) and \(n\geq 5\) such that \(n-2\leq g\leq \binom{n-1}{2}\) there is an irreducible component of the family of plane irreducible nodal curves of degree \(n\) and genus \(g\), having the expected number of moduli. In the range \(n-2\leq g\leq 3n/2-3\) (resp. \(2n-4\leq g\leq \binom{n-1}{2}\) this result was known (resp. independently proved) by \textit{P. Griffiths} and \textit{J. Harris} [Duke Math. J. 47, 233--272 (1980; Zbl 0446.14011)] (resp. by Coppens).
For \(r\geq 3\) the result is: for all \(g\) and \(n\geq r+1\) such that \(n-r\leq g\leq(r(n-r)-1)/(r-1)\) (resp. \(n-3\leq g\leq 3n-18\) if \(r=3)\) there is an open set of an irreducible component of the Hilbert scheme of \(\mathbb P^r\) parametrizing smooth irreducible curves of degree \(n\) and genus \(g\), which has the expected number of moduli. An immediate consequence is a special case of a theorem of \textit{D. Eisenbud} and \textit{J. Harris} [Invent. Math. 74, 371--418 (1983; Zbl 0527.14022)] on the existence of very ample line bundles of a prescribed type.
The proofs of the theorems are given by induction, the induction step being roughly as follows: Start with a particular curve \(C\) in \(\mathbb P^n\) whose existence is known, construct a new curve by adding a particular rational curve \(\gamma\) and get a new curve \(C'\) by flat smoothing of \(C\cup \gamma\). Hilbert-scheme; moduli space of curves Sernesi E. '' On the existence of certain families of curve .'', Inv. Math. 75 ( 1984 ), 125 - 171 . MR 728137 | Zbl 0541.14024 Families, moduli of curves (algebraic) On the existence of certain families 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 develop a technique that allows us to prove results about subvarieties of general type hypersurfaces. As an application, we use a result of \textit{H. Clemens} and \textit{Z. Ran} [Am. J. Math. 126, No. 1, 89--120 (2004; Zbl 1050.14035)] to prove that a very general hypersurface of degree \(\frac{3n+1}{2} \leq d \leq 2n - 3\) in \(\mathbb{P}^n\) contain lines but no other rational curves. Hypersurfaces and algebraic geometry, Parametrization (Chow and Hilbert schemes), Families, moduli of curves (algebraic), Grassmannians, Schubert varieties, flag manifolds Rational curves on general type 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 In the paper we discuss 7-dimensional orbits in \(\mathbb{C}^4\) of two families of nilpotent 7-dimensional Lie algebras; this is motivated by the problem on describing holomorphically homogeneous real hypersurfaces. Similar to nilpotent 5-dimensional algebras of holomorphic vector fields in \(\mathbb{C}^3 \), the most part of algebras considered in the paper has no Levi non-degenerate orbits. In particular, we prove the absence of such orbits for a family of decomposable 7-dimensional nilpotent Lie algebra (31 algebra). At the same time, in the family of 12 non-decomposable 7-dimensional nilpotent Lie algebras, each containing at least three Abelian 4-dimensional ideals, four algebras has non-degenerate orbits. The hypersurfaces of two of these algebras are equivalent to quadrics, while non-spherical non-degenerate orbits of other two algebras are holomorphically non-equivalent generalization for the case of 4-dimensional complex space of a known Winkelmann surface in the space \(\mathbb{C}^3\). All orbits of the algebras in the second family admit tubular realizations. homogeneous manifold; holomorphic function; vector field; Lie algebra; abelian ideal Almost homogeneous manifolds and spaces, Holomorphic functions of several complex variables, Lie algebras of vector fields and related (super) algebras, Families, moduli of curves (algebraic), Ideals and multiplicative ideal theory in commutative rings On degeneracy of orbits of nilpotent Lie algebras | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians The weighted Gaussian maps \(\gamma_{a,b}(X,L)\) are natural generalizations of the standard Gaussian maps. The paper under review gives conditions ensuring their surjectivity, following the approach of \textit{J. Wahl} in [J. Differ. Geom. 32, No. 1, 77--98 (1990; Zbl 0724.14022)] for the standard case. As the cokernel of the weighted Gaussian maps is related to the tangent space to the locus \(\mathbf{Th}_{g,h}^r\subset \mathcal{M}_g\) of curves \(C\) such that there is an \(h\)th root of \(K_C\) which admits at least \(r+1\) independent global sections, this yields a bound on the dimension of the Zariski tangent space to \(\mathbf{Th}_{g,h}^r\), which is sharp for \(r=0\). The case of components of \(\mathbf{Th}_{g,h}^r\) containing complete intersection curves is also discussed. Gaussian maps; weighted Gaussian maps; higher spin curves; higher theta-characteristics; complete intersections Families, moduli of curves (algebraic), Complete intersections On the surjectivity of weighted Gaussian maps | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians Let \(R_{3,n}\) denote the ring of ternary forms of degree \(n\) over the complex field \(\mathbb{C}\). Consider the action of \(\operatorname{SL} _3(\mathbb{C})\) on the ring \(R_{3,n}\). Explicit generators are known for \(n \leq 4\). While the cases \(n \leq 3\) are classically known, the case of \(n = 4\) was shown by \textit{J. Dixmier} [Adv. Math. 64, 279--304 (1987; Zbl 0668.14006)] and by Ohno (unpublished, see also \textit{A.-S. Elsenhans} [J. Symb. Comput. 68, Part 2, 109--115 (2015; Zbl 1360.13017)] and \textit{M. Girard} and \textit{D. R. Kohel} [Lect. Notes Comput. Sci. 4076, 346--360 (2006; Zbl 1143.14304)]). By the work of these authors it follows that the ring \(\mathbb{C}[R_{3,4}]^{\operatorname{SL} _3(\mathbb{C})}\) is generated by 13 elements, the so-called Dixmier-Ohno invariants of ternary quartics. The main result of the present paper is an explicit method that, given a generic tuple of Dixmier-Ohno invariants, reconstructs a corresponding plane quartic curve. The main technical tool is a method of \textit{J.-F. Mestre} [Prog. Math. 94, 313--334 (1991; Zbl 0752.14027)], see also the authors in [Open Book Ser. 1, 463--486 (2013; Zbl 1344.11049)]. A Magma package of the authors for reconstructing plane quartics from Dixmier-Ohno invariants is available under \url{https://github.com/JRSijsling/quartic\_reconstruction/}. plane quartic curves; invariant theory; Dixmier-Ohno invariants; moduli spaces; reconstruction Actions of groups on commutative rings; invariant theory, Geometric invariant theory, Families, moduli of curves (algebraic), Arithmetic ground fields for curves Reconstructing plane quartics from their invariants | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is 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 \(\mathcal M_{1,3}\) be the space of elliptic curves with three marked points \((E,p_1,p_2,p_3)\). \textit{C. T. McMullen} et al. introduced the \textit{flex locus} \(\mathcal F \subset \mathcal M_{1,3}\) consisting of triples where \(E\) admits a plane cubic model with projection \(\pi\) from a point \(s \in \mathbb P^2 - E\) such that \(p_1,p_2,p_3\) are collinear cocritical points of \(\pi\), meaning that the fiber of \(\pi\) is of type \(p_i+2q_i\) where \(q_i\) is a critical point [Ann. Math. 185, 957--990 (2017; Zbl 1460.14062)]. The flex locus is a primitive, totally geodesic subvariety. Let \(\overline{\mathcal F} \subset \overline{\mathcal M_{1,3}}\) be the closure in the Deligne-Mumford compactification. The author uses several test curves to compute the divisor class of \(\overline{\mathcal F} \subset \overline{\mathcal M_{1,3}}\) in terms of standard generators of \(\text{Pic}_{\mathbb Q} (\overline{\mathcal M_{1,3}})\).
The author then computes some tautological intersection numbers on the \textit{gothic locus} \(\Omega \mathcal G \subset \Omega \mathcal M_4 (2,2,2,0,0,0)\) [loc. cit.], an \(\text{SL}_2 (\mathbb R)\)-invariant subvariety. He achieves this by observing that away from a proper closed set, the gothic locus is isomorphic to \(\mathcal Q \mathcal F \subset \mathcal Q_{1,3}\), the locus of quadratic differentials \((E,q)\) over \(\mathcal F\) such that \(\text{div} (q) = z_1+z_2+z_3-p_1-p_2-p_3\) where \(z_1+z_2+z_3\) is a fiber of \(\pi\). Working with \(\mathcal Q \mathcal F \subset \mathcal Q_{1,3}\), he examines the projectivization of the closure \(\mathbb P \mathcal Q \overline{\mathcal F} \subset \mathbb P \overline{\mathcal Q_{1,3}}\). The latter has tautological bundle \(\mathcal O (-1)\) with first Chern class \(\eta\). Letting \(\lambda_1\) be the first Chern class of the Hodge bundle, he computes the intersection numbers \(\eta^2 \lambda_1\) and \(\eta^3\) on \(\mathbb P \mathcal Q \overline{\mathcal F}\), giving numerical confirmation of a conjecture of \textit{D. Chen} et al. [``Masur-Veech volumes and intersection theory on moduli spaces of abelian differentials'', Preprint, \url{arXiv:1912.02267}]. plane cubics; quadratic differentials; \({\text{SL}}_2({\mathbb{R}})\)-invariant varieties; Lyapunov exponents Families, moduli of curves (algebraic), Families, moduli of curves (analytic), Moduli of Riemann surfaces, Teichmüller theory (complex-analytic aspects in several variables) Dynamical invariants and intersection theory on the flex and gothic loci | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians Here we study the deformation theory of some maps \(f: X \rightarrow \mathbb P^{r}, r = 1,2,\) where \(X\) is a nodal curve and \(f|T\) is not constant for every irreducible component \(T\) of \(X\). For \(r = 1\) we show that the ``stratification by gonality'' for any subset of \({\overline {\mathcal M}}_g\backslash {\mathcal M}_g\) with fixed topological type behaves like the stratification by gonality of \({\mathcal M}_{g}\). stable curve; Brill-Noether theory; plane nodal curve; gonality Special divisors on curves (gonality, Brill-Noether theory), Families, moduli of curves (algebraic), Plane and space curves Gonality for stable curves and their maps with a smooth curve as their target | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians Let \(C\) be a smooth complex projective curve of genus \(g\). A point \(p \in C\) is subcanonical if the line bundle \(\mathcal{O}_C ((2g-2)p)\) is isomorphic to the canonical bundle \(\omega_C\). Those points are thence Weierstrass. If \(\mathcal{M}_{g,1}\) is the moduli space of pointed curves of genus \(g\), \(\mathcal{G}_g \subset \mathcal{M}_{g,1}\) is the subcanonical locus, described by pairs \((C,p)\) where \(p\) is a subcanonical point of \(C\). Then \(\mathcal{G}_g\) consists of three irreducible components, \(\mathcal{G}_g^{\text{hyp}}\), \(\mathcal{G}_g^{\text{odd}}\) and \(\mathcal{G}_g^{\text{even}}\). The last two components correspond to non-hyperelliptic curves \(C\) such that \(h^0 (C,(g-1)p)\) is odd and even, respectively.
The paper under review is devoted to study the dimension and irreducibility of subsets of \(\mathcal{G}_g^{\text{odd}}\) and \(\mathcal{G}_g^{\text{even}}\). In particular, the authors consider the subloci of \(\mathcal{M}_{g,1}\), \(\mathcal{G}_g^r\), consisting of non-hyperelliptic curves \(C\) such that \(h^0 (C,(g-1)p)\) is \(\geq r+1\) and \(\equiv r+1\) (mod 2). They obtain a general lower bound for the dimension of the irreducible components of \(\mathcal{G}_g ^r\) which they prove to be sharp for \(r \leq 3\). They conclude by applying the results in case \(r=2\) to prove an existence result of triply periodic minimal Riemann surfaces in the Euclidean space \(\mathbb{R}^3\). subcanonical point; Weierstrass point; minimal surface Riemann surfaces; Weierstrass points; gap sequences, Families, moduli of curves (algebraic), Minimal surfaces in differential geometry, surfaces with prescribed mean curvature, Differential geometry of immersions (minimal, prescribed curvature, tight, etc.) Subcanonical points on projective curves and triply periodic minimal surfaces in the Euclidean space | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians Let \(Y\) be a smooth projective curve degenerating to a reducible curve \(X\) with two components meeting transversally at one point. We show that the moduli space of vector bundles of rank two and odd determinant on \(Y\) degenerates to a moduli space on \(X\) which has nice properties, in particular, it has normal crossings. We also show that a nice degeneration exists when we fix the determinant. We give some conjectures concerning the degeneration of moduli space of vector bundles on \(Y\) with fixed determinant and arbitrary rank. smooth projective curve; moduli space of vector bundles of rank two; degeneration Nagaraj D.S., Seshadri C.S., Degenerations of the moduli spaces of vector bundles on curves. I, Proc. Indian Acad. Sci. Math. Sci., 1997, 107(2), 101--137 Vector bundles on curves and their moduli, Families, moduli of curves (algebraic), Algebraic moduli problems, moduli of vector bundles Degenerations of the moduli spaces of vector bundles on curves. I | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians Modular towers have been introduced by Fried in [\textit{M. D. Fried}, Recent developments in the inverse Galois problem. A joint summer research conference, July 17-23, 1993, University of Washington, Seattle, WA, USA. Providence, RI: American Mathematical Society. Contemp. Math. 186, 111--171 (1995; Zbl 0957.11047)]. In the paper under review, the author recalls how they are constructed and explains the main conjectures about them.
Let \(p\) be a prime number and \(G\) be a finite group. Assume that \(p\) divides \(|G|\) and \(G\) is \(p\)-perfect (\textit{i.e. generated by elements of order prime to \(p\)}). Let \(_{p}\tilde{G}\) be the universal \(p\)-Frattini cover of \(G\). Define a sequence of characteristic subgroups of \(_{p}\tilde{G}\) by
\[
\mathrm{ker}_{0} = \mathrm{ker}(_{p}\tilde{G}\to G),\ldots,\mathrm{ker}_{n} = \mathrm{ker}_{n-1}^p [\mathrm{ker}_{n-1},\mathrm{ker}_{n-1}], \ldots
\]
and the corresponding sequence of quotients \((_{p}^n\tilde{G})_{n\geq 0}\). Fix a set \({\mathbf C}\) of conjugacy classes of \(G\) of order prime to \(p\). It lifts to a projective system of conjugacy classes \(({\mathbf C}^n)_{n\geq 0}\). The modular tower associated with the triple \((G,p,{\mathbf C})\) is the tower of Hurwitz spaces \((H_{_{p}^n\tilde{G}}({\mathbf C}^n))_{n\geq 0}\) (coarse moduli space of \(_{p}^n\tilde{G}\)-covers of \({\mathbf P}^1\) with inertia canonical invariant \({\mathbf C}^n\)).
In the case where \(G\) is the dihedral group \(D_{p}\) of order \(2p\) (so that \(_{p}^n\tilde{G}\) is \(D_{p^n}\) and \(_{p}\tilde{G}\) is \(D_{p^\infty}\)) and \({\mathbf C}\) is four copies of the class of the non-trivial involution in \(D_{p}\), there exists a morphism from the modular tower to the tower of modular curves \((X_{1}(p^n))_{n\geq 0}\). The things we already know about this last tower give motivations for some Diophantine conjectures. For example, if \(F\) is a number field, we deduce from Faltings' theorem that \(X_{1}(p^n)(F)\) is empty when \(n\) is large enough. Conjecture 2.4 asserts that the same is true for modular towers. There is a counterpart in Galois theory. Let \(F\) be a number field and \(r_{0}\geq 3\) be an integer. Conjecture 2.3 asserts that only finitely many \(_{p}^n\tilde{G}\) can be regularly realized as Galois groups over \(F(T)\) with no more than \(r_{0}\) branch points.
At the end of the paper, the author describes the \(\ell\)-adic situation, which is quite different. Let \(k\) be a Henselian field for a rank one valuation. If \(k\) is of characteristic \(0\) and contains enough roots of unity, then modular towers of Harbater-Mumford type have projective systems of \(k\)-points (see [\textit{P. Dèbes} and \textit{B. Deschamps}, J. Reine Angew. Math. 574, 197--218 (2004; Zbl 1051.12003)]). modular tower; Hurwitz spaces; dihedral group; inverse Galois problem; rational points; moduli spaces Dèbes, Pierre: An introduction to the modular tower program. Sémin. congr. 13, 127-144 (2006) Inverse Galois theory, Universal profinite groups (relationship to moduli spaces, projective and moduli towers, Galois theory), Rational points, Families, moduli of curves (algebraic), Coverings of curves, fundamental group An introduction to the modular tower program | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is 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{F. Severi} claimed in the 1920s that the Hilbert scheme \({\mathcal H}_{d,g,r}\) of smooth irreducible non-degenerate curves \(C \subset \mathbb P^r\) of degree \(d\) and genus \(g\) is irreducible for \(d \geq g+r\) [Vorlesungen über algebraische Geometrie. Übersetzung von E. Löffler. Leipzig u. Berlin: B. G. Teubner (1921; JFM 48.0687.01)]. \textit{L. Ein} proved Severi's claim for \(r=3\) and \(r=4\) [Ann. Sci. Éc. Norm. Supér. (4) 19, No. 4, 469--478 (1986; Zbl 0606.14003); Proc. Symp. Pure Math. 46, 83--87 (1987; Zbl 0647.14012)], but there are counterexamples of various authors for \(r \geq 6\).
It has been suggested by \textit{C. Ciliberto} and \textit{E. Sernesi} [in: Proceedings of the first college on Riemann surfaces held in Trieste, Italy, November 9-December 18, 1987. Teaneck, NJ: World Scientific Publishing Co. 428--499 (1989; Zbl 0800.14002)] that Severi intended irreducibiity of the Hilbert scheme \({\mathcal H}^{\mathcal L}_{d,g,r} \subset {\mathcal H}_{d,g,r}\) of curves whose general member is linearly normal: indeed, the counterexamples above arise from families whose general member is not linearly normal.
Here the authors prove irreducibility for \(g+r-2 \leq d \leq g+r\) (the Hilbert scheme is empty for \(d > g+r\) by Riemann-Roch) and for \(d=g+r-3\) under the additional assumption that \(g \geq 2r+3\). This extends work of \textit{C. Keem} and \textit{Y.-H. Kim} [Arch. Math. 113, No. 4, 373--384 (2019; Zbl 1423.14028)]. Hilbert scheme; linear series; linearly normal curves Parametrization (Chow and Hilbert schemes), Families, moduli of curves (algebraic) On the Hilbert scheme of linearly normal curves in \(\mathbb{P}^r\) of relatively high 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 The present paper deals with the homology of the families of superelliptic curves
\[
\text{E}_n^d = \{(\text{P},z,y) \in \text{C}_n \times \text{D} \times \mathbb{C},\ y^d = (z-x_1) \dots (z-x_n)\},
\]
where D is the open unit disk in \(\mathbb{C}\), \(\text{C}_n\) is the configuration space of \(n\) distinct unordered points in D, and \(\text{P} = \{x_1, \dots, x_n\} \in \text{C}_n\). Given P, the curve
\[
\Sigma_n^d = \{(z,y) \in \text{D} \times \mathbb{C},\ y^d = (z-x_1) \dots (z-x_n)\}
\]
is a \(d\)-fold covering of D, totally ramified over P, and there is a fibration \(\pi : \text{E}_n^d \rightarrow \text{C}_n\) which maps \(\Sigma_n^d\) onto its set of ramification points. The case \(d=2\) of hyperelliptic curves was studied by the authors in [Isr. J. Math. 230, No. 2, 653--692 (2019; Zbl 1419.14039)]. Now, they consider a generic \(d\), and this paper is a natural sequel of the former. The main results are anticipated in Section 1, as Theorems 1.1--1.4. The first of them states that the space \( \text{E}_n^d\) is a classifying space for the complex braid group of type \(\text{B}(d,d,n)\). Then, the homology of the complex braid group of type \(\text{B}(d,d,n)\) over finite fields when \(d\) or \(n\) is odd is computed, and their stable homology groups are described. Explicit examples for \(3 \leq d \leq 6\) are given in the final section of the paper. Of course, when specializing the results to \(d=2\), the results of the earlier paper are recovered. superelliptic curves; integral homology; braid groups Coverings of curves, fundamental group, Families, moduli of curves (algebraic), Group actions on manifolds and cell complexes in low dimensions, Braid groups; Artin groups Families of superelliptic curves, complex braid groups and generalized Dehn twists | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is 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 0511.00010.]
The present paper is a brief but beautiful introduction to the geometric invariant theory (GIT) and also contains its applications to construct moduli spaces in algebraic geometry.
In the first two sections, the author sketches basics of GIT over the complex numbers. In {\S} 3 stable bundles on a smooth curve are considered and the author shows how to construct a moduli space of stable bundles. - In {\S} 4 the author shows that there are enough projective invariants of space curves of genus g and degree d, provided \(d\geq 2g\) and the curves are non-degenerate (theorem 4.2.). This result can be used to construct a moduli space \({\mathcal M}_ g\) of smooth curves of genus g.
In {\S} 5 and {\S} 6, the author studies the relationship between stable curves in the sense of Deligne and Mumford and those in the sense of GIT. The author shows that they are essentially the same (proposition 5.1 and proposition 6.1). The result implies that the compactification \(\bar {\mathcal M}_ g\) of \({\mathcal M}_ g\) given by Deligne and Mumford is projective. This fact was first proved by F. Knutsen in characteristic zero by a different method. In {\S} 7 the author indicates how GIT can be used to construct compactification of generalized Jacobians of stable curves. stable bundle; geometric invariant theory; moduli spaces; compactification of generalized Jacobians of stable curves D. Gieseker, Geometric invariant theory and applications to moduli problems, Invariant theory (Montecatini, 1982) Lecture Notes in Math., vol. 996, Springer, Berlin, 1983, pp. 45 -- 73. Algebraic moduli problems, moduli of vector bundles, Geometric invariant theory, Families, moduli of curves (algebraic), Group actions on varieties or schemes (quotients) Geometric invariant theory and applications to moduli problems | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians Let \(D_1\) and \(D_2\) be (associative) algebras finite-dimensional over their centres \(F_1\) and \(F_2\), respectively. Suppose that \(F_1\) and \(F_2\) contain as a subfield an algebraically closed field \(F\). As noted by the authors, the research presented in the paper under review has initially been motivated by the question of whether the tensor product \(D_1\otimes_FD_2\) is a domain. This question was posed by Schacher and studied, for instance, by \textit{G. M. Bergman} [Lect. Notes Math. 545, 32-82 (1976; Zbl 0331.16015)].
The paper gives an example where the answer is ``no'', as a part of which it obtains results of independent interest about division algebras and Brauer groups over curves; specifically, this includes a splitting criterion for certain Brauer group elements on the product of two curves over \(F\). This is preceded by a study of Picard group and Brauer group properties of \(F_1\otimes_FF_2\). At the same time, the authors show that the answer is ``yes'', provided that \(D_1=F_1\). Assuming in addition that \(F_1\) is endowed with a discrete valuation \(v\) and \(R\) is the valuation ring of \((F_1,v)\), they prove that \(D_1\otimes_FD_2\) is a domain whenever \(D_1\) is totally ramified at \(R\); in particular, this holds in case \(D_1/F_1\) has prime degree \(p\) different from the characteristic of \(F\), and \(D_1\) is ramified at \(R\). division algebras; tensor products; Schur indices; ramification; Picard groups; Brauer groups; products of curves; domains Louis Rowen and David J. Saltman, Tensor products of division algebras and fields, J. Algebra 394 (2013), 296 -- 309. Finite-dimensional division rings, Brauer groups of schemes, Picard groups, Brauer groups (algebraic aspects), Galois cohomology Tensor products of division algebras and fields. | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians Let K be an algebraically closed field of characteristic zero and T an integral algebraic scheme over K. Let \(\pi: X\to T\) be a family of projective curves with general fibre of geometric genus g. Let C be the fibre of \(\pi\) over a closed point \(0\in T\) and denote by \(C_ 1,C_ 2,...,C_ s\) the irreducible components of C, so that \(z(C)=\sum^{s}_{i=1}r_ iC_ i\) is the cycle associated with C. Suppose the \(C_ i\), \(i=1,...,s\), has geometric genus \(g_ i\) and also that \(g_ i>0\) if \(1\leq i\leq r\) and \(g_ i=0\) if \(i>r\) for some integer \(r\leq s\). Then the author proves the following relation: g-1\(\geq \sum_{1\leq i\leq r}r_ i(g_ i-1)+r-1\). family of projective curves; geometric genus Families, moduli of curves (algebraic) Variation du genre des fibrés d'une famille de courbes. (Variation of the genus of fibers of a family of curves) | 0 |
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