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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 Let \(\Gamma\) be an arithmetic group acting on the two-dimensional complex unit ball B. Special subdiscs D of B, called rational discs, project to algebraic curves on B/\(\Gamma\). These curves are called arithmetic curves. In the article there is proved a formula for the selfintersection number of an arithmetic curve on a certain smooth model of the Baily- Borel compactification B/\(\Gamma\), which is called the filled singularity resolution of B/\(\Gamma\) along D/\(\Gamma\). The proof of the selfintersection formula is a combination of Hirzebruch-Mumford's proportionality theory and a fine analysis of a stepwise resolution procedure for quotient singularities of surfaces. - The intersection formula is applied to the classification of Picard modular surfaces of Gauß and Eisenstein numbers. For this purpose the author uses a fine classification criterion for rational surfaces by means of Chern numbers and curve configurations. In the meantime the results have been effectively applied by the author to the study of automorphic forms, the monodromy group of an Euler partial differential equation (system) and of the integrals on Riemann surfaces of equation type \(y^ 3=x(x-1)(x-u)(x- v)\) and to related problems. The corresponding results can be found in the author's monograph ''Geometry and arithmetic around Euler partial differential equations'' (Mathematics and its applications, Dordrecht (1985)]. rational discs; selfintersection number of an arithmetic curve; filled singularity resolution; Picard modular surfaces; automorphic forms; monodromy group; integrals on Riemann surfaces Holzapfel, R.-P.: Arithmetic curves on ball quotient surfaces, Ann. Glob. Analysis and Geometry 1, o..2 (1983), 21-90. Special surfaces, Special algebraic curves and curves of low genus, Structure of modular groups and generalizations; arithmetic groups, Singularities of surfaces or higher-dimensional varieties, Fine and coarse moduli spaces, Global theory and resolution of singularities (algebro-geometric aspects) Arithmetic curves on ball quotient 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 To each closed Riemann surface \(X\), of genus \(g\), there is associated its Jacobian variety \(JX\), this being a \(g\)-dimensional principally polarized abelian variety. By fixing a canonical basis for its homology, there is a symmetric, \(g \times g\), matrix \(\Pi\) whose imaginary part is positive definite; called a period matrix of \(X\). The quotient torus \({\mathbb C}^{g}/({\mathbb Z}^{g} \oplus \Pi {\mathbb Z}^{g})\) can be identifies with \(JX\). Within this identification, one may construct theta functions (with characteristics \(\varepsilon, \varepsilon' \in {\mathbb R}^{g}\))
\[
\theta\begin{bmatrix} \varepsilon \\ \varepsilon' \end{bmatrix} (\zeta,\Pi)=
\]
\[
=\sum_{N \in {\mathbb Z}^{g}} \text{Exp}\left[2\pi i \left(\frac{1}{2} \left(N+\frac{\varepsilon}{2}\right)^{t} \Pi \left(N+\frac{\varepsilon}{2}\right) + \left(N+\frac{\varepsilon}{2}\right)^{t} \left(\zeta+\frac{\varepsilon'}{2}\right) \right)\right].
\]
When the characteristics are rationals, its values at \(\zeta=0\) are called theta constants. In [\textit{J. Thomae}, J. Reine Angew. Math. 71, 201--222 (1870; JFM 02.0244.01)] it was obtained formulae relating theta constants in the case \(X\) is a hyperelliptic Riemann surface. In later papers these formulae have been generalized to the so called non-singular \({\mathbb Z}_{n}\) curves. The paper under review consider the above for the arbitrary fully ramified \({\mathbb Z}_{n}\) curves (that is, every ramification point has maximal index \(n-1\)); these are represented by algebraic curves of the form \(w^{n}=\prod_{j=1}^{m}(x-a_{j})^{l_{j}}\), where each \(l_{j} \in \{1,\ldots,n-1\}\) is relatively prime with \(n\) and \(l_{1}+\cdots+l_{m}\) is either divisible by \(n\) or its is relatively prime to it. The main point in the paper is the construction of certain non-special divisors on \(X\), supported at fixed points of the automorphism \(\tau(x,w)=(x,e^{2 \pi i/n} w)\), and to consider certain powers of theta functions associated to such divisors. There is provided the details and computations needed to obtain the formulae. Also, it ends with a section concerning open questions and some conjectures. Riemann surface; jacobian variety; algebraic curve; theta function Jacobians, Prym varieties, Riemann surfaces; Weierstrass points; gap sequences, Riemann surfaces Thomae formulae for general fully ramified \(Z_n\) 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 separably closed field of characteristic \(p>0\) and \(g \geq 2\) be an integer. By \textit{P. Deligne} and \textit{D. Mumford} [Publ. Math., Inst. Hautes Étud. Sci. 36(1969), 75-110 (1970; Zbl 0181.488)] there exists a universal stable curve \(Z_ g \to H_ g\), where \(H_ g\) is a \(k\)-subscheme of a convenient Hilbert scheme such that every stable curve over \(k\) of genus \(g\) is isomorphic to a fiber of \(Z_ g \to H_ g\) and \(H_ g\) is geometrically irreducible and smooth over \(k\), moreover the set of \(x \in H_ g\) whose fiber in \(Z_ g\) is smooth is an open dense subset of \(H_ g\). Let \(\eta\) be the generic point of \(H_ g\) and \(L\) the algebraic closure of \(k(\eta)\). The generic curve of genus \(g\) is denoted by \(X=Z_ g \times_{H_ g} \text{Spec} L\). It is a proper, smooth and connected curve over \(L\). Given a scheme \(S\) of characteristic \(p\) and \(f:Z \to S\) any morphism of schemes, we denote by \(Z^{(p)}=Z \times_ SS\) with respect to the absolute Frobenius morphism \(S \to S\). Given a semi-stable curve \(Z\) over a field \(K\) of characteristic \(p>0\) the relative Frobenius \(F:Z \to Z^{(p)}\) induces a map \(F^*:H^ 1(Z^{(p)}, {\mathcal O}_{Z^{(p)}}) \to H^ 1(Z, {\mathcal O}_ Z)\). We say that \(Z\) is ordinary if \(F^*\) is bijective. The author's main result states that given any étale connected Galois covering \(Y\) of \(X\) with Galois group of order prime to \(p\) then \(Y\) is ordinary. In particular, \(X\) is ordinary. Furthermore, this result together with a result of \textit{R. M. Crew} [cf. Compos. Math. 52, 31-45 (1984; Zbl 0558.14009); corollary 1.8.3] which says that if \(Y\) is a complete nonsingular connected curve defined over an algebraically closed field \(k\) of characteristic \(p>0\) and \(X \to Y\) is a finite étale Galois covering of degree a power of \(p\), then \(X\) is ordinary if and only if \(Y\) is ordinary, implies that every étale abelian covering of a generic curve is ordinary. characteristic \(p\); absolute Frobenius; ordinary curve; étale abelian covering of a generic curve Finite ground fields in algebraic geometry, Coverings of curves, fundamental group Abelian étale coverings of generic curves and ordinarity | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is 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 0682.00008.]
Let G be a connected linear algebraic group and X be a normal G-variety over an algebraically closed field of characteristic zero. The authors give a new proof of the following result of \textit{Sumihiro}:
Let \(Y\subset X\) be an orbit in X. There is a finite-dimensional rational representation \(G\to GL(V)\) and a G-stable open neighborhood U of Y in X which is G-equivariantly isomorphic to a G-stable locally closed subvariety of the projective space P(V).
The main technical ingredients are G-linearizations of line bundles and the study of the Picard group of a linear algebraic group. Picard group of a linear algebraic group Knop, F., Kraft, H., Luna, D., et al.: Local properties of algebraic group actions, in: Algebraische Transformationsgruppen und Invariantentheorie (H. Kraft, P. Slodowy, T.A. Springer eds.), \textbf{63-76}, DMV Semin., 13, Birkhäuser, Basel-Boston-Berlin, 1989 Group actions on varieties or schemes (quotients), Representation theory for linear algebraic groups, Picard groups Local properties of algebraic 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 Let \(C\) be a smooth projective curve of genus \(g>0\). For any rank \(2\) vector bundle on \(C\) set \(S:= \mathbb {P}(C)\) and let \(l\) be the Segre invariant of \(E\), i.e. \(\deg (E)-2\deg (L)\), where \(L\) is a line subbundle of \(E\) with maximal degree, i.e. \(l= -C_0^2\), where \(C_0\) is a minimal section of the ruling of \(S\); call \(f\) the numerical class of a fiber of the ruling of \(S\). Let \(V_l\), \(l>0\), be the moduli space of bundles with Segre invariant \(l\). This paper studies in the case \(l>0\), i.e. \(E\) stable, the existence of bisecant and trisecant for \(S\), i.e. integral curves \(D\) numerically equivalent to \(kC_0+bf\), \(k=2,3\); let \(\delta _k(E)\) be the minimum of the integers \(b\) for which an integral \(D\) exists. They prove that \(\delta _2(E) = \lfloor (2g/3) -l\rfloor\) for a general \(E\in V_l\) when \(2g/3 < l\leq g\). They prove that \(\delta _3(E) = \lfloor -3g/4\rfloor\) (resp. \(\delta _3(E) = \lfloor -3(g-2)/4\rfloor\) when \(E\) is a general stable bundle and \(d-g\) is even (resp. odd). They obtain that for \(k=2,3\) and a general \(E\) the bundle \(\mathrm{Sym}^k(E)\) has the general Lange-stability; they also state a conjecture for the case \(k\geq 4\). vector bundles on a curve; ruled surfaces; moduli of vector bundles; bisecant curves; trisecant curves; Segre invariants; elementary transformation Divisors, linear systems, invertible sheaves, Rational and ruled surfaces, Vector bundles on curves and their moduli Bisecant and trisecant curves on ruled surfaces | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians Let W be a smooth complex projective 3-fold and \(B\subset W\) a reduced curve with irreducible components \(B_ 1,...,B_ m\). The author proves the existence of an irreducible normal surface \(T\subset W\) containing B, with Sing \(T\subset Sing B\) and such that one has an exact sequence \(0\to Pic W\to C(T)\to \oplus^{m}_{i=1}{\mathbb{Z}}[B_ i]\to 0,\) where C(T) means class group. As an immediate corollary one gets that any smooth connected curve D in \({\mathbb{P}}^ 3\) lies on a smooth surface \(S\subset {\mathbb{P}}^ 3\) with Picard number 2 such that Pic S is generated by D together with the hyperplane section. Another consequence says that any height-2 prime ideal Q in a regular factorial ring A which is a ring of quotients of some 3-dimensional finitely generated \({\mathbb{C}}\)-algebra, contains a height-1 prime ideal P such that A/P is normal and the class group C(A/P) is cyclicly generated by Q/P. class group of surfaces in 3-space; curves in threefolds; surface containing a given curve; Pic A. Buium , The automorphism group of a non-linear algebraic group , to appear. \(3\)-folds, Curves in algebraic geometry, Picard groups A note on the class group of surfaces in 3-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 \(C\) be a nodal algebraic curve and let \(\pi : \mathcal{C} \to B\) be a smoothing. Then there is a degree \(d\) Abel map
\[
\alpha^d : \mathcal{C}^d = \mathcal{C} \times_B \cdots \times_B \mathcal{C} \longrightarrow \overline{\mathcal{J}}
\]
to Esteves compactified Jacobian \(\overline{\mathcal{J}}\). Over the generic fiber of \(\pi\) this is just the usual Abel map of a smooth curve and hence well-defined. Over the special fiber however, \(\alpha^d\) will in general not be defined everywhere, i.e. \(\alpha^d\) is only a rational map.
The main theorem of this article employs tropical and toric geometry to describe blow-ups of \(\mathcal{C}^d\) that resolve the degree \(d\) Abel map. Moreover, the authors show that in degree 1 the Abel map is always a morphism and for biconnected \(C\) it is even injective.
Recall that the tropical Jacobian \(J(X)\) of a metric graph \(X\) is defined as the set of degree \(0\) divisors modulo linear equivalence. In previous work [\textit{A. Abreu} and \textit{M. Pacini}, Proc. Lond. Math. Soc. (3) 120, No. 3, 328--369 (2020; Zbl 1453.14082)] have endowed \(J(X)\) with a polyhedral structure using so-called \emph{pseudo divisors}. The tropical degree \(d\) Abel map
\[
\alpha_d^\mathrm{trop} : X^d \longrightarrow J(X)
\]
is not a map of polyhedral complexes. In the present article the authors give a unimodular triangulation of \(X^d\) that makes \(\alpha_d^\mathrm{trop}\) a map of polyhedral complexes. Via toric geometry this subdivision corresponds to blow-ups of \(\mathcal{C}^d\) and the authors show that these blow-ups resolve \(\alpha^d\). The key to this is a result from [\textit{A. Abreu} et al., Mich. Math. J. 64, No. 1, 77--108 (2015; Zbl 1331.14035)] that gives a criterion for resolving \(\alpha^d\) locally around a node in terms of certain numerical invariants of \(C\). In the bulk of the paper these numerical invariants are redefined on the tropical side and it is shown that the triangulation satisfies the criterion on the tropical side (and hence on the algebraic side as well).
The authors remark that there is also an induced subdivision of \(X^d\) by pulling back the polyhedral structure of \(J(X)\) along the Abel map. They describe an implementation of an algorithm to compute this. algebraic curve; tropical curve; Jacobian; Abel map; toric variety Families, moduli of curves (algebraic), Jacobians, Prym varieties, Applications of tropical geometry Abel maps for nodal curves via tropical geometry | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians Let \(X=\{P_ 1,...,P_ s\}\) be a set of distinct points in \({\mathbb{P}}^ n(k)\), 2\(\leq n\leq s\), \(k=\bar k\) an algebraically closed field. Let I be the ideal of these points in \(R=k[x_ 0,...,x_ n]\) and \(H_ X(-)\) the Hilbert function of the coordinate ring \(A=R/I\) of these points. The points are in generic position if \(H_ X(t)=\min\{(^{t+n}),s\}\), for all t. Almost all sets of s points in \({\mathbb{P}}^ n\) are in generic position and the problem considered in this paper is whether or not there is an open set in \(({\mathbb{P}}^ n)^ s\) on which the number of generators of the corresponding ideals has the constant value conjectured by \textit{A. V. Geramita} and \textit{F. Orecchia} in J. Algebra. 78, 36-57 (1982; Zbl 0502.14001). An affirmative answer to the conjecture is given for all s when \(n=2\) and for several infinite families of s for \(n>2.\)
These results give the complete minimal free resolution for a general set of s points in \({\mathbb{P}}^ 2\) and thereby also settle a conjecture of \textit{L. G. Roberts} (for \(n=2)\) [C. R. Math. Rep. Acad. Sci., Soc. R. Can. 3, 43-48 (1981; Zbl 0451.14017)]. As a byproduct to our investigations, we where able to make some comments on a problem posed by Abhyankar concerning the least degree of a nonsingular curve passing through a finite set of points in \({\mathbb{P}}^ n\). number of generators of ideals; Cohen-Macaulay type; Hilbert function of the coordinate ring; minimal free resolution; least degree of a nonsingular curve Geramita, A. V.; Maroscia, P., The ideal of forms vanishing at a finite set of points in \(\mathbb{P}^n\), J. Algebra, 90, 528-555, (1984) Relevant commutative algebra, Commutative rings and modules of finite generation or presentation; number of generators, Varieties defined by ring conditions (factorial, Cohen-Macaulay, seminormal), Singularities in algebraic geometry The ideal of forms vanishing at a finite set of points in \({\mathbb{P}}^ n\) | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians The authors develop a theory of zero divisors or zero currents for sections of a vector bundle under an orientation condition. An \(R^ n\)- valued function is said to be atomic if the function pulls back the basic forms \(dy^ I/| y|^ p\) on \(R^ n\) to locally Lebesgue integrable forms on the domain manifold in the range \(p=| I| \leq n-1\). Such a function automatically has a zero divisor or zero current. By considering vector bundles the concept of an atomic section is introduced. The main result is that the notion of zero divisor is independent of the choice of local frame. The introduction of the notion of an atomic section and its zero divisor is done by showing that under very mild geometric conditions a smooth \(R^ n\)-valued function is automatically atomic. Section 4 includes results about the exact nature of the divisors of an atomic section. Finally, in section 5 the authors prove that the divisor of an atomic section of a vector bundle \(E\), when considered as a cohomology class in \(H^ n(M,Z)\), is the Euler class of the bundle \(E\). zero divisors; zero currents for sections of a vector bundle; locally Lebesgue integrable forms; atomic section; divisor of an atomic section; Euler class F. R. Harvey and S. Semmes, Zero divisors of atomic functions , Ann. of Math. (2) 135 (1992), 567--600. JSTOR: Topology of vector bundles and fiber bundles, Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20], General theory of differentiable manifolds, Fiber bundles in algebraic topology Zero divisors of atomic functions | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians This paper is devoted to the study of the defining equations of blow-up algebras in some particular cases. Blow-up algebras often appear in algebraic geometry and represent fibrations of a variety.
More precisely, the author provides a generating set for the equations of Rees algebras of a Cohen-Macaulay domain associated to an ideal \(I\) of codimension two. \(I\) is required to satisfy some additional conditions and the result depends on the parity of the dimension of the base ring. The main aim of section 3 of the paper is to prove this central result. The generating set is given in terms of the jacobian dual of a presentation matrix of \(I\). The jacobian dual of a matrix is introduced and studied in section 2. Finally in section 4, some remarks are given concerning the degrees of the generators of the defining ideal whenever \(I\) is a codimension three Gorenstein ideal. defining equations of blow-up algebras; Rees algebras; Cohen-Macaulay domain; jacobian dual of a matrix; generators of the defining ideal DOI: 10.1016/0022-4049(95)00087-9 Relevant commutative algebra, Cohen-Macaulay modules, Associated graded rings of ideals (Rees ring, form ring), analytic spread and related topics, Special types (Cohen-Macaulay, Gorenstein, Buchsbaum, etc.), Low codimension problems in algebraic geometry Equations of blowups of ideals of codimension two and three | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians By a rational elliptic surface we mean a smooth complete complex surface that can be obtained from a pencil of cubic curves in \(\mathbb{P}^2\) with smooth members by successive blowing up (9 times) its base points. A more intrinsic characterization is to say that the surface is rational and admits a relatively minimal elliptic fibration possessing a section. Better yet: it is a smooth complete complex surface whose anticanonical system is base point free and defines a fibration. The description as a blown-up \(\mathbb{P}^2\) is not canonical (in general the possible choices are in bijective correspondence with a weight lattice of an affine root system of type \(\widehat E_8\)), but the last characterization makes it plain that the fibration is. The main goal of this paper is to investigate and describe the moduli space of these surfaces and certain compactification thereof.
In particular, we show that the moduli space of rational elliptic surfaces admitting a section is locally a complex hyperbolic variety of dimension 8. We compare its Satake-Baily-Borel compactification with a compactification obtained by means of geometric invariant theory, as considered by \textit{R. Miranda} [Math. Ann. 255, 379--394 (1981; Zbl 0438.14023)]. rational elliptic fibration; moduli; ball quotient; homology of cyclic coverings; central extension of a braid class group; Eisenstein curve; rational curves Gert Heckman and Eduard Looijenga, The moduli space of rational elliptic surfaces, Algebraic geometry 2000, Azumino (Hotaka), Adv. Stud. Pure Math., vol. 36, Math. Soc. Japan, Tokyo, 2002, pp. 185 -- 248. Moduli, classification: analytic theory; relations with modular forms, Elliptic surfaces, elliptic or Calabi-Yau fibrations, Automorphic forms in several complex variables, Coverings of curves, fundamental group The moduli space of rational elliptic surfaces. | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians Let \(X\) be a smooth complex algebraic surface. The Hilbert scheme \(X^{[n]}\) of \(n\) points on \(X\) is a smooth variety of dimension \(2n\). \textit{E. Carlsson} studied the generating series for the intersection pairings between the total Chern class of the tangent bundle and the Chern classes of tautological bundles on \((\mathbb C^2)^{[n]}\), proving that the reduced series \(\langle \text{ch}_{k_1} \dots \text{ch}_{k_N} \rangle^\prime\) is a quasi-modular form [Adv. Math. 229, 2888--2907 (2012; Zbl 1255.14005)]. \textit{A. Okounkov} conjectured that these reduced series are multiple \(q\)-zeta values [Funct. Anal. Appl. 48, 138--144 (2014; Zbl 1327.14026)]. \textit{Z. Qin} and \textit{F. Yu} [Int. Math. Res. Not. 2018, 321--361 (2018; Zbl 1435.14007)] proved the conjecture modulo lower weight terms via the reduced series
\[
\overline F^{\alpha_1,\dots,\alpha_N}_{k_1,\dots,k_N} (q) = (q;q)_\infty^{\chi (X)} \cdot \sum_n q^n \int_{X^{[n]}} (\Pi_{i=1}^N G_{k_i} (\alpha_i,n)) c(T_{X^{[n]}})
\]
where \(0 \leq k_i \in \mathbb Z\), \(\alpha_i \in H^* (X), (q;q)_\infty = \Pi_{n=1}^\infty (1-q^n)\) and \(G_{k_i}(\alpha_i, n) \in H^* (X^{[n]})\) are classes which play a role in the study of the geometry of \(X^{[n]}\) (see work of \textit{Z. Qin} [Hilbert schemes of points and infinite dimensional Lie algebras. Providence, RI: American Mathematical Society (AMS) (2018; Zbl 1403.14003)]).
In the paper under review, the authors further study the series \(\overline F^{\alpha_1,\dots,\alpha_N}_{k_1,\dots,k_N} (q)\). Defining functions \(\Theta_k^\alpha (q)\) depending on \(\alpha \in H^* (X)\) and \(k \geq 0\), they fix \(0 \leq k_1, \dots, k_N \in \mathbb Z\) and \(\alpha_1, \dots, \alpha_N \in H^* (X, \mathbb Q)\) and prove the following:
(1) If \(\langle K_X^2,\alpha_i \rangle =0\) and \(2|k_i\) for each \(i\), then the leading term \(\Pi_{i=1}^N \Theta_k^\alpha (q)\) of \(\overline F^{\alpha_1,\dots,\alpha_N}_{k_1,\dots,k_N} (q)\) is either \(0\) or a quasi-modular form of weight \(\sum (k_i+2)\).
(2) Suppose \(|\alpha_i|=4\) for each \(i\). If \(2|k_i\) for each \(i\), then \(\overline F^{\alpha_1,\dots,\alpha_N}_{k_1,\dots,k_N} (q)\) is a quasi-modular form of weight \(\sum (k_i+2)\). if \(2 \not |k_i\) for some \(i\), then \(\overline F^{\alpha_1,\dots,\alpha_N}_{k_1,\dots,k_N} (q)=0\).
These results are proved by relating the leading term of \(\overline F^{\alpha_1,\dots,\alpha_N}_{k_1,\dots,k_N} (q)\) for \(X\) to the leading term of \(\langle \text{ch}_{k_1} \dots \text{ch}_{k_N} \rangle^\prime\) for \(\mathbb C^2\) studied by Carlsson [loc. cit.]. Hilbert schemes of points on a surface; quasi-modular forms; multiple zeta value; generalized partition Parametrization (Chow and Hilbert schemes), Binomial coefficients; factorials; \(q\)-identities, Vertex operators; vertex operator algebras and related structures Hilbert schemes of points and quasi-modularity | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians By a result of \textit{B. Moonen} [Doc. Math. 15, 793--819 (2010; Zbl 1236.11056)], there are exactly 20 positive-dimensional families of cyclic covers of the projective line for which the Torelli image is open and dense in the associated Shimura variety. For each of these, we compute the Newton polygons, and the \(\mu \)-ordinary Ekedahl-Oort type, occurring in the characteristic \(p\) reduction of the Shimura variety. We prove that all but a few of the Newton polygons appear on the open Torelli locus. As an application, we produce multiple new examples of Newton polygons and Ekedahl-Oort types of Jacobians of smooth curves in characteristic \(p\). Under certain congruence conditions on \(p\), these include: the supersingular Newton polygon for genus 5, 6, 7; fourteen new non-supersingular Newton polygons for genus \(5-7\); eleven new Ekedahl-Oort types for genus \(4-7\) and, for all \(g \ge 6\), the Newton polygon with \(p\)-rank \(g-6\) with slopes \(1 / 6\) and \(5 / 6\). curve; cyclic cover; Jacobian; abelian variety; Shimura variety; PEL-type; moduli space; reduction; \(p\)-rank; supersingular; Newton polygon; \(p\)-divisible group; Kottwitz method; Dieudonné module; Ekedahl-Oort type Arithmetic aspects of modular and Shimura varieties, Curves over finite and local fields, Zeta functions and related questions in algebraic geometry (e.g., Birch-Swinnerton-Dyer conjecture), Modular and Shimura varieties, Families, moduli of curves (algebraic), Coverings of curves, fundamental group, Jacobians, Prym varieties Newton polygons arising from special families of cyclic covers of the projective line | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians The aim of this paper is to explain how some constructions on the cycle space (the Chow variety in the quasiprojective setting) allow one to pass from the \(n\)-convexity of a complex manifold \(Z\) to the 0-convexity of the topological space of compact \(n\)-cycles of \(Z\), \(C_n(Z),\) and from an \((n+ 1)\)-codimensional submanifold of \(Z\) having an ample normal bundle to a Cartier divisor of \(C_n(Z)\) having the same property. Constructions of the holomorphic, meromorphic and plurisubharmonic functions, and of the Kähler metric on \(C_n(Z)\) are considered. These tools are applied to the Hartshorne's conjecture. cycle space; Chow variety; \(n\)-convexity; ample normal bundle; topological space of compact \(n\)-cycles; analytic families of cycles; reduction of convexity; Cartier divisor; higher integration maps Barlet, D.: How to use the cycle cpace in complex geometry. Several Complex Variables MSRI Publications, vol. 37, pp. 25--42, 1999 Integration on analytic sets and spaces, currents, Divisors, linear systems, invertible sheaves, Foliations in differential topology; geometric theory How to use the cycle space in complex geometry | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians The paper is concerned with curves of genus two defined over a field of characteristic p\(>0\) whose Jacobian variety is a product of two supersingular elliptic curves. The authors obtain explicit formulas (depending, of course, on p) for the number of such curves with prescribed automorphism group. They first revise the theory of curves of genus two, their automorphisms, and criteria for them to be supersingular (in terms of the Hasse-Witt-matrix). Since the endomorphism ring of a supersingular elliptic curve is a definite quaternionic algebra it becomes possible to apply results on class numbers of quaternionic algebras to count supersingular curves. polarization; characteristic p; Jacobian variety; supersingular elliptic curves; prescribed automorphism group; curves of genus two; Hasse-Witt- matrix; class numbers of quaternionic algebras T.IBUKIYAMA, T.KATSURA, and F.OORT,\textit{Supersingular curves of genus two and class numbers}, Compositio Math. 57 (1986), no. 2, 127--152.MR827350 Jacobians, Prym varieties, Quaternion and other division algebras: arithmetic, zeta functions, Finite ground fields in algebraic geometry, Special algebraic curves and curves of low genus, Group actions on varieties or schemes (quotients), Families, moduli of curves (algebraic) Supersingular curves of genus two and class 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 It is known that any analytic subset of a real algebraic variety can be transformed into an algebraic subset by a smooth, but in general not analytical, diffeomorphism of the variety. The author substitutes smooth and analytic equivalences for almost analytic equivalence.
Main result: Let \(X\) be a compact affine smooth real algebraic variety, and \(V\) be a coherent closed analytic hypersurface in \(X\) with a finite set of singular points and \(\mathbb{Z}/2\)-homological to some algebraic subset in \(X\) (that is a necessary condition). Then there is a multiblow-up \(p:Y\to X\) along \(\hbox{Sing}(V)\) such that for any \(r\geq 0\) there exist a \(C^ r\)- diffeomorphism \(s_ r:X\to X\) and an analytic diffeomorphism \(t_ r:Y\to Y\) satisfying: (i) \(p\circ t_ r=s_ r\circ p\), (ii) \(s_ r(V)\) is algebraic. analytic subset of a real algebraic variety; diffeomorphism Real-analytic and semi-analytic sets Global almost analytic algebraicity of analytic sets | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is 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 note is devoted to the following result. Let \(f\) be an irreducible germ of a plane real curve, \(\mu\) its Milnor number, and \(B\) a Milnor ball for \(f\). Let \(\alpha\) and \(c\) be nonnegative integers such that \(\alpha+c\leq\mu/2\). Then there exists a deformation of \(f\) having \(\alpha\) ovals and \(c\) nondegenerate isolated double points in \(B\cup\mathbb{R}^2\), and without other singular points in \(B\). In the particular case \(\alpha=\mu/2\), the statement was proved by \textit{J.-J. Risler} [Invent. Math. 89, 119-137 (1987; Zbl 0672.14020)]. The proof of the result formulated above is obtained varying the parametrization of the germ. germ of a plane real curve; Milnor number Enumerative problems (combinatorial problems) in algebraic geometry, Topology of real algebraic varieties, Singularities of curves, local rings On the local Harnack's theorem | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians In this brief note the author constructs a certain filtration in the Galois group \(G_ K\) of the algebraic closure of an algebraic number field K as a pull-back of the filtration in the automorphism group Aut(G) where G is the maximal pro-\(\ell\)-quotient of the fundamental group of a smooth irreducible complete curve X of genus greater than 1. The pull- back is taken using the natural representation \({\tilde \rho}\): \(G_ K\to Aut(\pi_ 1(X\otimes_ K\bar K,\bar x))\) [cf. \textit{Y. Ihara}, Ann. Math., II. Ser. 123, 43-106 (1986; Zbl 0595.12003)]. The filtration on Aut(G) comes from the lower central series of G.
The result of the paper is that for any prime ideal of K not dividing \(\ell\) and modulo which X has stable reduction under certain conditions all ideals above it are not ramified in successive extensions of the tower of extensions of K given by the mentioned filtration of the Galois group \(G_ K\). This is an analog of the following result of Ihara (ibid.) in which a complete curve is replaced by the projective line without 3 points: Let \(X={\mathbb{P}}^ 1-\{0,1,\infty\}\) and \(\rho_{\ell}: Gal({\bar {\mathbb{Q}}}/{\mathbb{Q}})\to Out(\pi '_ 1(X\otimes_{{\mathbb{Q}}}{\bar {\mathbb{Q}}},*)\) be the representation of the Galois group into the outer automorphisms of pro-\(\ell\)-completion of the fundamental group of X. Then this representation is unramified outside of \(\ell\). filtration in the automorphism group; fundamental group of a smooth irreducible complete curve Oda, T.: A note on ramification of the Galois representation on the fundamental group of an algebraic curve. J. number theory 34, 225-228 (1990) Coverings of curves, fundamental group, Birational automorphisms, Cremona group and generalizations A note on ramification of the Galois representation on the fundamental group of an algebraic curve | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians The author gives a partial result concerning the real Jacobian conjecture in \(\mathbb{R}^{2}:\) if \(F=(f,g):\mathbb{R}^{2}\mathbb{\rightarrow R}^{2}\) is a polynomial map such that \(\det DF\) is nowhere zero, \(F(0,0)=(0,0),\) and the higher homogeneous terms of the polynomials \(ff_{x}+gg_{x}\) and \( ff_{y}+gg_{y}\) do not have real linear factors in common, then \(F\) is injective. real Jacobian conjecture; centre of a vector field; Hamiltonian vector field Jacobian problem, Linear first-order PDEs A sufficient condition in order that the real Jacobian conjecture in \(\mathbb{R}^2\) holds | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is 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 the canonical model of a Shimura variety associated to a reductive group \(G\). Let \(j: S\hookrightarrow S^*\) denote the open embedding into the canonical model of the Baily-Borel compactification. Let \(i:S_ 1\hookrightarrow S^*\) be the locally closed embedding of a boundary stratum; here \(S_1\) is the canonical model of a Shimura variety associated to another, explicitly given, reductive group \(G_1\). (This has been described in the author's dissertation [``Arithmetical compactification of mixed Shimura varieties'', Bonn. Math. Schr. 209 (1989; see the preceding review Zbl 0748.14007)].
An algebraic representation of \(G\) on a \(\mathbb{Q}_\ell\)-vector space \(V\) determines a smooth \(\ell\)-adic sheaf \({\mathcal V}\) on \(S\). The main result of the present article is a description of the \(\ell\)-adic sheaves \(i^*R^nj_*{\mathcal V}\) on \(S_1\). A simple group cohomological formula yields an algebraic representation of \(G_1\) and hence an \(\ell \)-adic sheaf on \(S_1\): this is canonically isomorphic to \(i^*R^nj_*{\mathcal V}\). The analogous statement in the analytic category is well-known and much easier to prove. canonical model of a Shimura variety; Baily-Borel compactification; \(\ell\)-adic sheaf R. Pink, On \textit{l}-adic sheaves on Shimura varieties and their higher direct images in the Baily-Borel compactification, Math. Ann. 292 (1992), no. 2, 197-240. Modular and Shimura varieties, Arithmetic aspects of modular and Shimura varieties, \(p\)-adic cohomology, crystalline cohomology, Cohomology of arithmetic groups On \(\ell\)-adic sheaves on Shimura varieties and their higher direct images in the Baily-Borel compactification | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians Given a family of elliptic curves \(E_ t\) and a 1-form \(\omega\) on \(E_ t\), the author gives an upper bound for the number of zeroes of a period \(I_{\omega}(t)\) of \(\omega\), on certain domains of the parameter, in terms of the degree of the polar divisor of \(\omega\). The proof uses the fact that \(I_{\omega}(t)\) satisfies a Picard-Fuchs equation, and applies to the associated Riccati equation the method developed by \textit{A. G. Khovanskij} [Sib. Mat. Zh. 25, No.3, 198-203 (1984) and Funkts. Anal. Prilozh. 18, No.2, 40-50 (1984)]. real zeroes; bound for the number of zeroes of a period; Picard-Fuchs equation G. S. Petrov, ''Number of zeros of complete elliptic integrals,'' Funkts. Anal.,18, No. 2, 73-74 (1984). Structure of families (Picard-Lefschetz, monodromy, etc.), Elliptic functions and integrals, Period matrices, variation of Hodge structure; degenerations On the number of zeroes of complete elliptic integrals | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians Let \(G\) be a semisimple group, \(B\subset G\) be a Borel subgroup, \({\mathcal B} =G/B\) be a generalized flag space (= a variety of Borel subgroups). We shall denote by \(X= X(G) \subset {\mathcal B}\) the closure of a general orbit of the maximal torus \(T \subset G\). In this paper, we are interested in the construction of the toric variety \(X\) and in the restriction homomorphism \(i^*: H^* ({\mathcal B}, \mathbb{Z}) \to H^* (X,\mathbb{Z})\). The computation of the mapping \(i^*\) in itself contains the problem of decomposition of \(X\) into Schubert cycles. It turns out that as a toric variety, \(X\) is generated by a fan of Weil chambers of the root system \(R^v\) dual to \(R(G)\). The image of the restriction homomorphism \(i^*\) coincides with the algebra of invariants \(H^* (X, \mathbb{Q})^W\) of the Weil group. This algebra has dimension \(2^r\), \(r= \text{rk }G\), and is arranged as follows. Denote by \({\mathcal L}_\omega\) a linear bundle on \({\mathcal B} =G/B\) induced by a character \(\omega: B\to\mathbb{C}^*\), and set \([\omega] =c_1 ({\mathcal L}_\omega \mid X)\). Then the algebra \(H^* (X,\mathbb{Q})^W\) is generated by generators \([\omega_i]\) and by quadratic relations \([\omega_i] \cdot [\alpha_i] =0\), \(i=1, \dots, r\), where \(\omega_i\) and \(\alpha_i\) are fundamental weights and simple roots. In these notations, the restriction of the Schubert cocycle \(P_w\), which is dual to \(S_w= \overline {BwB/B} \subset {\mathcal B}\), to the closure of the torus orbit is given by the formula
\[
[P_w] ={1\over m!} \sum [\omega_{\alpha_1}] [\omega_{\alpha_2}] \dots [\omega_{\alpha_m}],
\]
where the sum is taken over all reduced decompositions \(w= s_{\alpha_1} s_{\alpha_2} \dots s_{\alpha_m}\).
In the last section, a specialization of these theorems for the Grassmannian \(G^q_p\) is considered. This connects our results with the classical computational geometry. Here is a typical example:
Let \(X\subset \mathbb{P}^{d+1}\) be a general hypersurface of degree \(d\), and \(F(X)\) be the Fano scheme of lines situated on \(X\). Then the rational mapping \(F(X) \to\text{Conf}_{d+2} (\mathbb{P}^2)\) which assigns to each line \(l\subset X\) the arrangement of points cutted out on it by the coordinate hypersurfaces has degree \(d^{d+1}\).
Part of results of this paper was announced earlier [see \textit{A. A. Klyachko}, Funct. Anal. Appl. 19, 65-66 (1985); translation from Funkts. Anal. Prilozh. 19, No. 1, 77-78 (1985; Zbl 0581.14038)]. generalized flag space; variety of Borelian subgroups; Schubert cycles; toric variety; algebra of invariants; arrangement of points Klyachko, A.A.: Toric varieties and flag varieties. Tr. Mat. Inst. Steklova 208, 139--162 (1995) (in Russian), English transl.: Proc. Steklov Inst. Math. 208 (1995), 124--145 Toric varieties, Newton polyhedra, Okounkov bodies, Grassmannians, Schubert varieties, flag manifolds, Enumerative problems (combinatorial problems) in algebraic geometry Toric varieties and flag 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 Following the method of V. D. Goppa, we show explicitly, in the case of genus one, how to construct the parity-check matrix and the generator matrix for an elliptic code over a field of characteristic 2. elliptic curve; differential form; genus one; parity-check matrix; generator matrix; elliptic code over a field of characteristic 2 Linear codes (general theory), Combinatorial codes, Arithmetic ground fields for curves, Special algebraic curves and curves of low genus Un exemple de codes géométriques: les codes elliptiques. (An example of geometric codes: the elliptic codes) | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians See the preview in Zbl 0535.14015. hyperplane section; genus; moduli; semicontinuity; maximal rank conjecture; embedding of a general smooth curve; non special embeddings in projective 4-space Ballico E., Ellia P.: On postulation of curves in \$\$\{\(\backslash\)mathbb\{P\}\^4\}\$\$ . Math. Z. 188, 215--223 (1985) Families, moduli of curves (algebraic), Projective techniques in algebraic geometry, Embeddings in algebraic geometry, Algebraic moduli problems, moduli of vector bundles On postulation of curves in \({\mathbb{P}}^ 4\) | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians This paper studies some remarkable properties of abelian varieties arising from two classical objects in projective geometry. The first one is a Weddle surface: given a set B of 6 points in general position in \({\mathbb{P}}^ 3\), all singular points on (singular) quadrics through the points of B form a surface \(W_ B\). The other object is a Humbert curve: given \(B=\{p,p_ 1,...,p_ 5\}\) in \({\mathbb{P}}^ 3\) (6 points with one marked) this is the locus of points q such that the line pq is tangent to a twisted cubic through B. The two objects are intimately related, e.g. the projection of \(W_ B\) from the point p down to \({\mathbb{P}}^ 2\) has exactly the Humbert curve of B as branchlocus.
A Humbert curve \(X_ B\) has genus 5; on it the divisor D given by \(D=K- 3p-p_ 1-...-p_ 5\) (with K a canonical divisor) yields a nontrivial point of order 2 in \(Pic^ 0(X_ B)\). Associated to such a pair \(X_ B,D\) is an abelian variety of dimension 4, \(P(X_ B,D)\), the Prym variety of \((X_ B,D)\). A Weddle surface also gives rise to a 4- dimensional abelian variety: take the double cover of \({\mathbb{P}}^ 3\) branched along \(W_ B\). Blowing up the points over the set B gives a smooth 3-fold V; \textit{C. H. Clemens} [Adv. Math. 47, 107-230 (1983; Zbl 0509.14045)] has shown that its intermediate Jacobian J(V) is an abelian variety, and since \({\#}B=6\), \(\dim J(V)=10-6=4\). It follows from a theorem of \textit{A. Beauville} [Ann. Sci. Éc. Norm. Supér., IV. Sér. 10, 309-391 (1977; Zbl 0368.14018)] that \(P(X_ B,D)\) is isomorphic to J(V).
The author proves by comparing moduli of 6 points in \({\mathbb{P}}^ 3\) to moduli of Prym varieties and of abelian varieties that all sets B yield the same (upto isomorphism) abelian variety. His second result is a very strange one: a (symmetric) theta divisor on \(P(X_ B,D)\) has 10 singular points, which is the maximal number of singularities possible in the genus 5 case. (A general Prym variety of dimension 4 has a smooth theta divisor.) intermediate Jacobian; vanishing theta null; Weddle surface; Humbert curve; moduli of Prym varieties; theta divisor Varley, Robert, Weddle's surfaces, Humbert's curves, and a certain \(4\)-dimensional abelian variety, Amer. J. Math., 108, 4, 931-951, (1986) Algebraic moduli of abelian varieties, classification, \(4\)-folds, Special algebraic curves and curves of low genus, Special surfaces, Theta functions and abelian varieties Weddle's surfaces, Humbert's curves, and a certain 4-dimensional abelian variety | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians We study the functional codes of order \(h\) defined by G. Lachaud on a non-degenerate Hermitian variety, by exhibiting a result on divisibility for all the weights of such codes. In the case where the functional code is defined by evaluating quadratic functions on the non-degenerate Hermitian surface, we list the first five weights, describe the geometrical structure of the corresponding quadrics and give a positive answer to a conjecture formulated on this question by \textit{F. A. B. Edoukou} [J. Théor. Nombres Bordx. 21, No. 1, 131--143 (2009; Zbl 1183.94060)]. The paper ends with two conjectures. The first is about the divisibility of the weights in the functional codes. The second is about the minimum distance and the distribution of the codewords of the first \(2h+1\) weights. divisor of a code; functional codes; Hermitian surface; Hermitian variety; weights of codes Edoukou, F. A. B.; Ling, S.; Xing, C.: Structure of functional codes defined on non-degenerate Hermitian varieties, J. comb. Theory, ser. A 118, 2436-2444 (2011) Geometric methods (including applications of algebraic geometry) applied to coding theory, Applications to coding theory and cryptography of arithmetic geometry Structure of functional codes defined on non-degenerate Hermitian 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 [For the entire collection see Zbl 0624.00007.]
This note is a short summary of the author's thesis [Ouverts analytiques d'une courbe projective sur un corps valué complet ultramťrique alg\'briquement clos, Thèse Bordeaux 1987, see Ann. Inst. Fourier 37, No. 3, 39--66 (1987; Zbl 0616.14025)]. The results concern the genus of a one dimensional rigid analytic space and its embedding into a projective curve. genus of one-dimensional rigid analytic space; embedding into a projective curve Local ground fields in algebraic geometry, Arithmetic ground fields for curves, Non-Archimedean function theory Analytic open sets of an algebraic curve in rigid geometry | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians See the preview in Zbl 0566.58028. bigonal construction; double cover of a hyperelliptic curve; Prym; varieties; geodesic flow; abelian varieties S. Pantazis, ''Prym varieties and the geodesic flow on SO(n),'' Math. Ann.,273, No. 2, 297-315 (1986). Dynamical systems of geometric origin and hyperbolicity (geodesic and horocycle flows, etc.), Geodesic flows in symplectic geometry and contact geometry, Abelian varieties and schemes, Coverings of curves, fundamental group, Dynamics induced by flows and semiflows Prym varieties and the geodesic flow on \(SO(n)\) | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians Let \(f: X\to C\) be a double covering of smooth projective curves. Let \(P\in C\) be a ramification point of \(f\) with \(f(P')=P\). Let \(H(P)\) and \(H(P')\) be the Weierstrass semigroup of \(P\) and \(P'\), respectively. The authors pose the problem of the clasification of all \(H(P')\) for all smooth plane curves. They solved it here when \(C\) has degree \(6\). They did the case \(\deg (C)=5\) in [\textit{S. J. Kim} and \textit{J. Komeda}, Kodai Math. J. 38, No. 2, 270--288 (2015; Zbl 1327.14159)] and did some other cases in [\textit{S. J. Kim} and \textit{J. Komeda}, Bull. Korean Math. Soc. 55, No. 2, 611--624 (2018; Zbl 1401.14158)]. numerical semigroup; Weierstrass semigroup of a point; double cover of a curve; plane curve of degree 6 Riemann surfaces; Weierstrass points; gap sequences, Plane and space curves, Coverings of curves, fundamental group, Commutative semigroups Double covers of plane curves of degree six with almost total flexes | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is 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 algebraic variety, E and F vector bundles over X of ranks e and f, respectively. For a bundle map \(\phi:\quad E\to F,\) let I(k) be the k-singular locus of \(\phi\), i.e. \(I(k):=\{x\in X;\) rank\((\phi)_ x\leq k\}.\) The paper gives the fundamental class of I(k) as polynomial in the Chern classes of E and F in the case cod\(_ XI(k)=(e-k)(f-k)\) or \(I(k)=\emptyset\); cf. \textit{J. Harris} and \textit{L. Lu} for a similar result [Invent. Math. 75, 467-475 (1984; Zbl 0542.14015)]. As an application, the genus of the ''divisor singular loci'' of dimension r and degree d of a general curve of genus g is computed in the case \(g-(r+1)(g-d+r)=1\). If \(r=1\), \(d=m-2\), and \(g=2m+1\), that has been done by \textit{G. R. Kempf} [Compos. Math. 55, 157-162 (1985)]. curves of special divisors; Schubert variety; bundle map; Chern classes G. P. Pirola, Chern character of degeneracy loci and curves of special divisors. \textit{Ann. Mat. Pura Appl. (4)}\textbf{142} (1985), 77-90 (1986). Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20], Characteristic classes and numbers in differential topology, Divisors, linear systems, invertible sheaves, Determinantal varieties, Families, moduli of curves (algebraic), Low codimension problems in algebraic geometry Chern character of degeneracy loci and curves of special 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 A complete linear series \(g^ r_ d\) on a smooth curve \(C\) is said to be primitive if both it and its dual \(| K_ C-g^ r_ d|\) are base-point free. The authors define the Clifford sequence \(S(C)\) of \(C\) as the set of Clifford indices (defined as \(d-2r)\) of all nontrivial primitive series on \(C\). For example, the Clifford sequence for the general curve of genus \(g\) is \(S(C)=\{c,c+1,\dots,g-3\}\) (where \(c=[{g- 1\over 2}])\), while for a hyperelliptic curve it is \(S(C)=\{0\}\). The primitive length of \(C\) is defined to be the cardinality \(\ell(C)\) of \(S(C)\). The following result about the primitive length are proved:
For sufficiently high genus, a characterization for curves with \(\ell\leq 3\) is given. More precisely, they are hyperelliptic (if \(\ell=1\) and \(g\leq 7)\) or bielliptic (if \(\ell=2\) and \(g\geq 11)\), while there are not curves with \(\ell=3\) and \(g\geq 19\).
For any fixed positive integer \(g'\) the primitive length of genus \(g\) curves that admit a double cover onto a curve of genus \(g'\) is bounded. Given a positive integer \(\ell\), the genus of curves that are not a double cover and have primitive length \(\ell\) is bounded.
In the course of the proof, the authors also obtain some results about the dimension of certain varieties \(W^ r_ d\) of special divisors. For instance, H. Marten's theorem is refined. primitive length of a smooth curve; complete linear series; Clifford sequence 3. Coppens, M., Keem, C., Martens, G.: Primitive linear series on curves. Manuscripta Math. \textbf{77}, 237-264 (1992)CKM Special algebraic curves and curves of low genus, Divisors, linear systems, invertible sheaves Primitive linear series 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 The study of toric varieties is a highly interesting part of algebraic geometry, having deep connections with polytopes, polyhedra, combinatorics, commutative algebra, symplectic geometry and topology while being of unexpected applications in such diverse areas as physics, coding theory, algebraic statistics and geometric modeling. The concreteness of toric varieties enables one to grasp the meaning of the powerful techniques of modern algebraic geometry firmly, providing a fertile testing ground for general theories.
This book is a good introduction to this rich subject, providing more details with numerous examples, figures and exercises than existing books [\textit{G. Ewald}, Combinatorial convexity and algebraic geometry. Graduate Texts in Mathematics. 168. New York, NY: Springer (1996; Zbl 0869.52001); \textit{W. Fulton}, Introduction to toric varieties. Annals of Mathematics Studies. 131. Princeton, NJ: Princeton University Press (1993; Zbl 0813.14039); \textit{T. Oda}, Convex bodies and algebraic geometry. An introduction to the theory of toric varieties. Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge, Bd. 15. Berlin etc.: Springer-Verlag (1988; Zbl 0628.52002)].
The book, consisting of 15 chapters, begins with affine toric varieties in Chapter 1 and projective toric varieties in Chapter 2. Chapter 3 is concerned with normal varieties, though the definition of variety does not assume normalcy. Chapter 4 considers Weil divisors and Cartier divisors, which coincide on a smooth variety, but whose relationship is more complicated for a normal variety. It is shown that normal varieties are the natural setting to develop a theory of divisors and divisor classes. Chapter 5 demonstrates that the classical construction \(\mathbb{P}^{n}\)\ can be generalized to any toric vaiety \(X_{\Sigma}\). Chapter 6 relates Cartier divisors to invertible sheaves on \(X_{\Sigma}\). The structure of the nef cone and its dual called the Mori cone is described in detail. In Chapter 7 the authors extend the relation between polytopes and projective toric varieties to that between polyhedra and projective toric morphisms \(\phi:X_{\Sigma}\rightarrow U_{\Sigma}\). Projective bundles over a toric variety are discussed, so that smooth projective toric varieties of Picard number \(2\)\ are classified. In Chapter 8 Weil divisors are related to reflexive sheaves of rank one, where Zariski \(p\)-forms are defined. Chapter 9 is devoted to sheaf cohomology. Chapter 10 is concerned with the structure of \(2\)-dimensional normal toric varieties (toric surfaces). Their singularities are described, and smooth complete toric surfaces are classified. Chapter 11 establishes the existence of toric resolutions of singularities for toric varieties of all dimensions. The goal in Chapter 12 is to understand some topological invariants of a toric variety \(X\) with applications to polytopes. Chapter 13 proves the Hirzebruch-Riemann-Roch theorem for a line bundle \(\mathcal{O}_{X}(D)\) on a smooth complete toric variety \(X_{\Sigma}\). Chapters 14 and 15 study the GIT (Geometric Invariant Theory) quotients \(\mathbb{C} ^{r}//_{\chi}G\) as \(\chi\in\widehat{G}\) varies. The full story of what happens as \(\chi\) varies is controlled by the secondary fan, which is the main topic of the last section of Chapter 14. The aim in Chapter 15 is to understand the structure of the GKZ (Gel'fand, Kapranov and Zelevinsky) cones and what happens to the associated toric varieties as one moves around the secondary fan. The book is accompanied by three appendices on the history of toric varieties, computational methods and spectral sequences. toric variety; Weil divisor; Cartier divisor; convexity; Picard number; polytope; polyhedron; sheaf cohomology; Hizrebruch-Jung continued fraction; Gröbner fan; McKay correspondence; Rees algebra; multiplier ideal; Hirzebruch-Riemann-Roch theorem; Chow ring; intersection cohomology; invariant theory; GKZ cone; secondary fan; spectral sequence D. A. Cox, J. B. Little, and H. K. Schenck, \textit{Toric varieties}, Graduate Studies in Mathematics, 124, American Mathematical Society, Providence, RI, 2011.Zbl 1223.14001 MR 2810322 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to algebraic geometry, Toric varieties, Newton polyhedra, Okounkov bodies, Intersection theory, characteristic classes, intersection multiplicities in algebraic geometry, (Equivariant) Chow groups and rings; motives Toric varieties | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians [For the entire collection see Zbl 0683.00010.]
This is an exposition of the results of the author on the link at infinity of complex affine plane curves. For example the author proves that in many cases the topology of a complex curve \(V\subset {\mathbb{C}}^ 2\) is determined by its link at infinity. link at infinity; topology of a complex curve Curves in algebraic geometry, Topological properties in algebraic geometry On the topology of curves in complex surfaces | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians We compute the minimum non-zero norm of vectors in the lattice given by the Jacobian of the curve \(y^3= x^4-1\). The result shows that the Klein curve is not extremal for the Jacobian systole problem in genus 3. Picard curve; Fermat curve; Bolza curve; Jacobian; Klein curve; systole J.R. Quine, Jacobian of the Picard curve, in \(Extremal Riemann Surfaces (San Francisco, CA, 1995)\). Contemporary Mathematics, vol. 201 (American Mathematical Society, Providence, RI, 1997), pp. 33-41 Compact Riemann surfaces and uniformization, Special algebraic curves and curves of low genus, Lattices and convex bodies in \(n\) dimensions (aspects of discrete geometry) Jacobian of the Picard curve | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians The classification of the singularities lying on a normal surface in \(P^ 3\) of small degree is a subject with classical roots. For instance one knows all possible configurations of singularities lying on a cubic surface. In this paper the author classifies completely the singularities of a normal quartic surface (resp. of a sextic curve in \(P^ 2)\) having one of the following prescribed singularities: a simple elliptic singularity of type \(\tilde E_ 7\) or \(\tilde E_ 6\), a cusp singularity of type \(T_{2,4,5}\) or \(T_{3,3,4}\), a unimodular exceptional singularity of type \(Z_{11}\) or \(Q_{10}\) (resp. \(\tilde E_ 7\), \(T_{2,4,5}\), or \(Z_{11})\). In an earlier paper of the same author the classification of the singularities of normal quartic surfaces and of sextic curves with one of the singularities \(\tilde E_ 8\), \(T_{2,3,7}\) or \(E_{12}\) was given [see the author, ''On quartic surfaces and sextic curves with singularities of type \(\tilde E_ 8\), \(T_{2,3,7}\), \(E_{12}\)'' (Preprint 1983)]. Finally, we note that this paper gives no details concerning the proofs of the results. singularities of a normal quartic surface; sextic curve; simple elliptic singularity; cusp singularity; unimodular exceptional singularity , On quartic surfaces and sextic curves with certain singularities, Proc. Japan Acad., 59, Ser. A, (1983) 434-437. Singularities of surfaces or higher-dimensional varieties, Singularities of curves, local rings, Singularities in algebraic geometry On quartic surfaces and sextic curves with certain singularities | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians The paper states some nonemptiness and connectedness results on the moduli spaces of spinor bundles over a real algebraic curve. Relations with the moduli spaces of real algebraic curves are also discussed. nonemptiness; connectedness; moduli spaces of spinor bundles over a real algebraic curve S. M. Natanzon, ?Spinor bundles over real algebraic curves,? Usp. Mat. Nauk,44, No. 3, 165-166 (1989). Families, moduli of curves (algebraic), Real algebraic and real-analytic geometry, Spin and Spin\({}^c\) geometry, Topological properties in algebraic geometry Spinor bundles over real algebraic curves | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians The paper, written by two distinguished model theorists, arose from certain model-theoretic problems and uses a hard machinery borrowed from stability theory. On the other hand, the paper deals with classical objects of algebraic geometry, and forms an important and profound contribution to the understanding of those objects. The goal of the paper is to characterize the Zariski topologies over an algebraically closed field in terms of general dimension-theoretic properties. The authors define a Zariski geometry to be a set \(X\), together with a collection of compatible topologies of \(X^n\) for each \(n\), such that a dimension can be assigned to be closed sets, satisfying certain natural conditions. Any smooth algebraic variety is then a Zariski geometry, as is any compact complex manifold if the closed subsets of \(X^n\) are taken to be closed holomorphic subvarieties. However, in the paper only one-dimensional Zariski geometries are considered. If \(X\) arises from an algebraic curve, there exists a family of plane curves \({\mathcal C}\) over \(X\) such that (i) for any generic pair of points, there is a curve in \({\mathcal C}\) passing through both points, and (ii) for any two points, there is a curve in \({\mathcal C}\) separating them. An abstract Zariski geometry \(X\) with this property is called very ample; if there is \({\mathcal C}\) with (i) only, \(X\) is called ample. The main result of the paper is that for any very ample Zariski geometry \(X\) there is a smooth curve \(C\) over an algebraically closed field \(F\) such that \(X\) and \(C\) are isomorphic as Zariski geometries. Moreover, \(F\) and \(C\) are unique, up to a field isomorphism and an isomorphism of curves over \(F\). Nonample Zariski geometries are well understood: in the paper of \textit{E. Hrushovski} and \textit{J. Loveys} [``Locally modular strongly minimal sets'' (to appear)]\ it is shown that such geometries are in a sense degenerate or else closely related to the linear geometries, where \(X\) is an arbitrary skew field and the closed subsets of \(X^n\) are given by linear equations. In the paper under review, ample but not very ample Zariski geometries are also analyzed in detail. Such a geometry is shown to be in a certain sense a finite cover of the projective line over an algebraically closed field. However, no analog of the Riemann existence theorem is valid here; there exist finite covers of the projective line which do not arise from algebraic curves. An excellent brief exposition of the ideas and results of the paper is given in the authors' research announcement [Bull. Am. Math. Soc., New Ser. 28, 315-323 (1993; Zbl 0781.03023)]. stability theory; Zariski topologies over an algebraically closed field; Zariski geometry; dimension; smooth algebraic variety; algebraic curve; ample; finite covers of the projective line Hrushovski E. and Zilber B., Zariski geometries, J. Amer. Math. Soc. 9 (1996), 1-56. Models of other mathematical theories, Foundations of algebraic geometry, Other classical first-order model theory, Classification theory, stability, and related concepts in model theory, Rational and birational maps Zariski geometries | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is 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 \(d\) be a positive integer and \(N\) a prime such that \(N > (60d-24)\). Let \(J\) be the Eisenstein quotient of the Jacobian \(J_ 0(N)\) of the modular curve \(X_ 0(N)\) [\textit{B. Mazur}, Publ. Math., Inst. Hautes Étud. Sci. 47(1977), 33--186 (1978; Zbl 0394.14008)]. Assuming that \(N > (1 + \sqrt {pd})^ 2\) and there exist \(d\) weight-two cusp forms \(f_ 1, \dots, f_ d\) attached to \(J\) satisfying the linear independence conditions mod \(P\), the author proves that there does not exist any elliptic curve with a point of order \(N\) rational over any field of degree \(d\).
Using this result and ideas of Mazur (as well as results of Manin, Kolyvasin-Logachev, Faltings and Frey), \textit{L. Merel} has recently proved that for each \(d\) there exists a positive integer \(B(d)\) such that for each elliptic curve \(E\) over a number field \(K\) of degree \(d\) each torsion point in \(E(K)\) has order \(\leq B(d)\). In other words, he has solved the famous boundedness torsion problem for elliptic curves. torsion of elliptic curves; Jacobian; modular curve Kamienny, S., \textit{torsion points on elliptic curves over fields of higher degree}, Int. Math. Res. Not. IMRN, 6, 129-133, (1992) Elliptic curves, Arithmetic ground fields for curves, Arithmetic ground fields for abelian varieties, Rational points, Elliptic curves over global fields Torsion points on elliptic curves over fields of higher 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 origin of what is currently meant by the notion of Hodge theory can be traced back to \textit{W. V. D. Hodge}'s fundamental work accomplished in the 1930s. In modern terminology, Hodge prepared the ground for describing the De Rham cohomology algebra of a Riemannian manifold in terms of its harmonic differential forms. In the following two decades, Hodge's decomposition principle has been extended to the (then) new sheaf-theoretic and cohomological framework of Hermitean differential geometry, complex-analytic geometry, and transcendental algebraic geometry. The names of G. De Rham, A. Weil, K. Kodaira, and many others stand for the tremendous progress achieved during this period, in particular with regard to deformation and classification theory in these areas. The special algebraic structures (Hodge structures) arising from Hodge decompositions and their generalizations have led to a rather independent field of research in geometry, precisely to the so-called Hodge theory, which represents a powerful and indispensible toolkit for contemporary complex geometry, general algebraic geometry, and -- nowadays -- also for mathematical physics. The vast activity in Hodge theory and its related areas, especially during the recent twenty years, is not reflected in the current textbook literature, at least not comprehensively or in an updated form compiling the various recent aspects and applications, so that a panoramic overview of the present state of art must be regarded as a highly welcome (and needed) service to the mathematical community.
A conference on the present state of Hodge theory, serving exactly that purpose, took place at the University of Grenoble (France) in November 1994. The book under review grew out of the series of lectures which the authors delivered at this meeting. The aim of the text is to develop a number of fundamental concepts and results of classical and modern Hodge theory, and in this the book is prepared for students and non-expert researchers in the field, who wish to get acquainted in depth with the subject, and obtain a profound up-to-date knowledge of its present level of development. -- The material is divided into three main parts, each of which is written by different authors and devoted to various central and complementary aspects of the theory.
Part I, written by \textit{J.-P. Demailly}, is entitled ``\(L^2\)-Hodge theory and vanishing theorems''. The author discusses in detail two fundamental applications of Hilbert \(L^2\)-space methods to complex analysis and algebraic geometry, respectively. This part adopts basically the analytic viewpoint and consists, on its side, of two chapters. Chapter 1 provides an introduction to standard complex Hodge theory, including the basics on Hermitean and Kähler geometry, differential operators on vector bundles, Hodge decomposition, Hodge degeneration, the spectral sequence of Hodge-Frölicher, Gauss-Manin connexion, and the deformation behavior of the Hodge groups (after Kodaira). Chapter 2 is devoted to \(L^2\)-estimates for the \(\overline \partial \)-operator and the resulting vanishing theorems for cohomology groups of Kähler manifolds and projective varieties. The main topics here are the classical methods of Oka, Bochner, and Hörmander in pseudo-convex analysis, their consequences for cohomology vanishing, as well as the more recent but already well-known fundamental contributions by the author himself towards the interpretation of the great vanishing theorems of A. Nadel and of Kawamata-Viehweg. -- The concluding two sections of this chapter deal with the property of very-ampleness of line bundles on projective varieties. The first central result discussed here is the author's analytic approach to the famous conjecture of Fujita, culminating in an improvement of \textit{Y.-T. Siu}'s very recent theorem on an effective bound for very-ampleness [cf. ``Effective very ampleness'', Invent. Math. 124, No. 1-3, 563-571 (1996)]. The second central result is an effective version of the classical ``Big embedding theorem of Matsusaka'', whose surprisingly simple proof is due to the author himself (1996), based on some foregoing work of \textit{Y.-T. Siu} [Ann. Inst. Fourier 43, No. 5, 1387-1405 (1993; Zbl 0803.32017)], and methodically related to the effective bound for very-ampleness discussed before. These two last sections provide a particularly up-to-data account on the newest developments in analytical Hodge theory and its (algebraic) applications.
Part II of the text, written by \textit{L. Illusie}, is entitled ``Frobenius and Hodge degeneration''. These notes aim at introducing non-specialists to those methods and techniques of algebraic geometry over a field of characteristic \(p > 0\), which have been used by P. Deligne and the author to give an algebraic proof of the Hodge degeneration and the Kodaira-Akizuki-Nakano vanishing theorem for smooth projective varieties in characteristic zero. Basically, this part of the book is a careful, detailed introduction to the important work ``Relèvements modulo \(p^2\) et décomposition du complexe de De Rham'' [Invent. Math. 89, 247-270 (1987; Zbl 0632.14017)] by \textit{P. Deligne} and \textit{L. Illusie}. Here the reader is assumed to bring along some basic knowledge of the theory of algebraic schemes and of homological algebra (in categories). After recalling the basics on schemes, differentials and the algebraic De Rham complex in characteristic \(p>0\), the author discusses the following topics: smoothness and coverings, the Frobenius morphism and the Cartier isomorphism, derived categories and spectral sequences, decomposition theorems, vanishing theorems in characteristic \(p\), degeneration theorems, the standard techniques for passing from characteristic \(p\) to characteristic zero, and the proof of the above mentioned degeneration and vanishing theorems. The concluding section of this part points to some recent developments and open problems concerning Hodge theory in characteristic \(p\).
Also this part is essentially self-contained, and most proofs are given in detail. Some proofs are -- quite naturally -- at least outlined, assuming the reader to follow the precise hints to the related textbook literature (mostly EGA) and original papers.
Part III of the book, written by \textit{J. Bertin} and \textit{C. Peters}, is entitled ``Variations of Hodge structures, Calabi-Yau manifolds, and mirror symmetry''. It consists again of two main chapters, whose interrelation is beautifully explained in a comprehensive introduction. -- Chapter I is devoted to the comparatively elementary part of the theory of variation of Hodge structures and its applications in complex algebraic geometry. This includes detailed descriptions of the Hodge bundles, the Hodge filtrations, the De Rham cohomology sheaves, the Gauss-Manin connexion in its general setting (after Katz and Oda) and with its transversality property (due to Griffiths), variations and infinitesimal variations of Hodge structures, the Griffiths period domains for polarized Hodge structures, mixed Hodge structures, limits of Hodge structures (after Deligne), the Picard-Lefschetz theory and the local monodromy theorem, Deligne's degeneration criteria for Hodge spectral sequences, and a brief discussion of the method of vanishing cycles. At the end, the authors give a sketch of the use of Higgs bundles for the construction of variations of Hodge structures, mainly by following Simpson's approach [cf.: \textit{C. T. Simpson}, Proc. Int. Congr. Math., Kyoto 1990, Vol. I, 747-756 (1991; Zbl 0765.14005)], as well as some comments on M. Saito's work on Hodge modules, intersection cohomology, and \({\mathcal D}\)-modules in algebraic analysis. -- Chapter II reflects the fact that Calabi-Yau manifolds, their Hodge theory, and their mirror symmetry have recently gained enormous significance in both algebraic geometry and theoretical physics, particularly in constructing two-dimensional conformal quantum field theories. The material presented here covers the fundamental facts on Calabi-Yau manifolds, their construction and deformation theory, and their mirror properties. After a digression on the cohomology of hypersurfaces (after Griffiths and Dimca), which is used for the description of the link between the Picard-Fuchs equation and the variation of Calabi-Yau structures, the variation of Hodge structures for families of Calabi-Yau threefolds, their Yukawa couplings, and their mirror symmetries are explained in more depth. The interested reader can find a very complete and comprehensive account on this subject in the recent monography ``Symétrie miroir'' by \textit{C. Voisin} [Panoramas et Synthèses, No. 2 (1996; see the preceding review)]. In a concluding section, the authors discuss (following an idea of P. Deligne) a possible approach to mirror symmetry via a certain duality between variations of Hodge structures for Calabi-Yau threefolds. A rich bibliography enhances this very systematic and lucid treatise.
Altogether, the present book, in all its three parts, which consistently refer to each other, may be regarded as a masterly introduction to Hodge theory in its classical and very recent, analytic and algebraic aspects. Aimed to students and non-specialists, it is by far much more than only an introduction to the subject. The material leads the reader to the forefront of research in many areas related to Hodge theory, and that in a detailed and highly self-contained manner. As such, this text is also a valuable source for active researchers and teachers in the field, in particular due to the utmost carefully arranged index at the end of the book. characteristic \(p\); De Rham cohomology algebra; Hodge theory; vanishing theorems; very-ampleness of line bundles; Hodge degeneration; De Rham complex; Frobenius morphism; Cartier isomorphism; variation of Hodge structures; Gauss-Manin connexion; period domains; Picard-Lefschetz theory; Higgs bundles; Calabi-Yau manifolds; two-dimensional conformal quantum field theories; Picard-Fuchs equation; Yukawa couplings; mirror symmetries J. BERTIN, J.-P. DEMAILLY, L. ILLUSIE, C. PETERS. Introduction à la théorie de Hodge. Panorama et synthèses, publications SMF (1996). Transcendental methods, Hodge theory (algebro-geometric aspects), Vanishing theorems in algebraic geometry, Research exposition (monographs, survey articles) pertaining to algebraic geometry, Variation of Hodge structures (algebro-geometric aspects), Characteristic \(p\) methods (Frobenius endomorphism) and reduction to characteristic \(p\); tight closure, Hodge theory in global analysis, Structure of families (Picard-Lefschetz, monodromy, etc.), String and superstring theories; other extended objects (e.g., branes) in quantum field theory, Calabi-Yau manifolds (algebro-geometric aspects) Introduction to Hodge 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 \(G\) be a simple and simply connected algebraic group. The authors compute the Picard group of the moduli stack of quasi-parabolic \(G\)-bundles over a smooth, complete complex curve \(X\). Quasi parabolic \(G\)-bundles are defined with respect to \(n\) distinct points of \(X\), each one labelled by a parabolic subgroup of \(G\) containing the same Borel subgroup. The proof requires the uniformization theorem which describes the stack as double quotient of certain infinite dimensional algebraic groups. Basic facts about stacks and Lie theory needed in the proofs are clearly presented in this paper. Generators for the Picard group are explicitly computed when \(G\) is classical or \(G_2\), by constructing a pfaffian line bundle. This construction is tricky and requires explicit computations.
An application is the construction of a square root of the dualizing bundle of the stack for \(n=0\). The authors find also a canonical isomorphism between the space of global sections of the above stack and the corresponding space of conformal blocks of Tsuchiya, Ueno and Yamada which appears in conformal field theory. A consequence of this isomorphism is a generalization of the Verlinde formula. For an account about the Verlinde formula and its relation to mathematical physics see the survey of \textit{C. Sorger} [Sém. Bourbaki, Vol. 1994/95, Astérisque 237, 87-114, Exp. No. 794 (1996; Zbl 0878.17024)].
At the end the authors determine that the Picard group of the moduli space of \(G\)-bundles over curves is isomorphic to \({\mathbb{Z}}\), a result found independently by \textit{S. Kumar} and \textit{M. S. Narasimhan} [Math. Ann. 308, No. 1, 155-173 (1997; Zbl 0884.14004)]. For \(G=SL(r)\) this is a classical result of Drezet and Narasimhan. The assumption of simple connectedness of \(G\) has been removed in a subsequent paper [see \textit{A. Beauville, Y. Laszlo} and \textit{C. Sorger}, Composit. Math. 112, No. 2, 183-216 (1998)]. moduli stack; parabolic bundles over a complex curve; Picard group; uniformization; pfaffian line bundle; conformal blocks; Verlinde formula Y.~Laszlo, C.~Sorger, The line bundles on the moduli of parabolic \(G\)-bundles over curves and their sections, {\em Ann.~Sci.~Éc.~Norm.~Supér.~(4)} 30(4): 499--525, 1997 Picard groups, Algebraic moduli problems, moduli of vector bundles, Vector bundles on curves and their moduli The line bundles on the moduli of parabolic \(G\)-bundles over curves and their sections | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians Let C be a non rational algebraic projective smooth curve defined or \({\mathbb{C}}\). For any \(\alpha\in Pic^ 2(C)\) we consider the map \(\Phi_{\alpha}\) between the symmetric product \(C^{(2)}\) of the curve and its Jacobian, \(J(C)\), such that \(\Phi_{\alpha}(P,Q)= {\mathcal O}_{P+Q}\otimes \alpha^{-1}\) for all \((P,Q)\in C^{(2)}\). Let A be a proper Abelian subvariety of \(J(C)\).
We prove that \(\dim[\Phi_{\alpha}^{-1}(A)]=1\) if and only if there exist two curves K and \(K'\) and two non constant maps \(f: C\to K\) and \(h: K\to K'\) such that \(\deg(h)=2\), and
\[
A\supseteq (h\circ f)^*[J(K')] + \{\text{connected component of }\ker(f_*)\};
\]
\((f_*\), \((h\circ f)^*\) are the induced maps between Jacobians). We also prove that if \(\dim[\Phi_{\alpha}^{-1}(A)]<1\) for every \(\alpha\) and A, then \(h^ 1(-D)=0\) for every effective connected reduced divisor D of \(C^{(2)}\). Finally, we use our results to show that there are no genus 9 curves in the generic Abelian variety of dimension 5; this improves all bounds we know about this subject. abelian subvariety of Jacobian; symmetric correspondences; vanishing theorems; projective smooth curve; symmetric product Alzati, A; Pirola, GP, On abelian subvarieties generated by symmetric correspondences, Math. Z., 205, 333-342, (1990) Jacobians, Prym varieties, Algebraic theory of abelian varieties, Vanishing theorems in algebraic geometry, Vanishing theorems On abelian subvarieties generated by symmetric correspondences | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians [For the entire collection see Zbl 0679.00007.]
Let X be a smooth complete complex algebraic variety on which a complex affine algebraic group G acts with a dense orbit \(\Omega\simeq G/H\). We consider the case that X-\(\Omega\) is a G-homogeneous divisor. \textit{D. Akhiezer} [Ann. Global Anal. Geom. 1, 49-78 (1983; Zbl 0537.14033)] and \textit{A. Huckleberry}, \textit{D. Snow} [Osaka J. Math. 19, 763-786 (1982; Zbl 0507.32023)] classified the couples (G,H) such that G/H can be completed by homogeneous divisors, and proved the following facts. For such couples (G,H), every maximal compact subgroup K of G acts on G/H with an orbit of real codimension one as a rule. In particular, when G and H are reductive, then G/H is the complexification of a compact isotropic Riemannian homogeneous space. The purpose of this paper is to recover most of the results of the above papers by algebraic methods, i.e., embedding theory of spherical homogeneous spaces; they are the spaces G/H where G is reductive connected, and a Borel subgroup B of G has a dense orbit in G/H. smooth complete complex algebraic variety; complex affine algebraic group; dense orbit; homogeneous divisors; compact isotropic Riemannian homogeneous space; embedding theory of spherical homogeneous spaces M. Brion,On spherical varieties of rank one, CMS Conf. Proc.10 (1989), 31--41. Linear algebraic groups over the reals, the complexes, the quaternions, Homogeneous spaces and generalizations, Homogeneous complex manifolds, Divisors, linear systems, invertible sheaves, Group actions on varieties or schemes (quotients), Differential geometry of homogeneous manifolds On spherical varieties of rank 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 [For the entire collection see Zbl 0721.00009.]
During the last years several researchers have considered the problem of finding polynomial-time sequential algorithms for the computation of the topology of a real algebraic plane curve. The authors divide these algorithms into two main groups: those coding real algebraic numbers by means of isolating intervals and those coding them à la Thom. The aim of this note is to survey the state of the art as far as the second group is concerned. In particular, the authors introduce two new algorithms, the algorithm computing the topology of a real algebraic plane curve with the lowest running time, while the other one has been successfully implemented. Recently, it has been noted that the ground algorithm used to code real algebraic numbers à la Thom has a complexity smaller than the one used for the complexity estimates given by \textit{M. F. Roy}. In the first part of this note the authors give a proof of this fact and, using it, they obtain new upper bounds for the running times of the auxiliary procedures introduced by Roy, and which they use in their algorithms. Notice that, in both cases, the running times are much lower than those of the corresponding fastest algorithms using isolating intervals.
In the second part of this note the authors introduce some modifications to Roy's algorithms, obtaining: (i) an algorithm \(\text{CAD}_ 0\); (ii) an algorithm \(\text{TOP}_ 0\); and (iii) another algorithm for the computation of the topology, \(\text{TOP}_ 2\). They conclude that \(\text{CAD}_ 0\) is the fastest algorithm computing an \(F\)-invariant c.a.d. of \(R^ 2\), while \(\text{TOP}_ 0\) is the fastest algorithm computing the topology of \(C\). They remark that, although these all are similar in spirit, there is a great difference between Roy's algorithm and \(\text{TOP}_ 2\) on one side, and \(\text{CAD}_ 0\) and \(\text{TOP}_ 0\) on the other side: while the first ones freely use points of the real spectrum of \(R[X,Y]\), the second ones only use real algebraic numbers to get the same information. polynomial time sequential algorithms; topology of a real algebraic plane curve Cucker, F.; Vega, L. Gonzalez; Roselló, F.: On algorithms for real algebraic plane curves. Progress in mathematics 94 (1991) Computational aspects of algebraic curves, Topology of real algebraic varieties, Special algebraic curves and curves of low genus On algorithms for real algebraic plane curves | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians For a prime p, a natural projection of modular curves \(X_ 1(p)/{\mathbb{Q}}\to X_ 0(p)/{\mathbb{Q}}\) defines a Galois covering with the group \(G\cong ({\mathbb{Z}}/p{\mathbb{Z}})^ x/\{\pm 1\}\). The jacobian variety \(J_ 1(p)/{\mathbb{Q}}\) of \(X_ 1(p)/{\mathbb{Q}}\) decomposes according to characters of G. Let \(J^{(i)}\) denote the factor which corresponds to the character of order i of G. - A condition for the finiteness of \(J^{(\pi)}(K)\) is given, where \(J^{(\pi)}\) is a certain Eisenstein quotient of \(J^{(i)}\) and K is an imaginary quadratic field (theorem 3.4). It is shown that \(X_ 1(p)(K)\) consists of the cusps lying over 0 under some conditions (theorems 4.1 and 4.2). When \(i=2\), \(p\equiv 1 mod 4\), \(p>17\), the author proves unconditionally that indices of subgroups generated by cuspidal divisor classes in \(J^{(2)}({\mathbb{Q}})_{tors}\) or in \(J^{(2)}(K)_{tors}\), \(K={\mathbb{Q}}(\sqrt{p})\) are powers of 2 (theorem 5.1). Eisenstein ideals; Mordell-Weil groups; Eisenstein quotient of jacobian variety; modular curves; Galois covering; cusps Kamienny, S., Rational points on modular curves and abelian varieties, J. Reine Angew. Math., 359, 174-187, (1985) Rational points, Special algebraic curves and curves of low genus, Holomorphic modular forms of integral weight, Algebraic theory of abelian varieties Rational points on modular curves and abelian varieties | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians Let \(y^ 3-p_ 4(x)=0\) be the affine model of the Picard curve \(C\), over the perfect field \(K\) of characteristic \(p>3\) \((\deg(p_ 4(x))=4)\). In \(\S 1\), the author obtains explicit expressions for the coefficients of the Hasse-Witt matrix of \(C\), in the terms of the coefficients of \(p_ 4(x)\). In \(\S 3\), he reformulates the results of \textit{N. Yui} [J. Algebra 52, 378-410 (1978; Zbl 0404.14008)] for the particular case of Picard curves. As a consequence, the author obtains an explicit characterization of the ordinary (resp. supersingular) Jacobian varieties of Picard curves in characteristic \(p>3\), in the terms of the (well- defined) coefficient of the Hasse-Witt matrix.
It is obtained also an explicit formula for the number of \(K\)-rational points on a Picard curve defined over the field \(K=GF(p)\), \(p>3\). number of \(K\)-rational points; characteristic \(p\); Hasse-Witt matrix; Picard curve J. Estrada Sarlabous, On the Jacobian varieties of Picard curves defined over fields of characteristic \?>0, Math. Nachr. 152 (1991), 329 -- 340. Finite ground fields in algebraic geometry, Jacobians, Prym varieties, Enumerative problems (combinatorial problems) in algebraic geometry, Special algebraic curves and curves of low genus, Rational points On the Jacobian varieties of Picard curves defined over fields of characteristic \(p>0\) | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is 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 0752.00024; for the reviews of the preceding two articles of the conference see ibid. 1-29 (1992; Zbl 0756.14011) and 31-45 (1992; Zbl 0760.14009)].
Let \(X=G/K\) be a hermitian symmetric domain corresponding to the reductive Lie group \(G\), and let \(\Gamma\subset G\) be a torsionfree arithmetic subgroup. Using a \(G\)-invariant metric on \(X\) the smooth algebraic quotient variety \(\Gamma\backslash X\) is also a complete Riemannian manifold. The Baily-Borel compactification \(\overline{\Gamma\backslash X}\) is a normal projective variety. On the one hand, for each local system \(E\) on \(\Gamma\backslash X\) (coming from a finite dimensional representation of \(G\) restricted to \(\Gamma)\) there is a well-defined \(L^ 2\)-cohomology \(H_{(2)}^*(\Gamma\backslash X;E)\). On the other hand one disposes on the intersection cohomology \(IH^*(\overline{\Gamma\backslash X};E)\). The Zucker conjecture asserts that both cohomologies are isomorphic.
For \(\mathbb{Q}\)-rank 1 groups, especially for \(G=GU(n,1)\), the conjecture was proved by \textit{S. Zucker} [Invent. Math. 70, 169-218 (1982; Zbl 0508.20020)]. A general proof was found by \textit{F. Looijenga} [Compos. Math. 67, No. 1, 3-20 (1988; Zbl 0658.14010)]. The author's lecture gives some intuition behind Looijenga's proof in the ball case \(G=GU(n,1)\) with special regard to \(GU(2,1)\). The main tools are: stratifications of algebraic varieties, local Hecke correspondences around cusps and the following decomposition theorem [\textit{A. A. Beilinson}, \textit{J. Bernstein} and \textit{P. Deligne}, ``Faisceaux pervers'', Astérisque 100 (1982; Zbl 0536.14011)]:
Let \(\pi:Y\to X\) be a (stratified) proper algebraic map, \(IC^*(Y)\) the sheaf complex of intersection cohomology. In the derived category of sheaves on \(X\) there is an isomorphism \(R\pi_ *(IC^*(Y))\cong\bigoplus^ r_{i=0}\bigoplus_ jIC^*(\overline X_ i;L^ i_ j)[\ell^ i_ j]\), where \([\ell^ i_ j]\) are shifts by integers, \(L^ i_ j\) are local systems on the strata \(X_ i\) of \(X\). If all the \(L^ i_ j\) are irreducible, then the decomposing factors are uniquely determined by \(\pi\) and the stratifications of \(X\) and \(Y\).
The following steps of proof are explained in more details with special regard to the ball example: Reduce the problem to a local verification near each singular point \(p\in Y=\overline{\Gamma\backslash X}\), using sheaf theory and the axiomatic characterization of intersection cohomology. --- Obtain as explicit as possible a description of the topology of the link \(L\) of the singular stratum \(S\) which contains the point \(p\). --- Decompose the intersection cohomology of \(L\) into subspaces according to the rates of growth of representative differential forms. --- Realize this decomposition as the eigenspace decomposition of \(IH^ k(L)\) under the action of certain geometrically defined ``local Hecke correspondences'' \(\Phi_ a:L\to L\). Reduce the problem to proving that the intersection cohomology classes which must vanish are those of ``weight'' \(\geq c\) and degree \(k<c=\text{codim} S\).
Resolve the singularities \(\pi:\tilde Y\to Y\) and apply the decomposition theorem to exhibit \(IH^*(Y)\) as a subspace of \(H^*(\tilde Y)\) both locally and globally, i.e. \(IH^*(L)\subset H^*(\pi^{-1}(L))\). Observe that the local Hecke correspondence \(\Phi_ a\) also acts locally on \(\tilde Y\) and that it decomposes \(H^*(\pi^{-1}(L))\) into eigenspaces as well. Apply techniques of variations of Hodge structures to see that \(H^ k(\pi^{-1}(L))\) has no classes of weight \(>k\), and so the same is true for \(IH^ k(L)\). Thus, for \(k<c\), the group \(IH^ k(L)\) has no classes of weight \(\geq c\). Picard modular spaces; \(L^ 2\)-cohomology; hermitian symmetric domain; quotient variety; Baily-Borel compactification; intersection cohomology; Zucker conjecture; variation of Hodge structtures Goresky, M., \(L^2\) cohomology is intersection cohomology, (Langlands, R.P.; Ramakrishnan, D., The zeta functions of Picard modular surfaces, (1992), Centre de Recherches Mathématiques Montréal) Other algebro-geometric (co)homologies (e.g., intersection, equivariant, Lawson, Deligne (co)homologies), Modular and Shimura varieties \(L^ 2\)-cohomology is intersection cohomology | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians We study the collection of homological equivalence relations on a fixed curve. We construct a moduli space for pairs consisting of a curve of genus \(g\) and a homological equivalence relation of degree \(n\) on the curve, and a classifying set for homological equivalence relations of degree \(n\) on a fixed curve, modulo automorphisms of the curve. We identify a special type of homological equivalence relations, and we characterize the special homological equivalence relations in terms of the existence of elliptic curves in the Jacobian of the curve. algebraic curve; homological equivalence relation; Jacobian variety; elliptic curve Algebraic moduli of abelian varieties, classification, Homotopy theory and fundamental groups in algebraic geometry, Subvarieties of abelian varieties, Families, moduli of curves (algebraic) Homological equivalence relations on an algebraic curve | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians We determine explicitly the formal moduli space of certain complete topological modules over a topologically finitely generated local \(k\)- algebra \(R\), not necessarily commutative, where \(k\) is a field. The class of topological modules we consider includes all those of finite rank over \(k\) and some of infinite rank as well, namely those with a Schauder basis in the sense of \S1. This generalizes the results of the second author [Ph. D. thesis, Brandeis 1987], where the result was obtained in a different way in case the ring \(R\) is the completion of the local ring of a plane curve singularity and the module is \(k^ n\).
Along the way, we determine the ring of infinite matrices which correspond to the endomorphisms of the modules with Schauder bases. We also introduce functions called ``growth functions'' to handle explicit epsilonics involving the convergence of formal power series in non- commuting variables evaluated at endomorphisms of our modules. The description of the moduli space involves the study of a ring of infinite series involving possibly infinitely many variables and which is different from the ring of power series in these variables in either the wide or the narrow sense. Our approach is beyond the methods of \textit{M. Schlessinger} [Trans. Am. Math. Soc. 130, 208-222 (1968; Zbl 0167.495)] which were used in \textit{P. Shukla}'s thesis cited above and is more conceptual. growth functions; formal moduli space; complete topological modules; completion of the local ring of a plane curve singularity; convergence of formal power series in noncommuting variables Noncommutative algebraic geometry, Power series rings, Topological and ordered rings and modules, Valuations, completions, formal power series and related constructions (associative rings and algebras), Formal power series rings, Normed modules and Banach modules, topological modules (if not placed in 13-XX or 16-XX) Formal moduli of modules over local \(k\)-algebras | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians Let A be an abelian variety over the field of complex numbers \({\mathbb{C}}\) such that dim A\(>1\) and its endomorphism algebra \(E=End A\otimes {\mathbb{Q}}\) is an imaginary quadratic field. We have two embeddings, \(\sigma\) and \(\tau\), say, of E into \({\mathbb{C}}\). E acts on the Lie algebra Lie(A) of A; let \(n_{\sigma}\sigma +n_{\tau}\tau\) be the character of this action. Here \(n_{\sigma}\), \(n_{\tau}\) are positive integers and \(n_{\sigma}+n_{\tau}=\dim A\). Assume that \(n_{\sigma}\) and \(n_{\tau}\) are relatively prime (e.g., dim A is prime). Under this assumption the author computes explicitly the Hodge group of A in terms of E and proves that all Hodge classes on A are linear combinations of products of classes of divisors. In particular, all Hodge classes are algebraic [the case of prime dim A was treated earlier by \textit{S. G. Tankeev} in Math. USSR, Izv. 20, 157-171 (1983); translation from Izv. Akad. Nauk SSSR, Ser. Mat. 46, No.1, 155-170 (1982; Zbl 0587.14005)]. Notice that for K3 surfaces the Hodge group was explicitly computed by the reviewer [J. Reine Angew. Math. 341, 193-220 (1983; Zbl 0506.14034)]. Hodge conjecture; (1,1) criterion; abelian variety; endomorphism algebra; Hodge group; classes of divisors K. A. Ribet, Hodge classes on certain types of abelian varieties, Amer. J. Math. 105 (1983), 523--538. JSTOR: Transcendental methods, Hodge theory (algebro-geometric aspects), Analytic theory of abelian varieties; abelian integrals and differentials, Divisors, linear systems, invertible sheaves Hodge classes on certain types of Abelian varieties | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians This is Volume II of a study in derived algebraic geometry, and depends heavily on Volume I [Zbl 1408.14001]. In short, Volume I defines the different categories and the functors on them, needed for derived algebraic geometry. Then it tells which properties these functors have, and how to work with them. Volume I contains explanations of why we need the higher geometry: It is needed so that the functors have adjoint functors. It considers Ind-coherent sheaves and categories of correspondences. So Volume I gives the basic theory of derived algebraic geometry based on $(\infty,2)$-categories, and Volume II gives essential results from this theory, solving algebraic geometric problems by representation theory extended to derived theory. \par The geometric objects of interest in Volume II are the class of geometric objects named inf-schemes, with their corresponding categories of ind-coherent sheaves. These are the categories that appear in representation theoretic situations. \par Volume II consists of two parts. Part I has four chapters, and starts with a recalling (from Volume I) of inf-schemes, and a slight functorial interpretation of deformation theory. Also, the theory of Ind-coherent sheaves on inf-schemes are recalled. The introduction ends with an interesting discussion linking Crystals and $D$-modules. Chapter 1 give a proper definition of deformation theory, using representation theory, (or even higher category theory): Push-out of (derived) schemes, (pro)-cotangent and tangent spaces are defined on their infinitesimal objects, their properties are given, infinitesimal cohesiveness, and the the higher category definition of deformation theory. After this, chapter 1 gives the consequences of admitting deformation theory; categories with deformation theory is a concept influencing the rest of Volume II. For instance, a subsection on isomorphism properties is given; the property of having deformation theory can be used to show that certain maps between prestacks are isomorphisms, and the same technique is used to prove a criterion for being locally almost of finite type. The final section of chapter 1 considers square-zero extensions of prestacks. Chapter 2 start with a functorial definition of ind-schemes connected to inductive limits. The basic results of this category is then summed up and proved, and the different concepts are combined to (Ind)-inf-schemes and nil-closed embedding (to mention its link to deformation theory). Chapter three grows on the previous chapter. It considers Ind-coherent sheaves on ind-schemes involving proper base change for Ind-Schemes, and the concept corresponding to coherent sheaves, InCoh on (ind)-inf-schemes and the direct image functor for ind-inf-schemes. The formalism of correspondences is extended to inf-schemes and the self-duality and multiplicative structure of IndCoh ond ind-inf-schemes are considered. Chapter 3 looks deeply into Ind-coherent sheaves on ind-inf-schemes: Their proper base-change, their inductive extension IndCoh on (ind)-inf schemes, and the direct image functor for ind-inf-schemes. The formalism of correspondences is extended to inf-schemes, and the authors apply self-duality and the multiplicative structure of IndCoh on ind-inf-schemes. The final chapter of part I, chapter 4, is an application of crystals: First, the book defines crystals on prestacks and inf-schemes. The category of crystals on a prestack $\mathcal X$ is defined to be IndCoh on the corresponding prestack $\mathcal X_{dR}.$ This is also seen as a functor out of the category of correspondences, and the introduction of crystals means that the forgetful functor $\text{Crys}(\mathcal Z)\rightarrow\text{IndCoh}(Z)$ has a left adjoint. This is provided that $\mathcal Z$ is a prestack admitting deformation theory. Finally, in this chapter this is compared to the classical theory of $D$-modules. \par Part II is named formal geometry. The introduction to this chapter more or less indicates that in geometry, formal can be thought of as convergent sequences, thereby as inductive limits, and so completing up to analytic geometry. Going from geometry to algebra, Lie algebras comes up as the kernel of the exponential map. From this it also follows that a Lie algebroid corresponds to a functor of Lie algebras, leading to infinitesimal differential geometry. Chapter 5 is short and concise, and defines formal moduli and formal moduli problems, by introducing functors to groupoids. From this reviewer's viewpoint, this is a vital organ of the book, and readers should pay attention. Chapter 6 on Lie algebras and co-commutative co-algebras considers algebras over operads, Koszul duality, Associative algebras, universal enveloping algebras and ways to construct it. The chapter Introduces modules and proves the Poincare-Birkhoff-Witt (PBW) theorem, and defines commutative co-algebras and bialgebras. Chapter 7 goes into formal groups and formal Lie algebras. It tells how formal moduli problems leads to co-algebras, it confirms that inf-affines is what one would think it is, and it introduces modules over formal groups and formal Lie algebras. These modules leads to actions of formal groups on prestacks. Chapter 8 is dedicated to the study of Lie algebroids, in derived geometry: The inertial group, the basic structure, examples, results about modules on Lie algebroids and the universal enveloping algebra, square-zero extensions and Lie algebroids, IndCoh of a square-zero extension. Also there are results about global sections of a Lie algebroid, and Lie algebras as modules over a monad. Chapter 8 ends with a comment on the relation to classical Lie algebroids and gives an application of ind-coherent sheaves on push-outs. Chapter 9 makes sense of (infinitesimal) differential geometry in the context of derived algebraic geometry. The notions include deformations to the normal cone of a closed embedding, the notion of the $n$-th infinitesimal neighbourhood of a scheme embedded in another, the PBW filtration on the universal enveloping algebra of a Lie algebroid over a smooth scheme and the Hodge filtration (a.k.a. de Rham resolution) of the dualizing $D$-module. \par The two books together, and in particular this volume II, is written in a totally categorical language. This says that there are no explicit definitions, and the objects under study are given by their functorial properties. This makes the two books rather hard to read, especially as the notation must be stored by imagination, but then after all it illustrates advanced material in a relatively compact way. The ideas presented by the volumes are truly magnificent, and it is proved that the properties we want from a derived algebraic geometry holds. Thus it is magnificent both as an introduction to the power of derived algebraic geometry, and as a reference work. derived scheme; connective pro-cotangent space; connective deformation theory; almost of finite type (pro-)quasicoherent sheaf; anchor map; Chevalley functor; ind scheme; cocommutative Hopf algebra; cocommutative bi-algebra; co-operad; composition monoidal structure; crystal; de Rham prestack; differential of $x$; exponential map; filtered object; formal moduli problem; formally smooth; Hodge filtration; ind-inf-scheme; inertia object; Lie operad; left-Lax equivariance; $n$-coconnective ind-scheme; pro-cotangent complex; reduced indscheme; shifted anchor map; smooth of relative dimension $n$; splitting of a Lie algebroid; universal envelope of a Lie algebra; Verdier duality; Weil restriction; zero Lie algebroid Research exposition (monographs, survey articles) pertaining to algebraic geometry, Generalizations (algebraic spaces, stacks), Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20], Differentials and other special sheaves; D-modules; Bernstein-Sato ideals and polynomials, Double categories, \(2\)-categories, bicategories, hypercategories, Nonabelian homotopical algebra A study in derived algebraic geometry. Volume II: Deformations, Lie theory and formal geometry | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians [For the entire collection see Zbl 0741.00012.]
We present a deterministic algorithm to compute the number of \(\mathbb{F}_ q\)-rational points of every elliptic curve defined over a finite field \(\mathbb{F}_ q\), with \(j\)-invariant 0 or 1728, and which is given by a Weierstrass equation. This algorithm takes \(O(\log^ 4p)\) bit operations, where \(p\) is the characteristic of \(\mathbb{F}_ q\). rational points of elliptic curve defined over a finite field; deterministic algorithm Computational aspects of algebraic curves, Number-theoretic algorithms; complexity, Analysis of algorithms and problem complexity, Finite ground fields in algebraic geometry An algorithm to compute the number of points of elliptic curves with invariant 0 or 1728 over a finite field | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians Let \(\pi : Z \rightarrow X\) be a Galois covering of smooth projective curves with Galois group the Weyl group of a simple and simply connected Lie group \(G.\) For any dominant weight \(\lambda\) consider the curve \(Y = Z/\text{Stab}(\lambda)\). The Kanev correspondence defines an abelian subvariety \(P_{\lambda}\) of the Jacobian of \(Y\). The authors compute the type of the polarization of the restriction of the canonical principal polarization of \(\text{Jac}(Y)\) to \(P_{\lambda}\) in some cases. In particular, in the case of the group \(E_{8}\) they obtain families of Prym-Tyurin varieties. The main idea is the use of an abelianization map of the Donagi-Prym variety to the moduli stack of principal \(G-\)bundles on the curve \(X\). Galois covering of smooth projective curves; abelian subvariety of the Jacobian; canonical principal polarization; Prym-Tyurin varieties; Donagi-Prym variety Lange (H.) and Pauly (C.).- Polarizations of Prym varieties for Weyl groups via Abelianization, Journal of the European Mathematical Society, Volume 11, No. 2, p. 315-349 (2009). Zbl1165.14026 MR2486936 Jacobians, Prym varieties, Theta functions and abelian varieties, Coverings of curves, fundamental group, Vector bundles on curves and their moduli, Algebraic moduli problems, moduli of vector bundles Polarizations of Prym varieties for Weyl groups via abelianization | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians Simple parametrizations of complex curve singularities of arbitrary embedding dimension are classified. A parametrization is called simple, if there are only finitely many isomorphism classes in the versal deformation. This is a more restricted definition of simpleness for parametrizations compared to the definitions of \textit{J. W. Bruce} and \textit{T. J. Gaffney} [J. Lond. Math. Soc., II. Ser. 26, 465--474 (1982; Zbl 0575.58008)] and \textit{C. G. Gibson} and \textit{C. A. Hobbs} [Math. Proc. Camb. Philos. Soc. 113, No. 2, 297--310 (1993; Zbl 0789.58013)], since it allows neighbouring singulairites with more irreducible components.
This leads to a smaller list than that obtained by looking at the neighbours in the space of multi-germs with a fixed number of branches. The list of simple parametrizations of plane curves is the A-D-E-list. For space curves the list of \textit{M. Giusti} [Proc. Symp. Pure Math. 40, 457--494 (1983; Zbl 0525.32006)] (complete intersections) together with the list of \textit{A. Frühbis-Krüger} [Commun. Algebra 27, No. 8, 3993--4013 (1999; Zbl 0963.14011)] (determinantal codimension 2 singularities) is obtained. In these cases the classification coincides with the classification of simple curves defined by equations [\textit{V. I. Arnol'd}, Proc. Steklov Inst. Math. 226, 20--28 (1999; Zbl 0991.32015); translation from Tr. Mat. Inst. Steklova 226, 27--35 (1999); Giusti, loc. cit.; Frühbis-Krüger, loc. cit.]. simple singularities; A-D-E-list; curve singularities; classification of simple singularities; parametric curves Singularities of curves, local rings, Local complex singularities Simple curve singularities | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians Let \(k\) denote an algebraically closed field. Let \(Y\) be a nonsingular projective curve over \(k\) and let \(f:X\rightarrow Y\) be a semistable curve of genus \(g\geq 2\) such that \(X\) is smooth and the generic fiber of \(f\) is a smooth hyperelliptic curve. When \(k=\mathbb C\), \textit{M.~Cornalba} and \textit{J.~Harris} [Ann. Sci. Éc. Norm. Supér., IV. Sér. 21, No.3, 455--475 (1988; Zbl 0674.14006)] established a formula for the class of the \((8g+4)\)th tensor power of the Hodge bundle of this family in terms of the classes of the singular fibers. This formula was extended to the case when the characteristic of \(k\) is greater than two by \textit{I.~Kausz} [Compos. Math. 115, No.1, 37--69 (1999; Zbl 0934.14015)]. The author shows that the Cornalba-Harris formula also holds when the characteristic of \(k\) is two. The strategy of the proof is to use the result in characteristic zero and a compactification \(\overline{\mathcal I}_g\) of the algebraic stack (over \(\mathbb Z\)) of smooth hyperelliptic curves of genus \(g\) to derive the result in all characteristics. A key element in the proof is a result of \textit{S. Maugeais} [Espace de modules des courbes hyperelliptiques stables et une inégalité de Cornalba-Harris-Xiao, preprint, \texttt{http://arxiv.org/abs/math.AG/0107015}] that is used to establish the irreducibility of the specialization of \(\overline{\mathcal I}_g\) to characteristic two. algebraic stack; Hodge class Yamaki K.: Cornalba-Harris equality for semistable hyperelliptic curves in positive characteristic. Asian J. Math. 8(3), 409--426 (2004) Families, moduli of curves (algebraic), Special algebraic curves and curves of low genus, Picard groups Cornalba-Harris equality for semistable hyperelliptic curves in positive characteristics | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians The problem of compactifying the (generalized) Jacobian of a singular curve has been studied since \textit{J. Igusa} [Am. J. Math. 78, 171--199, 745--760 (1956; Zbl 0074.15803)]. He constructed a compactification of the Jacobian of a nodal and irreducible curve \(X\) as limit of the Jacobians of smooth curves approaching \(X.\) Igusa also showed that his compactification does not depend on the considered family of smooth curves. When the curve \(X\) is reducible and nodal, \textit{T. Oda} and \textit{C. S. Seshadri} [Trans. Am. Math. Soc. 253, 1--90 (1979; Zbl 0418.14019)] produced a family of compactified Jacobians \(\text{Jac}_{\phi}\) parameterized by an element \(\phi\) of a real vector space. \textit{C. S. Seshadri} [``Fibrés vectoriels sur les courbes algébriques'', Astérisque 96 (1982; Zbl 0517.14008)] dealt with the general case of a reduced curve considering sheaves of higher rank as well. \textit{L. Caporaso} [J. Am. Math. Soc. 7, No. 3, 589--660 (1994; Zbl 0827.14014)] showed how to compactify the relative Jacobian over the moduli of stable curves and described the boundary points of the compactified Jacobian of a stable curve \(X\) as invertible sheaves on certain Deligne-Mumford semistable curves that have \(X\) as a stable model. Most of the above papers are devoted to the construction of the compactified Jacobian of a curve, not to describe it.
In the paper under review, the author gives an explicit description of the structure of these Simpson schemes, \(\text{Jac}^{d}(X)_{s},\) and \(\overline{\text{Jac}^{d}(X)}\) of any degree \(d,\) for \(X\) a polarized curve of the following types: tree-like curves and all reduced and reducible curves that can appear as singular fibers of an elliptic fibration. moduli of stable curves; Deligne-Mumford semistable curves; Simpson scheme López~Martín, A.C., Simpson Jacobians of reducible curves, J. reine angew. math., 582, 1-39, (2005) Families, moduli of curves (algebraic), Jacobians, Prym varieties, Vector bundles on curves and their moduli, Picard groups, Schemes and morphisms, Families, moduli of curves (analytic) Simpson Jacobians of reducible curves | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians In this paper, we consider morphisms of algebraic stacks \(\mathcal X \to \mathcal Y\) which are torsors under a group stack \(\mathcal G\). We show that line bundles on \(\mathcal Y\) correspond exactly with \(\mathcal G\)-linearized line bundles on \(\mathcal X\) (with a suitable definition of a \(\mathcal G\)-linearization). We use this fact to determine the precise structure of the Picard group of the moduli stack of \(G\)-bundles on an algebraic curve when \(G\) is any group of type \(A_n\). linearization; morphisms of algebraic stacks; Picard group; torsors; moduli stack on an algebraic curve Laszlo Y., Linearization of group stack actions and the Picard group of the moduli of SLr/{\(\mu\)}s-bundles on a curve, Bull. Soc. Math. France, 1997, 125(4), 529--545 Vector bundles on curves and their moduli, Picard groups, Families, moduli of curves (algebraic), Group actions on varieties or schemes (quotients) Linearizaton of group stack actions and the Picard group of the moduli of \(SL_r/\mu_s\)-bundles on a curve | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians This paper describes the group of line bundles, or the Picard group, of the moduli space of line bundles on smooth curves as well as some closely related moduli spaces. The Picard group had been studied before, notably in [\textit{J. Ebert} and \textit{O. Randal-Williams}, Doc. Math., J. DMV 17, 417--450 (2012; Zbl 1273.14058); \textit{C. Fontanari}, Atti Accad. Naz. Lincei, Cl. Sci. Fis. Mat. Nat., IX. Ser., Rend. Lincei, Mat. Appl. 16, No. 1, 45--59 (2005; Zbl 1222.14055)], and [\textit{A. Kouvidakis}, J. Differ. Geom. 34, No. 3, 839--850 (1991; Zbl 0780.14004)], but this is the first paper to carefully treat stack-theoretic issues.
Recall the moduli space of degree \(d\) line bundles on smooth curves of genus \(g\), denoted \(\mathcal{J}ac_{d,g}\) in the paper, exists as an algebraic stack. This stack has the property that every stabilizer group contains the multiplicative group \(\mathbb{G}_{m}\) (acting by scalar multiplication on line bundles), and a natural rigidification construction removes the \(\mathbb{G}_{m}\)s to produce a new stack \(\mathcal{J}_{d,g}\). The stabilizer groups of the new stack \(\mathcal{J}_{d,g}\) are finite groups, in fact \(\mathcal{J}_{d,g}\) is a Deligne-Mumford stack, but the stabilizers can be nontrivial, so \(\mathcal{J}_{d,g}\) is not an algebraic variety. The nontrivial stabilizer groups can, however, be removed to produce a third object, the coarse moduli space of \(\mathcal{J}_{d,g}\) which is a quasi-projective variety. In the literature, all of these spaces are often called the universal Jacobian, although in the present paper the term is reserved for \(\mathcal{J}ac_{d,g}\).
The universal Jacobian is studied alongside certain moduli spaces parameterizing line bundles on (possible unstable) nodal curves that satisfy the balancedness condition (a numerical condition on the multidegree). Similar to the previous situation, one considers the moduli stack \(\overline{\mathcal{J}ac}_{d,g}\) of these objects as well as a Deligne-Mumford stack \(\overline{\mathcal{J}}_{d,g}\) obtained by removing \(\mathbb{G}_{m}\)s from stabilizers and a projective variety \(\overline{J}_{d,g}\) obtained by removing all stabilizers. All of these spaces are often called the compactified universal Jacobian as \(\overline{J}_{d,g}\) is proper.
The main results of this paper compute the Picard groups of these stacks when \(g \geq 3\). The results are easiest to state for \(\mathcal{J}ac_{d,g}\) and \(\overline{\mathcal{J}ac}_{d,g}\). The universal family of curves over \(\mathcal{J}ac_{d,g}\) admits two natural line bundles: the universal line bundle \(\mathcal{L}\) and the relative dualizing sheaf \(\omega\). Theorem A states that the Picard group of \(\mathcal{J}ac_{d,g}\) is freely generated by the determinants of cohomology of \(\mathcal{L}\), \(\omega\), and \(\mathcal{L} \otimes \omega\). Furthermore, the Picard group of \(\overline{\mathcal{J}ac}_{d,g}\) is freely generated by these line bundles together with the line bundles associated to the irreducible components of the boundary (i.e. the complement of \(\mathcal{J}ac_{d,g}\) in \(\overline{\mathcal{J}ac}_{d,g}\)).
The Picard groups of \(\mathcal{J}_{d,g}\) and \(\overline{\mathcal{J}}_{d,g}\) are described in Theorem B. This result is more complicated to state because the universal family of curves over \(\mathcal{J}_{d,g}\) does not admit a universal family of line bundles. The result states that the Picard group of \(\mathcal{J}_{d,g}\) is freely generated by the determinant of cohomology of \(\omega\) and a line bundle that is more complicated to describe but is similar to a certain explicit linear combination of the determinants of cohomology of \(\mathcal{L}\) and \(\mathcal{L} \otimes \omega\). These line bundles together with the line bundles associated to the irreducible components of the boundary freely generate the Picard group of \(\overline{\mathcal{J}}_{d,g}\).
The analogous results for \(J_{d,g}\) and \(\overline{J}_{d,g}\) were known by [Zbl 1222.14055], and the authors also relate the descriptions in that work to their Theorems A and B in Theorem C.
The main theorems are proven by using Kouvidakis's work [Zbl 0780.14004] to compute the Picard group of \(\mathcal{J}_{d,g}\) and then relating the other Picard groups of interest to \(\text{Pic}(\mathcal{J}_{d,g})\). Results similar to the results about \(\mathcal{J}ac_{d,g}\) and \(\mathcal{J}_{d,g}\) were proven in [Zbl 1273.14058], which appeared shortly after a preliminary of the present paper was made publically available. The authors of that paper compute the Picard groups of topological stacks that are expected to be topological models of \(\mathcal{J}ac_{d,g}\) and \(\mathcal{J}_{d,g}\). The proofs there are very different from the proofs in the present paper. In [Zbl 1273.14058] the main results are proven using algebraic topology, especially ideas from homotopy theory, while the present paper uses algebraic geometry. The relation between the results of the two papers is carefully described in Section 1.1 of the present paper and Section 4.5 of [Zbl 1273.14058]. Brauer group; Picard group; gm-gerbe; universal Jacobian stack and scheme; compactified universal Jacobian Melo, M.; Viviani, F., The Picard group of the compactified universal Jacobian, Doc. Math., 19, 457-507, (2014) Families, moduli of curves (algebraic), Jacobians, Prym varieties, Picard groups, Generalizations (algebraic spaces, stacks), Geometric invariant theory The Picard group of the compactified universal Jacobian | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians The theory of \(\mathbb Q\)-Cartier divisors on the space of \(n\)-pointed, genus 0, stable maps to projective space is considered. Generators and Picard numbers are computed. A recursive algorithm computing all top intersection products of \(\mathbb Q\)-divisors is established. As a corollary, an algorithm computing all characteristic numbers of rational curves in \(\mathbb P^r\) is proven (including simple tangency conditions). Computations of these characteristic numbers are carried out in many examples. The degree of the 1-cuspidal rational locus in the linear system of degree \(d\) plane curves is explicitly evaluated. enumerative geometry; Cartier divisors; \(n\)-pointed, genus 0, stable maps; intersection products of \(\mathbb{Q}\)-divisors; characteristic numbers of rational curves; 1-cuspidal rational locus R. Pandharipande, Intersections of \(\({ Q}\)\)-divisors on Kontsevich's moduli space \(\({\overline{M}}_{0, n}({ P}^{r}, d)\)\) and enumerative geometry. Trans. Am. Math. Soc. 351(4), 1481-1505 (1999) Enumerative problems (combinatorial problems) in algebraic geometry, Families, moduli of curves (algebraic), Birational geometry, Intersection theory, characteristic classes, intersection multiplicities in algebraic geometry, Divisors, linear systems, invertible sheaves, Picard groups Intersections of \(\mathbb{Q}\)-divisors on Kontsevich's moduli space \(\overline M_{0,n}(\mathbb P^r,d)\) and enumerative geometry | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians The author constructs a compactification for the relative degree-\(d\) Picard variety associated to a family of (proper) stable curves. The problem arises because the Picard functor neither is proper nor separated when fibers of \(X/S\) are not smooth. For instance one can find families of invertible sheaves specializing to sheaves not locally free when the central fiber has nodes. The author uses a new functor to solve this problem. One crucial idea is that in this new functor such families specialize to invertible sheaves on the curve resulting from replacing the node (in the central fiber) by \(\mathbb{P}^1\). [The same idea was used by \textit{D. Gieseker} for rank-2 vector bundles on nodal curves; cf. J. Differ. Geom. 19, 173-206 (1984; Zbl 0557.14008).]The proof relies on GIT (=``geometric invariant theory'') theory. moduli problem; semistable curves; geometric invariant theory; compactification; Picard variety Lucia Caporaso, A compactification of the universal Picard variety over the moduli space of stable curves , J. Amer. Math. Soc. 7 (1994), no. 3, 589-660. JSTOR: Families, moduli of curves (algebraic), Algebraic moduli problems, moduli of vector bundles, Picard groups A compactification of the universal Picard variety over 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 In the paper under review, the author studies the Picard group of the moduli spaces of certain vector bundles over nodal curves. Let \(Y\) be an irreducible reduced curve over the field of complex numbers such that \(Y\) has at most ordinary nodes as singularities. Fix a line bundle \(\mathcal L\) on \(Y\). Let \(U'_{\mathcal L}(n, d)\) (respectively, \({U'}_{\mathcal L}^{s}(n, d)\)) be the moduli space of semistable (respectively, stable) vector bundles of rank-\(n\) with fixed determinant \(\mathcal L\) on \(Y\). Let \(g\) (respectively, \(g_Y\)) be the geometric (respectively, arithmetic) genus of \(Y\). The main result in the paper says that if \(g \geq 2\), then \(\text{Pic }{U'}_{\mathcal L}(n, d) \cong \text{Pic }{U'}_{\mathcal L}^{s}(n, d) \cong \mathbb{Z}\) and \({U'}_{\mathcal L}(n, d)\) is locally factorial except possibly in case \(g =n=2\) and \(2|d\). Moreover, if \(g_Y=n=2\), then \(\text{Pic }{U'}_{\mathcal L}(n, d) \cong \mathbb{Z}\). These generalize the well-known results when the curve \(Y\) is non-singular. The main technique in the proof is to carry out various codimension computations among the relevant moduli spaces. Other interesting results concerning semistable sheaves over singular curves are also obtained. Picard group; moduli space; vector bundle; nodal curves Bhosle U.: Picard groups of the moduli spaces of vector bundles. Math. Ann. 314(2), 245--263 (1999) Picard groups, Families, moduli of curves (algebraic), Algebraic moduli problems, moduli of vector bundles, Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20], Vector bundles on curves and their moduli, Singularities of curves, local rings Picard groups of the moduli spaces of vector bundles | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians Trigonal curve is a smooth projective curve with \(3:1\) map onto a curve which is isomorphic to \({\mathbb P}^1\). The isomorphism is not part of the data. The stack \({\mathcal T}_g\) of genus \(g\) smooth trigonal curves over fixed field \(k\) of characteristic not equal 2 or 3 is considered. The objects over \(k\)-scheme \(S\) are families \(C \to P \to S\) where \(P\to S\) is a smooth conic bundle, \(C\to P\) smooth \(3:1\) cover and their composite \(C\to S\) is a family of genus \(g\) curves. The forgetful morphism \((C \to P \to S) \mapsto (C\to S)\) takes \({\mathcal T}_g\) to the stack \({\mathcal M}_g\) of smooth genus \(g\) curves. One of central results of the paper is that this morphism is a locally closed immersion whenever \(g\geq 5\). Another central result is a description of the stack \({\mathcal T}_g\) as a quotient \([X_g/\Gamma_g]\) of an appropriate scheme \(X_g\) by the action of a certain algebraic group \(\Gamma_g\). The third central result is a computation of the Picard group of \({\mathcal T}_g\) for \(g\neq 1\). The authors give a description of the stack of vector bundles over a conic as a quotient stack what is of independent interest besides of being one of main tools of the work together with the result of \textit{R. Miranda} [Am. J. Math. 107, 1123--1158 (1985; Zbl 0611.14011)] ``that describes a flat finite triple cover of a scheme \(S\) as given by a locally free sheaf \(E\) of rank two on \(S\), with a section of \(\text{Sym}^3 E \otimes E^{\vee}\)'' (cited from the abstract). Also the stack \(\hat{\mathcal T}_g\) of triple covers which contains \({\mathcal T}_g\) as an open substack, is examined. In particular, the locus of singular curves in \(\hat{\mathcal T}_g\) is analyzed. trigonal curves; algebraic stack; stack of smooth curves; Picard group of a stack; stack of vector bundles on a conic Bolognesi, M; Vistoli, A, Stacks of trigonal curves, Trans. Am. Math. Soc., 364, 3365-3393, (2012) Families, moduli of curves (algebraic), Generalizations (algebraic spaces, stacks), Picard groups Stacks of trigonal curves | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians Denote by \(V_{d, \delta}\) the Severi variety of reduced and irreducible curves of degree \(d\) in \(\mathbb{P}^ 2\) with exactly \(\delta\) nodes. If \(g = {(d-1) (d-2) \over 2} - \delta \), then there is a map \(V_{d, \delta} \to {\mathcal M}_ g\), where \({\mathcal M}_ g\) is the moduli space of curves of genus \(g\). The map is defined by sending a point representing a \(\delta\)-nodal curve to the point representing its normalization. For \(\delta\) sufficiently large with respect to \(d\), a dimension count shows that the map is dominant. The purpose of this paper is to relate \(\text{Pic}_ \mathbb{Q} (V_{d, \delta})\), the rational Picard group of \(V_{d, \delta}\) to \(\text{Pic} _ \mathbb{Q} ({\mathcal M}_ g)\), the rational Picard group of \({\mathcal M}_ g\). Specifically we prove the following theorem.
If \(d > 2g + 2\), and \(g = {(d-1) (d-2) \over 2} - \delta > 3\), then \(\text{Pic}_ \mathbb{Q} (V_{d, \delta}) = 0\) if and only if \(\text{Pic}_ \mathbb{Q} ({\mathcal M}_ g) = \mathbb{Q}\).
Since \(\text{Pic}_ \mathbb{Q} ({\mathcal M}_ g)=\mathbb{Q}\) by Harer's theorem [\textit{J. Harer}, Invent. Math. 72, 221-239 (1983; Zbl 0533.57003)] this theorem answers, for \(\delta\) large with respect to \(d\), a conjecture of Harris [cf. \textit{S. Diaz} and \textit{J. Harris}, Trans. Am. Math. Soc. 309, No. 1, 1-34 (1988; Zbl 0677.14003)]\ in the affirmative. Severi variety; rational Picard group D. Edidin, Picard groups of Severi varieties, 22 (1994), 2073-2081. Picard groups, Families, moduli of curves (algebraic) Picard groups of Severi 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 If \(V_{d, \delta}\) denotes the variety of irreducible plane curves of degree \(d\) with exactly \(\delta\) nodes as singularities, \textit{S. Diaz} and \textit{J. Harris} [Trans. Am. Math. Soc. 309, No. 1, 1--34 (1988; Zbl 0677.14003) and in Algebraic Geometry, Proc. Conf., Sundance 1986, Lect. Notes Math. 1311, 23--50 (1988; Zbl 0677.14004)] have conjectured that \(\text{Pic}(V_{d, \delta})\) is a torsion group. In this note we study rational equivalence on some families of singular plane curves and we prove, in particular, that \(\text{Pic}(V_{d,1})\) is a finite group, so that the conjecture holds for \(\delta=1\). Actually the order of \(\text{Pic}(V_{d,1})\) is \(6(d-2) (d^ 2 - 3d + 1)\), the group being cyclic if \(d\) is odd and the product of \(\mathbb Z_ 2\) and a cyclic group of order \(3(d-2) (d^ 2 - 3d + 1)\) if \(d\) is even. Picard groups; intersection rings; rational equivalence on families of singular plane curves J. M. Miret and S. Xambó-Descamps, Rational equivalence on some families of plane curves, Ann. Inst. Fourier (Grenoble) 44 (1994), no. 2, 323 -- 345 (English, with English and French summaries). Families, moduli of curves (algebraic), Picard groups, (Equivariant) Chow groups and rings; motives, Projective techniques in algebraic geometry, Enumerative problems (combinatorial problems) in algebraic geometry, Singularities of curves, local rings Rational equivalence on some families of plane curves | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians We introduce and study the GIT CONE of \(\bar{M}_{0,n}\), which is generated by the pullbacks of the natural ample line bundles on the GIT quotients \((\mathbb P^1)^n//\mathrm{SL}(2)\). We give an explicit formula for these line bundles and prove a number of basic results about the GIT cone.
As one application, we prove unconditionally that the log canonical models of \(\bar{M}_{0,n}\) with a symmetric boundary divisor coincide with the moduli spaces of weighted curves or with the symmetric GIT quotient, extending the result of \textit{M. Simpson} [``On log canonical models of the moduli space of stable pointed curves'', \url{arXiv:0709.4037}] (see also a different proof by \textit{M. Fedorchuk} and \textit{D. I. Smyth} [J. Algebr. Geom. 20, No. 4, 599--629 (2011; Zbl 1230.14034)]). Alexeev, Valery; Swinarski, David, Nef divisors on \(\overline{M}_{0, n}\) from GIT, (Geometry and Arithmetic, EMS Ser. Congr. Rep., (2012), Eur. Math. Soc. Zürich), 1-21 Geometric invariant theory, Families, moduli of curves (algebraic), Divisors, linear systems, invertible sheaves, Picard groups Nef divisors on \(\bar{M}_{0,n}\) from GIT | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is 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 smooth curves of genus \(g\), and let \(\overline{\mathcal M}_{g}\) be the Deligne-Mumford compactification of \({\mathcal M}_{g}\). The slope \(s(D)\) of an effective divisor \(D\) in \(\overline{\mathcal M}_{g}\) is defined as the smallest rational number \(a/b \geq 0\) such that the divisor \(a\lambda-b(\delta_{0}+\delta_{1}+\dots \delta_{[g/2]})-[D]\) is an effective combination of the boundary divisor classes \(\delta_{0}, \delta_{1}, \dots ,\delta_{[g/2]}\); here \(\lambda\) is the class of the Hodge line bundle. The slope \(s_{g}\) of the moduli space \(\overline{\mathcal M}_{g}\) is defined as \(s_{g}:=\inf \{s(D): D\in \text{Eff}(\overline {\mathcal M}_{g})\}\).
The slope conjecture of Harris and Morrison predicts that \(s_{g}\geq 6+12/g+1\). The slope conjecture was shown to be true for all \(g\leq 12\), \(g\neq 10\) by \textit{J. Harris} and \textit{I. Morrison} [Invent. Math. 99, 321--355 (1990; Zbl 0705.14026)] and \textit{S.-L. Tan} [Int. J. Math. 9, 119--127 (1998; Zbl 0930.14017)].
The aim of the paper under review is to prove two statements: first that the Harris-Morrison slope conjecture fails to hold on \(\overline{\mathcal M}_{10}\) and second, that in order to compute the slope of \(\overline{\mathcal M}_{g}\) for \(g \leq 23\), one only has to look at the coefficients of the classes \(\lambda\) and \(\delta_{0}\) in the standard expansion in terms of the generators of the Picard group. The authors use the fact that the condition that a smooth curve of genus \(g\) lies on a \(K3\) surface is divisorial (only) for \(g=10\), therefore one obtains an effective divisor \(K\) on the moduli space \({\mathcal M}_{10}\). Then the authors compute the class of the closure \(\overline{K}\) of \(K\) in \(\text{Pic}(\overline{\mathcal M}_{10})\), obtaining the following formula:
\[
[\overline{K}]=7\lambda-\delta_{0}-5\delta_{1}-9\delta_{2}-12\delta_{3}-14\delta_{4}-B_{5}\delta_{5},
\]
with \(B_{5}\geq 6\). The first two coefficients in the above formula were computed by \textit{F. Cukierman} and \textit{D. Ulmer} [Compos. Math. 89, No. 1, 81--90 (1993; Zbl 0810.14012)]. From the formula one computes \(s(\overline{K})=7\), which is strictly smaller than the bound \(78/11\) predicted by the slope conjecture, so \(\overline{K}\) gives the promised counterexample.
To compute the class of \(\overline{K}\) the authors show that \(K\) has four incarnations as a geometric subvariety of the moduli space \({\mathcal M}_{10}\). In particular, \(K\) can be thought of as either (1) the locus of curves \([C] \in {\mathcal M}_{10}\) for which the rank 2 Mukai type Brill-Noether locus \(\{E \in \text{SU}_{2}(C, K_{C}): h^{0}(C,E)\geq 7\}\) is non empty, or (2) the locus of curves \([C] \in {\mathcal M}_{10}\) with a non-surjective Wahl map \(\Psi_{K_{C}}: \Lambda^{2}H^{0}(C,K_{C}) \to H^{0}(C,3K_{C})\) (this second characterization is due to F. Cukierman and D. Ulmer). Note that these characterizations of \(K\), unlike the original definition of \(K\), can be extended to other genera \(g \geq 13\). Finally, the authors give a counterexample to a hypothesis formulated by Harris and Morrison that the Brill-Noether divisors are essentially the only effective divisors on the moduli space of curves having minimal slope \(6+12/g+1\). Farkas, Gavril; Popa, Mihnea, Effective divisors on \(\overline{\mathcal{M}}_g\), curves on \(K3\) surfaces, and the slope conjecture, J. Algebraic Geom., 14, 2, 241-267, (2005) Families, moduli of curves (algebraic), Special divisors on curves (gonality, Brill-Noether theory), \(K3\) surfaces and Enriques surfaces, Algebraic moduli problems, moduli of vector bundles, Picard groups Effective divisors on \(\overline{\mathcal M}_g\), curves on \(K3\) surfaces, and the slope conjecture | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians The spin moduli space \(S_ g\) in genus \(g\) is the space parametrizing all couples (smooth genus \(g\) algebraic curve \(C\), theta characteristic on \(C)\). It has a natural structure of an algebraic variety and a well behaved compactification of \(S_ g\) was introduced by the author in ``Moduli of curves and theta characteristics'' (Lectures on Riemann surfaces, World Scientific, Singapore 1989). There were described also natural classes in the Picard group of this compactification. The answer to the problem whether these classes generate the Picard group is still not known. The author gives a complete answer to the question of what relations they satisfy. Finally it is shown that the Picard group of spin moduli space contains 4 torsion. generation of Picard group; spin moduli space; algebraic curve; theta characteristic M. Cornalba, A remark on the Picard group of spin moduli space,Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei (9) Mat. Appl. 2 (1991), 211--217. Families, moduli of curves (algebraic), Picard groups, Algebraic moduli problems, moduli of vector bundles, Theta functions and abelian varieties A remark on the Picard group of spin moduli space | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians We study a compactification of the variety \(U_{d,m}, 1<m<d\), of plane curves of degree \(d\) with an ordinary singular point of multiplicity \(m\) as a unique singularity, given by means of a projective bundle \(X_{d,m}\). We show that the boundary \(X_{d,m}-U_{d,m}\) consists of two irreducible components of codimension 1: \(A_{d,m}\), the closure of the variety of curves with two singular points, one of multiplicity \(m\) and another of multiplicity 2, and \(B_{d,m}\), the closure of the variety of curves with a non-ordinary singular point of multiplicity \(m\). We determine the relations that express the classes of \(A_{d,m}\) and \(B_{d,m}\) in terms of a basis of the group \(\text{Pic}(X_{d,m})\). From this we describe the Picard group of the variety \(U_{d,m}\), obtaining that it is a finite group of order \(3(m-1)(d-m)[2d^{2}-(m+4)d-(m^{2}-2m-2)]\). Picard group; intersection product Picard groups, Families, moduli of curves (algebraic), Intersection theory, characteristic classes, intersection multiplicities in algebraic geometry, Plane and space curves, Enumerative problems (combinatorial problems) in algebraic geometry On the variety of plane curves of degree \(d\) with a singular point of multiplicity \(m\) | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians The Picard sheaves have been studied from differents aspects. \textit{Y. Li} [Int. J. Math. 2, 525-550 (1991; Zbl 0751.14019)] proves the stability of the Picard bundle \({\mathcal W}\) over the moduli space \({\mathcal M}(n,d)\) of stable bundles of rank \(n\) and degree \(d\). In general the restrictions of stable bundles need not be stable. In this paper we study the restriction \({\mathcal W}_\xi\) of the Picard bundle \({\mathcal W}\) to the subvariety \({\mathcal M}(n,\xi)\) of stable bundles with fixed determinant \(\xi\). We give a condition to get polystability. If such a condition is satisfied for rank 2 then \({\mathcal W}_\xi\) is stable and the connected component of the moduli space of stable bundles over \({\mathcal M}(2,\xi)\) with the same Hilbert polynomial as \({\mathcal W}_\xi\) containing \({\mathcal W}_\xi\) is isomorphic to the Jacobian \(J\) of the curve. Picard sheaves; polystability; moduli space of stable bundles; Jacobian Brambila-Paz L, Hidalgo-Solís L and Muciño-Raymondo J, On restrictions of the Picard bundle. Complex geometry of groups (Olmué, 1998) 49--56;Contemp. Math. 240;Am. Math. Soc. (Providence, RI) (1999) Vector bundles on curves and their moduli, Picard groups, Fine and coarse moduli spaces, Families, moduli of curves (algebraic) On restrictions of the Picard bundle | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians We compute the Picard group of the moduli stack of smooth curves of genus \(g\) for \(3\le g\le 5\), using methods of equivariant intersection theory. We base our proof on the computation of some relations in the integral Chow ring of certain moduli stacks of smooth complete intersections. As a byproduct, we compute the cycle classes of some divisors on \(\mathcal{M}_g\). Families, moduli of curves (algebraic), (Equivariant) Chow groups and rings; motives, Picard groups, Arithmetic ground fields for curves, Positive characteristic ground fields in algebraic geometry Picard group of moduli of curves of low genus in positive characteristic | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians Denote by \(M_g\) the coarse moduli space of smooth complex projective curves of genus \(g\) and let \(\overline M_g\) be its Deligne-Mumford compactification by means of stable curves. Although the rigorous construction of these classifying spaces was only achieved as late as in the 1960s, the existence of \(M_g\) was intuitively taken for granted since B. Riemann's pioneering work on Abelian functions in 1857. In fact, various geometric properties of the space \(M_g\) have been established already in the late nineteenth and early twentieth century, mainly by diverse renowned exponents of the schools of algebraic geometry in Italy and in Germany. Especially the determination of the birational type of the space \(M_g\) has been a subject of intensive study during the last 100 years. As early as in 1915, F. Severi proved that \(M_g\) is unirational for \(g\leq 10\) and he made the conjecture that \(M_g\) is unirational (or even rational!) for all genera \(g\).
Severi's conjecture remained as an open problem until 1982, when J. Harris and D. Mumford showed that \(M_g\) is a variety of general for \(g\geq 24\) thereby proving that Severi's conjecture is maximally wrong for sufficiently large genera. In the sequel, various partial results have been obtained as for the intermediate, still unsettled cases \(11\leq g\leq 23\). More precisely, the unirationality of the moduli spaces \(M_g\) for \(11\leq g\leq 14\) could be verified by \textit{E. Sernesi} [Ann. Sc. Norm. Super. Pisa, Cl. Sci., IV. Ser. 8, 405--439 (1981; Zbl 0475.14024), \textit{M. C. Chang} and \textit{Z. Ran} (1984), and \textit{A. Verra} [Compos. Math. 141, No. 6, 1425--1444 (2005; Zbl 1095.14024)], respectively. On the other hand, M. C. Chang and Z. Ran also proved that the Kodaira dimension of the spaces \(M_{15}\) and \(M_{16}\) is \(-\infty\).
The paper under review provides a very comprehensive and detailed overview of these fascinating developments and results, with some improvements of arguments and alternative proofs. Moreover, as a new additional result in this context, the author gives a proof of the fact that the moduli spaces \(M_{22}\) and \(\overline M_{22}\) are again of general type. As for the precise contents, the present article consists of seven sections treating the following topics:
After an utmost profound and enlightening introduction, with numerous historical remarks and bibliographic references, Section 2 gives a description of various approaches to prove the unirationality of \(M_g\), ranging from Severi's work to Verra's recent results on the birational geometry of \(M_g\) for \(g\leq 14\) [cf. \textit{A. Verra}, Compos. Math. 141, No. 6, 1425--1444 (2005; Zbl 1095.14024)]. Section 3 discusses the structure of the Picard group of the moduli stack \(\overline{\mathcal M}_g\) whereas Section 4 explains the calculation of the canonical class of the moduli space \(M_g\) in terms of the generators of \(\text{Pic}(\overline{\mathcal M}_g)\). This section also exhibits some slope estimates for \(\overline M_g\) together with their relations Gromov-Witten theory. Chapter 6 presents another novelty, namely a different proof of the Eisenbud-Harris-Mumford theorem stating that the moduli space \(M_g\) is of general type for genus \(g\geq 23\). Using his method of syzygies of canonical curves (instead of admissible covers, Hurwitz divisors, or limit linear series), the author is able to derive a much shorter, more direct and technically simpler proof of this meanwhile classical theorem. Section 6 is devoted to a systematic study of the geometry of curves lying on a \(K3\) surface, which is then used to construct effective divisors on \(\overline M_g\) of small slope, on the one hand, and to compute the class of certain effective divisors on \(M_g\) by means of Koszul cohomology of line bundles on curves [cf. \textit{G. Farkas}, Duke Math. J. 135, No. 1, 53--98 (2006; Zbl 1107.14019)] on the other.
Finally, the special case of \(g= 22\) is investigated in Section 7, where the author calculates the class of a particular, geometrically defined Koszul divisor \(\overline D_{22}\) on \(\overline M_{22}\). According to the previous results developed in Section 6, this original andsubtle calculation implies the fact that the moduli space \(\overline M_{22}\) is of general type.
As for complete details of a more general construction within this framework, the reader is referred to the author's recent paper ``Koszul divisors on moduli spaces of curves'' [Am. J. Math. 131, No. 3, 819--867 (2009; Zbl 1176.14006)].
All together, this masterly written treatise contains a wealth of (old and new) material about the birational geometry of moduli spaces of curves, including numerous novel ideas and methods developed by the author himself during the last few years. moduli spaces of curves; rationality questions; special divisors; Brill-Noether theory; unirational varieties; varieties of general type; Koszul cohomology; \(K3\) surfaces \textsc{G. Farkas}, Birational aspects of the geometry of \(M_{g}\), In: Surveys in Differential Geometry. Vol. XIV. Geometry of Riemann Surfaces and Their Moduli Spaces, 57--110 Surv. Differ. Geom., vol. 14, Int. Press, Somerville, MA, 2009 Families, moduli of curves (algebraic), Rationality questions in algebraic geometry, Special divisors on curves (gonality, Brill-Noether theory), Divisors, linear systems, invertible sheaves, Picard groups, \(K3\) surfaces and Enriques surfaces Birational aspects of the geometry of \(\overline{\mathcal M}_g\) | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians Let \(M_ g\) be the moduli space of genus g smooth curves (g\(\geq 3)\), \(k_ g\) the function field and \({\mathcal C}_ g\) the universal curve over \(k_ g\). The Picard group Pic(\({\mathcal C}_ g)\) is generated by the canonical divisor class. This is A. Franchetta's conjecture [\textit{A. Franchetta}, Matematiche 9, 126-147 (1954; Zbl 0057.129)] which is proved as a consequence of J. Harer's results [\textit{J. Harer}, Invent. Math. 72, 221-239 (1983; Zbl 0533.57003)]. The aim of this paper is to prove that the rational points group \underbar{Pic}\({}_{{\mathcal C}_ g/k_ g}(k_ g)\) of the Picard scheme \underbar{Pic}\({}_{{\mathcal C}_ g/k_ g}\) of \({\mathcal C}_ g\) is also generated by the canonical divisor class. We use Franchetta's conjecture and canonical singular reducible curves of arithmetical genus g in \({\mathbb{P}}^{g-1}\) already used by \textit{F. Enriques} and \textit{O. Chisini} when they proved that the degree of any divisor on \({\mathcal C}_ g\) is a multiple of 2g-2 [''Lezioni sulla teoria geometrica delle equazione e delle funzioni algebriche'' (Bologna 1915 and 1918)]. moduli space of smooth curves; rational points group of the Picard scheme; canonical divisor class DOI: 10.1007/BF01389421 Picard groups, Families, moduli of curves (algebraic), Rational points Conjecture de Franchetta forte. (Strong Franchetta conjecture) | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians We denote by \({\mathcal M}^ 0_ g\) the moduli space of smooth curves of genus \(g\) \((g\geq 3)\) without automorphisms, and by \({\mathcal C}_ g@>\pi>>{\mathcal M}^ 0_ g\) the universal curve over \({\mathcal M}^ 0_ g\). For any integer \(d\), we denote by \(\psi_ d:{\mathcal T}^ d_ g\to{\mathcal M}^ 0_ g\) the universal Picard (Jacobian) variety of degree \(d\); the fiber \(J^ d(C)\) over a point \([C]\) of \({\mathcal M}^ 0_ g\) parametrizes line bundles on \(C\) of degree \(d\), modulo isomorphism. We describe a group \({\mathcal N}({\mathcal T}^ d_ g)\) (which we call the relative Néron-Severi group of \({\mathcal T}^ d_ g)\) defined to be the group of line bundles on \({\mathcal T}^ d_ g\), modulo the relation that two line bundles are equivalent if their restrictions to the fibers of the map \(\psi_ d\) are algebraically equivalent.
Lemma: The Néron-Severi group of the Jacobian of a curve \(C\) with general moduli is generated by the class \(\theta\) of its theta divisor.
We can define an embedding of groups \(\varphi_ d:{\mathcal N}({\mathcal T}^ d_ g)\hookrightarrow\mathbb{Z}\). To describe the group \({\mathcal N}({\mathcal T}^ d_ g)\) is equivalent to finding the generator \(k^ d_ g\) of the image of the map \(\varphi_ d\).
Theorem: For \(d=0,\ldots,g-1\) the numbers \(k^ d_ g\) are given by the following formula: \(k^ d_ g=(2g-2)/\text{g.c.d}(2g-2,g+d-1)\). universal Picard variety; universal Jacobian variety; moduli space of smooth curves; relative Néron-Severi group Kouvidakis, A., The Picard group of the universal Picard varieties over the moduli space of curves, J. Differential Geom., 34, 3, 839-850, (1991) Picard groups, Jacobians, Prym varieties, Families, moduli of curves (algebraic) The Picard group of the universal Picard varieties over 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 The algebraic curves of degree \(d\) in \(\mathbb P^2\) are parametrized by a projective space \(\mathbb P^N\). Denote by \(V=V(d,\delta)\) the locally closed subspace of \(\mathbb P^N\) parametrizing the nodal curves of degree \(d\) and geometric genus \(g\), where each such curve is singular at exactly \(\delta =1/2 (d-1)(d-2)-g\) nodes. The classical Severi problem was to show that \(V\) is irreducible when nonempty; it was solved several years ago by \textit{J. Harris} [Invent. Math. 84, 445--461 (1986; Zbl 0596.14017)]. Given the irreducibility, we can investigate the structure of suitable full or partial compactifications of \(V\); these are called Severi varieties. The closure in \(\mathbb P^N\), of course, is one, but the boundary of \(V\) in this closure seems forbiddingly complex, because our knowledge of the possible degenerations is quite incomplete. Therefore attention has turned to more accessible possibilities. The object of this fine paper [and its sequel, part II of this paper in Algebraic geometry, Proc. Conf., Sundance Utah 1986, Lect. Notes Math. 1311, 23--50 (1988; Zbl 0677.14004)] is to study, instead, a partial compactification, denoted \(W\), of \(V\), obtained by adjoining the codimension 1 equisingular strata of the closure of \(V\) in \(\mathbb P^N\), and then normalizing. First the authors show that \(W\) is smooth, and then they work out some natural relations in \(\operatorname{Pic}(W)\), using techniques inspired in part by the enumerative geometry of moduli spaces.
Based on the authors' previous results [cf. ``Ideals associated to deformations of singular plane curves'', Trans. Am. Math. Soc. 309, No. 2, 433--468 (1988; Zbl 0707.14022)], the image of \(W\) in \(\mathbb P^N\) is the union of \(V\) and the following codimension 1 strata: CU, the locus of reduced and irreducible curves of genus \(g\) with \(\delta-1\) nodes and one cusp; TN, the locus of reduced and irreducible curves of genus \(g\) with \(\delta-2\) nodes and one tacnode; TR, the locus of reduced and irreducible curves of genus \(g\) with \(\delta-3\) nodes and one ordinary triple point; and \(\Delta\), the locus of reduced curves of geometric genus \(g-1\) with at most two irreducible components and \(\delta+1\) nodes.
Of these four codimension one subschemes of the closure of \(V\), the authors suspected (but did not know then) that the first three are irreducible. After the present article appeared, their irreducibility was established, through a generalization of Harris' irreducibility theorem, by \textit{Ziv Ran} [``Families of plane curves and their limits: Enriques' conjecture and beyond'', Ann. Math. (2) 130, No. 1, 121--157 (1989; Zbl 0704.14018)]. In contrast, \(\Delta\) is reducible, and its components play important parts in the theory.
A complete set of generators for \(\operatorname{Pic}(W)\) is at present unknown, but there are many natural candidates, among them the boundary components mentioned already. The loci of curves with hyperflexes, with flecnodes, with flex bitangents, with nodal tangents tangent to the curve elsewhere, with binodal tangents, with tritangent lines, with three collinear nodes, also come into consideration. Their classes are extrinsic, because they reflect properties of the embeddings of the given curves in the plane. Other potential generators (also extrinsic) depend on the position of given curves relative to points and lines in \(\mathbb P^2\), and so are important in classical enumerative geometry. These include the divisor of curves through a given point, those tangent to a given line, those with a node on a given line, and so on.
We can also consider several intrinsic divisor classes, analogous to the intrinsic classes on the moduli space of stable curves. Over \(W\), there is a universal family \(\mathcal C\) of curves of genus \(g\), and we have two natural divisor classes on \(\mathcal C\): the first Chern class of the relative dualizing sheaf of \(\mathcal C/W\), and the pullback of \(c_1(\mathcal O_{\mathbb P^2}(1))\). We obtain three classes, A, B and C, in \(\operatorname{Pic}(W)\) by taking pairwise products on \(\mathcal C\) and then pushing down.
These intrinsic classes bring some order into the seeming chaos of potential generators, at least over \(\mathbb Q\), because each extrinsic class described above can be expressed (as the authors show) as a rational linear combination of A, B, and C, together with either \(\Delta\) or suitable components of \(\Delta\). As a corollary, the authors show that the original variety \(V\) of node curves is affine. Further, they conjecture that A, B, C, and the components of \(\Delta\) generate \(\operatorname{Pic}(W)\) over \(\mathbb Q\). This conjecture is equivalent to the assertion that \(\operatorname{Pic}(V)\) is torsion.
Partial compactifications of spaces of plane curves are useful for solving enumerative problems [see for example \textit{S. L. Kleiman} and the reviewer ``Enumerative geometry of nodal plane cubics'' in Algebraic geometry, Proc. Conf., Sundance/Utah, Lect. Notes Math. 1311, 156--196 (1988; Zbl 0678.14013) and ``Enumerative geometry of nonsingular plane cubics'', Algebraic geometry, Proc. Conf. Sundance/Utah 1988, Contemp. Math. 116, 85--113 (1991; Zbl 0753.14045)]. In the latter, we need a different partial compactification, here denoted \(W+\), of the variety \(V\) of nonsingular cubics, in order to obtain proper intersections with the variety of cubics tangent to a given line, because the dual curve also needs to be parametrized by the partial compactification. On \(W^+\), compared to the Severi variety W which it dominates, the locus of triple line degenerations blows up to a divisor. This gives an additional boundary component, crucial for checking the classical enumerative results. Much of the divisor structure of \(W^+\) is reflected in \(W\), however, and this will hold more generally as we pass to plane curves of higher degree.
This article represents a valuable step forward in the study of plane curves, not just for its results, but for the links it provides between the geometry of moduli, the study of the authors' Severi varieties, and the structure of the somewhat more elaborate parameter spaces needed for an enumerative geometry, envisaged classically but still not carried out, for plane curves of any degree. enumerative geometry of moduli spaces of curves; generators of Picard group; nodal curves; Severi problem; Severi varieties; divisor classes S. Diaz - J. Harris, Geometry of Severi varieties, Trans. Amer. Math. Soc. 309 (1988) 1-34. Zbl0677.14003 MR957060 Families, moduli of curves (algebraic), Picard groups, Enumerative problems (combinatorial problems) in algebraic geometry, Curves in algebraic geometry, Divisors, linear systems, invertible sheaves, Projective techniques in algebraic geometry Geometry of the Severi variety | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians In this paper, the author provides an explicit description of all components of the Brill-Noether loci for a general \(k\)-gonal curve. Having such a curve with its map to the projective line, the push forward of the corresponding line bundle is a rank \(k\) vector bundle. It is shown that the components are arised from the stratification of the Picard variety by the splitting type of the push-forward bundle. The main idea then to show that these strata are smooth of expected dimension is degeneration to a chain of elliptic curves, each together with a degree \(k\) map to the projective line. Brill-Noether loci; \(k\)-gonal curve; degeneration Divisors, linear systems, invertible sheaves, Picard groups, Riemann-Roch theorems, Families, moduli of curves (algebraic), Elliptic curves, Subvarieties of abelian varieties A refined Brill-Noether theory over 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 The paper contains new results on the classification of curves in the blowing up \(S_r\) of the projective plane at \(r\) general points (any characteristic). The author considers curves of arithmetic genus \(p_a\geq 2\), whose class \(D\) in the Picard group has \(\chi(D)>0\). He shows that for any \(p_a\) there are only finitely many multi-integers \((a_0,\dots,a_r)\) with \(a_i\geq 2\) for each \(i\), corresponding to a class of genus \(p_a\) in the natural identification \(\text{Pic} (S_r)={\mathbb Z}^{r+1}\).
For \(p_a=2\) and \(p_a=3\), the author presents a classification of curves of genus \(p_a\), modulo the Weyl group of automorphisms of \(S_r\). He shows that for any allowed multi-integer there exist reduced, irreducible curves in the corresponding class; he also proves the non-singularity of these curves, when expected, except for one case in genus \(3\). curves on rational surfaces; Picard group Families, moduli of curves (algebraic), Rational and ruled surfaces, Picard groups Classification of curves on generic rational surfaces | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians The first edition of this well-known and very popular standard text on compact complex surfaces was published in 1984, back then written by \textit{W. Barth}, \textit{C. Peters} and \textit{A. Van de Ven} (1984; Zbl 0718.14023). Apart from the modern treatises on complex surfaces by \textit{I. R. Shafarevich}, \textit{B. G. Averbukh}, \textit{J. R. Vajnberg}, \textit{A. B. Zizhchenko}, \textit{Yu. I. Manin}, \textit{B. G. Mojshezon}, \textit{G. N. Tyurina}, \textit{A. N. Tyurin} and \textit{A. Beauville} [Astérisque 54, 1--172 (1978; Zbl 0394.14014)], the first edition of the book under review offered the only up-to-date account of the subject in textbook form. Moreover, for almost twenty years it has been by far the most comprehensive textbook on complex surfaces from the modern point of view. The first edition contained eight main chapters on about 300 pages, concluding with the classification of K3 surfaces and Enriques surfaces.
The book under review is the second, substantially enlarged edition of this standard text, this time with K. Hulek as fourth co-author. In fact, in the two decades after the appearance of the first edition of the book, several crucial developments in the theory of complex surfaces have taken place, and the authors have taken the opportunity to update the original text by including some of those recent achievements.
The most important progress in the theory of complex surfaces has been made in regard of a better understanding of their differentiable structure (as real 4-manifolds), not at least in view of their appearance in mathematical physics. The new invariants discovered by Donaldson, on the one hand, and by Seiberg and Witten, on the other hand, stand for these spectacular recent developments. Other far-reaching achievements have been obtained by means of the study of nef-divisors on surfaces, parallel to progress made in the birational classification of higher-dimensional algebraic varieties, and also the Kähler structures on complex surfaces are now much better understood. Finally, I. Reider's new approach to adjoint mappings, Bogomolov's inequality for the Chern classes of rank-2 vector bundles on surfaces, and the mirror symmetry properties of K3 surfaces represent just as important new insights in the theory of complex surfaces.
Well, all these recent developments have been worked into the present new edition, in one way or another, and the text has grown to 436 pages, that is by more than forty percent. Apart from the correction of some minor irregularities in the first edition, the authors have left the well-proven original text basically intact. The enlargement of the material has been contrived by the addition of the new Chapter 9 entitled ``Topological and Differentiable Structure of Surfaces'', including an introduction to Donaldson and Seiberg-Witten invariants (mainly by instructive examples), and by substantially enhancing Chapter 4 (``Some General Properties of Surfaces''). This chapter comes now with twelve (instead of eight) sections and includes the above-mentioned topics such as the nef cone, Bogomolov's inequality, Reider's method, and the existence of Kähler metrics on surfaces. There are also some other refining polishings and rearrangements in Chapter 5 (``Examples'') and Chapter 8 (``K3-Surfaces and Enriques Surfaces''). As to the contents of Chapter 8, three sections on special topics have been added, too, discussing the mirror symmetry phenomenon for projective K3-surfaces, special curves on K3-surfaces and an application to hyperbolic geometry, respectively.
Needless to say, the bibliography has been updated and tremendously enlarged, thereby reflecting the vast activity in the field during the past twenty years.
Now as before, the text is enriched by numerous instructive examples, but there are still no exercises for self-control, stimulus for further reading, or different outlooks.
All in all, the second, enlarged edition of this (meanwhile classic) textbook on complex surfaces has gained a good deal of topicality and disciplinary depth, while having maintained its high degree of systematic methodology, lucidity, rigor, and cultured style. No doubt, this book remains a must for everyone dealing with complex algebraic surfaces, be it a student, an active researcher in complex geometry, or a mathematically ambitioned (quantum) physicist. complex algebraic surfaces; classification of algebraic surfaces; deformation theory; topology of surfaces; fibrations; surfaces of general type; K3-surfaces; Enriques surfaces; period mappings; period domains; Donaldson invariants; Seiberg-Witten invariants Barth, W. P.; Hulek, K.; Peters, C. A. M.; Ven, A. van de., \textit{Compact Complex Surfaces}, Vol. 4 of \textit{Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge}, Springer-Verlag, Berlin Moduli, classification: analytic theory; relations with modular forms, Research exposition (monographs, survey articles) pertaining to algebraic geometry, Research exposition (monographs, survey articles) pertaining to several complex variables and analytic spaces, Compact complex surfaces, Transcendental methods of algebraic geometry (complex-analytic aspects), Picard groups, Families, moduli of curves (algebraic), Complex-analytic moduli problems, \(K3\) surfaces and Enriques surfaces, Topology of surfaces (Donaldson polynomials, Seiberg-Witten invariants), Calabi-Yau manifolds (algebro-geometric aspects) Compact complex surfaces | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians Denote by \(M_{g,n}\) the moduli space of \(n\)-pointed smooth algebraic curves over the field of complex numbers, and by \(\overline M_{g,n}\) its compactification via stable curves of this type. Let \({\mathcal M}_{g,n}\) and \(\overline{\mathcal M}_{g,n}\) be the respective moduli stacks.
Over the past thirty years, the study of the enumerative geometry of the spaces \(M_{g,n}\) and \(\overline M_{g,n}\) has played a central role in the classification theory of algebraic curves, in connection with which the calculation of the rational (co)homology of these spaces proved to be of crucial significance. In this regard, the computation by \textit{J. Harer} [Invent. Math. 72, 221--239 (1983; Zbl 0533.57003)] of the second homology groups of \(M_{g,0}\), accomplished already in the early 1980s by using rather intricate topological and analytical methods of Teichmüller theory, was certainly a major step forward in the moduli theory of algebraic curves and compact Riemann surfaces. However, Harer's approach relies entirely on transcendental concepts and tools, without reflecting the algebro-geometric nature of his result as distinct as it would be desirable.
In view of this peculiar fact, the authors of the article under review explain the possibility to reduce the transcendental part of Harer's involved calculation to another important result of his, thereby providing an alternative proof of Harer's theorem on the second homology groups of \(M_{g,0}\), which appears much more algebro-geometric in nature. More precisely, \textit{J. L. Harer's} so-called homological vanishing theorem [Invent. Math. 84, 157--176 (1986; Zbl 0592.57009)] asserts that the homology of \(M_{g,n}\) vanishes above a certain explicit degree, \(c(g,n)\). This result, a proof of which is outlined in the present survey, is combined with a series of additional algebro-geometric arguments to obtain an explicit calculation of the cohomology groups \(H^1(M_{g,n};\mathbb{Q})\) and \(H^2(M_{g,n};\mathbb{Q})\) in various cases with respect to the parameters \(g\) and \(n\). Although these are meanwhile classical results [\textit{D. Mumford}, J. Anal. Math. 18, 227--244 (1967; Zbl 0173.22903); \textit{J. Harer}, Invent. Math. 72, 221--239 (1983; Zbl 0533.57003)], the authors' approach sheds some important new light on the overall picture, first and foremost from the viewpoint of algebraic geometry.
In addition, the authors also describe how the first and second rational cohomology groups of the moduli spaces \(\overline M_{g,n}\) of stable \(n\)-pointed curves, of genus \(g\) can be calculated. Here they follow the presentation of an earlier work of theirs [Publ. Math., Inst. Hautes Étud. Sci. 88, 97--127 (1998; Zbl 0991.14012)], in which the third and the fifth cohomology groups were also exhibited and shown to always vanish.
As for the basic, largely new methods described in the paper under review, the authors investigate the boundary strata in \(\overline M_{g,n}\) by means of the corresponding moduli stack \(\overline{\mathcal M}_{g,n}\) and certain graphs attached to stable \(n\)-pointed curves, study then some natural (or tautological) classes in the cohomology ring of \(\overline{\mathcal M}_{g,n}\) and various relations satisfied by them, and finally establish a version of Deligne's ``Gysin spectral squence'' in Hodge theory for the pair consisting of \(\overline M_{g,n}\) and its boundary \(\partial M_{g,n}\) to calculate the cohomology of the open variety \(M_{g,n}=\overline M_{g,n}\setminus M_{g,n}\) in terms of the cohomology of the boundary, strata described before. In the special case of genus \(g= 0\), S. Keel's approach to calculate the Chow ring \(A^*(\overline M_{0,n})= H^*(\overline M_{0,n}; \mathbb{Z})\) is suitably modified (and simplified) in order to deal with the relevant divisor classes in the cotext of the present article.
It should be pointed out that most of the material discussed in the present, highly enlightening survey article can also be found, in the meantime, in the recent monograph ``Geometry of algebraic curves. Volume II'' by \textit{E. Arbarello}, \textit{M. Cornalba} and \textit{P. A. Griffiths} [Grundlehren der Mathematischen Wissenschaften 268. Berlin: Springer (2011; Zbl 1235.14002)], where Chapter XIX is particularly devoted to the (co)homology of moduli spaces of curves, though in a much wider context, in greater generality and in a more elaborate presentation. algebraic curves; Riemann surfaces; moduli spaces; cohomology rings; Chow rings; divisor classes; homology groups; Picard groups; Teichmüller theory; Gysin spectral sequence Arbarello E., Grundlehren der Mathematischen Wissenschaften 268, in: Geometry of Algebraic Curves (2010) Families, moduli of curves (algebraic), (Equivariant) Chow groups and rings; motives, Divisors, linear systems, invertible sheaves, Picard groups, Other algebro-geometric (co)homologies (e.g., intersection, equivariant, Lawson, Deligne (co)homologies), Teichmüller theory for Riemann surfaces Divisors in the moduli spaces of curves | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians We compute the Picard group of the moduli stack of stable hyperelliptic curves of any genus, exhibiting explicit and geometrically meaningful generators and relations. moduli; hyperelliptic curves; Picard group Cornalba, M., The Picard group of the moduli stack of stable hyperelliptic curves, Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei (9) Mat. Appl., 18, 1, 109-115, (2007) Families, moduli of curves (algebraic), Algebraic moduli problems, moduli of vector bundles, Picard groups, Vector bundles on curves and their moduli The Picard group of the moduli stack of stable hyperelliptic curves | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians Let us recall that the mapping class group \(\mathop{Mod}_g\) is the group of isotopy classes of orientation preserving homeomorphisms of a fixed closed oriented genus \(g\) surface \(\Sigma_g\). The paper under review deals with the homology of finite-index subgroups \(\Gamma\) of \(\mathop{Mod}_g\) containing the Torelli group \(\mathcal{I}_g\) of homeomorphisms that act trivially on the first homology group of \(\Sigma_g\). In view of the construction of the moduli space of smooth complex projective curves as a quotient of Teichmüller space by the action of \(\mathop{Mod}_g\), these homology groups can also be interpreted geometrically as the homology groups of finite covers of the moduli space of curves, i.e. moduli spaces of curves with level structures.
The main result of the paper under review is that the second homology group of \(\Gamma\) with rational coefficients is always one-dimensional for \(g\geq 5\). In view of \textit{R. Hain}'s results [In: Current topics in complex algebraic geometry. New York, NY: Cambridge University Press. Math. Sci. Res. Inst. Publ. 28, 97--143 (1995; Zbl 0868.14006)] this implies the one-dimensionality of the rational Picard group of the corresponding finite covers of the moduli space of curves. For instance, this applies to moduli spaces of curves of genus \(g\) with level \(n\) structures and to the moduli space of curves of genus \(g\) with spin structures, for \(g\geq 5\).
The results and the methods of the paper under review are used in the subsequent paper [\textit{A. Putman}, Duke Math. J. 161, No. 4, 623--674 (2012; Zbl 1241.30015)] to study the second cohomology group of \(\Gamma\) with integral coefficients, i.e. the integral Picard group of finite covers of \(\mathcal{M}_g\). moduli space of curves; mapping class group; level structures; Picard group; Torelli group; group cohomology Ivanov, N. V.: Subgroups of Teichmüller modular groups. Translations of Mathematical Monographs \textbf{115}. AMS (1992) Families, moduli of curves (algebraic), Moduli of Riemann surfaces, Teichmüller theory (complex-analytic aspects in several variables), Picard groups, General geometric structures on low-dimensional manifolds The second rational homology group of the moduli space of curves with level structures | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians Let \({\mathcal S}_g\) be the moduli space of smooth \textit{spin curves} of genus \(g\), i.e., of pairs \([C,\eta]\), where \([C] \in {\mathcal M}_g\) is a smooth curve of genus \(g\) and \(\eta \in \text{Pic}^{g-1}(C)\) is a \textit{theta-characteristic}, i.e., \({\eta}^{\otimes 2} \simeq {\omega}_C\). The forgetful map \(\pi : {\mathcal S}_g \rightarrow {\mathcal M}_g\) is finite, of degree \(2^{2g}\), and \({\mathcal S}_g\) is the union of two connected components \({\mathcal S}_g^+\) and \({\mathcal S}_g^-\), of relative degrees \(2^{g-1}(2^g+1)\) and \(2^{g-1}(2^g-1)\), corresponding to even and odd theta-characteristics, i.e., to the parity of \(\text{h}^0(C,\eta)\). \textit{M. Cornalba} [Moduli of curves and theta-characteristics. Proceedings of the first college on Riemann surfaces held in Trieste, Italy, November 9-December 18, 1987. Teaneck, NJ: World Scientific Publishing Co. 560--589 (1989; Zbl 0800.14011)] constructed a compactification \({\overline{\mathcal S}}_g\) of \({\mathcal S}_g\) over \({\overline{\mathcal M}}_g\) as the coarse moduli space of the stack \({\overline{\mathbf S}}_g\) of stable spin curves of genus \(g\) over the stack \({\overline{\mathbf M}}_g\) of stable curves of genus \(g\). A \textit{stable spin curve} is a triple \((X,\eta,\beta)\), where \(X\) is the total transform of a stable curve \(C\) of (arithmetic) genus \(g\) under the blow-up of a set \(N\subseteq \text{Sing}\, C\), \(\eta\) is a line bundle of degree \(g-1\) on \(X\) such that \(\eta \, | \, E \simeq {\mathcal O}_E(1)\) for every exceptional component \(E\) of \(X\) and \(\beta : {\eta}^{\otimes 2} \rightarrow {\omega}_X\) is a morphism which is generically non-zero along each non-exceptional component of \(X\). Recall that if \(C\) is an irreducible component of \(X\) then \({\omega}_X\, | \, C \simeq {\omega}_C(D)\), where \(D\) is the sum of the points of \(C \cap \overline{(X\setminus C)}\).
Let \({\Delta}_0\) be the component of the boundary \({\overline{\mathcal M}}_g \setminus {\mathcal M}_g\) whose general member is of the form \([C]\), where \(C\) is an irreducible curve with only one node \(x\). The fibre \(({\pi}^+)^{-1}([C])\subset {\overline{\mathcal S}}_g^+\setminus {\mathcal S}_g^+\) has two types of elements. The first type consists of pairs \([C,\eta]\), where \(\eta \in \text{Pic}^{g-1}(C)\), \({\eta}^{\otimes 2} \simeq {\omega}_C\) and \(\text{h}^0(C,\eta)\) is even. The second type consists of pairs \([X,\eta]\), where \(X = C^{\nu}\cup E\) is the total transform of \(C\) under the blow-up of \(x\) (\(C^{\nu}\) being the normalization of \(C\), which is a smooth curve of genus \(g-1\)) and \(\eta \in \text{Pic}^{g-1}(X)\) such that \(\eta \, | \, C^{\nu}\) is an even theta-characteristic on \(C^{\nu}\) and \(\eta\, | \, E \simeq {\mathcal O}_E(1)\). Accordingly, \(({\pi}^+)^{-1}({\Delta}_0)\subset {\overline{\mathcal S}}_g^+\setminus {\mathcal S}_g^+\) has two components \(A_0\) and \(B_0\). If \({\delta}_0 = [{\Delta}_0] \in \text{Pic}({\overline{\mathcal M}}_g)\), \({\alpha}_0 = [A_0] \in \text{Pic}({\overline{\mathcal S}}_g^+)\) and \({\beta}_0 = [B_0] \in \text{Pic}({\overline{\mathcal S}}_g^+)\) then \(({\pi}^+)^{\ast}({\delta}_0) = {\alpha}_0 + 2{\beta}_0\).
Now, let \(d, r > 0\) be two integers such that the \textit{Brill-Noether number} \(\rho (g,r,d) := g-(r+1)(g+r-d)\) is zero. Assume. moreover, that \(d\) is even, \(d = 2i\). The author considers the locus \({\mathcal U}^r_{g,d} \subset {\mathcal S}_g^+\) consisting of spin curves \([C,\eta]\) with the property that \(\exists \, L \in W^r_d(C)\) such that \(\eta \otimes L^{-1} \simeq {\mathcal O}_C(D-E)\), where \(D\) and \(E\) are effective divisors of degrees \(g-i-1\) and \(i\), respectively. According to a result of \textit{G. Farkas, M. Mustaţă} and \textit{M. Popa} [Ann. Sci. École Norm. Sup. 36, No. 4, 553--581 (2003; Zbl 1063.14031)], the last condition is equivalent to the determinantal condition \(\text{H}^0(C,{\wedge}^iM_C \otimes \eta \otimes L) \neq 0\), where \(M_C\) is the kernel of the evaluation morphism \(\text{H}^0(C,{\omega}_C)\otimes {\mathcal O}_C \rightarrow {\omega}_C\). Using this interpretation, the author constructs an open subset \({\overline{\mathbf M}}^0_g\) of \(({\pi}^+)^{-1}({\mathbf M}_g \cup {\Delta}_0)\), with complement of codimension \(\geq 2\), such that \({\overline{\mathcal U}}^r_{g,d} \cap {\overline{\mathbf M}}^0_g\) is the push-forward by a finite morphism of the degeneracy locus of a morphism between two vector bundles of the same rank. Based on this description, the author determines the coefficients of \(({\pi}^+)^{\ast}(\lambda)\) (where \(\lambda \in \text{Pic}({\overline{\mathcal M}}_g)\) is the Hodge class), \({\alpha}_0\) and \({\beta}_0\) in the expression of \([{\overline{\mathcal U}}^r_{g,d}]\) as a linear combination of the elements of the basis of \(\text{Pic}({\overline{\mathcal S}}^+_g)_{\mathbb Q}\) consisting of above mentioned classes and of the classes of the other boundary divisors. A similar result holds for \({\overline{\mathcal S}^{\, -}_g}\).
As a particular case of the above result one gets that:
\[
{\overline{\mathcal U}}^{g-1}_{g,2g-2} \equiv 2{\overline{\Theta}}_{\text{null}} \equiv \frac{1}{2}({\pi}^+)^{\ast}(\lambda) - \frac{1}{8}{\alpha}_0 - 0\cdot {\beta}_0 - \cdots \, \in \text{Pic}({\overline{\mathcal S}}^+_g)
\]
where \({\Theta}_{\text{null}}\) is the irreducible divisor on \({\mathcal S}^+_g\) consisting of the pairs \([C,\eta]\) with \(\text{H}^0(C,\eta) \neq 0\). One recovers, in this way, the main calculation from G. Farkas [Adv. Math. 223, No. 2, 433--443 (2010; Zbl 1183.14020)] used to prove that \({\overline{\mathcal S}}^+_g\) is a variety of general type for \(g > 8\).
The author also offers another way of computing the class \([{\overline{\Theta}}_{\text{null}}]\) by expressing \({\overline{\Theta}}_{\text{null}}\) as the push-forward of a degeneracy locus of a morphism of vector bundles of the same rank defined over a Hurwitz stack of coverings.
In the final part of the paper, the author studies the divisor \({\overline{\Theta}}_{g,1}\) on the ``universal curve'' \({\overline{\mathcal M}}_{g,1}\), where \({\Theta}_{g,1}\) consists of the classes \([C,q]\in {\mathcal M}_{g,1}\) such that \(q \in \text{supp}(\eta)\) for some odd theta-characteristic \(\eta\) on \(C\). He computes the coefficients of the expression of \([{\overline{\Theta}}_{g,1}]\) as a linear combination of the elements of the usual basis of \(\text{Pic}({\overline{\mathcal M}}_{g,1})_{\mathbb Q}\) and shows that \({\overline{\Theta}}_{g,1}\) is \textit{big} for \(g \geq 3\). The divisor \({\overline{\Theta}}_{g,1}\) is similar to the divisor \({\overline{\mathcal W}}_g\) of Weierstrass points on \({\overline{\mathcal M}}_{g,1}\), considered by \textit{F. Cukierman} [Duke Math. J. 58, No. 2, 317--346 (1989; Zbl 0687.14026)]. projective curve; theta-characteristic; moduli space; Deligne-Mumford stack; divisors on moduli spaces; Brill-Noether divisors Farkas, G., Brill-Noether geometry on moduli spaces of spin curves, (Classification of algebraic varieties, EMS ser. congr. rep., (2011), Eur. Math. Soc. Zürich), 259-276 Families, moduli of curves (algebraic), Divisors, linear systems, invertible sheaves, Picard groups, Stacks and moduli problems, Special divisors on curves (gonality, Brill-Noether theory) Brill-Noether geometry on moduli spaces 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 Let \(k\) be an algebraically closed field and let \({\mathcal Pic}_{d,g,n}\) be the universal Picard stack parametrizing families of \(n\)-pointed smooth curves of genus \(g\) over \(k\) endowed with line bundles of relative degree \(d\). Let \({\overline{\mathcal M}}_{g,n}\) be the moduli stack of stable curves of genus \(g\) with \(n\) marked (smooth) points.
In the paper under review, which is part of the author's Ph. D. thesis under the direction of L. Caporaso, one constructs an algebraic (Artin) stack \({\overline {\mathcal P}}_{d,g,n}\) with a universally closed map onto \({\overline{\mathcal M}}_{g,n}\), containing \({\mathcal Pic}_{d,g,n}\) as a dense open substack. This construction extends the results of \textit{L. Caporaso} [Am. J. Math. 130, No. 1, 1--47 (2008; Zbl 1155.14023)] who considered the case \(n = 0\) and \((d-g+1,2g-2) = 1\), and of the author herself [Math. Z. 263, No. 4, 939--957 (2009; Zbl 1183.14039)] who considered the case \(n = 0\) and \(d\) arbitrary. The strategy of construction is to start with the case \(n = 0\) and to construct \({\overline {\mathcal P}}_{d,g,n}\) by induction on the number \(n\) of marked points, following the lines of \textit{F. Knudsen}'s construction of \({\overline{\mathcal M}}_{g,n}\) from [Math. Scand. 52, No. 2, 161--199 (1983; Zbl 0544.14020)].
To be more precise, \({\overline {\mathcal P}}_{d,g,n}\) is, by definition, the stack parametrizing families of genus \(g\) \(n\)-pointed \textit{quasistable} curves endowed with a relative degree \(d\) \textit{balanced} line bundle. The notion of \(n\)-pointed quasistable curve generalizes the notion of quasistable curve (which is a semistable curve such that two exceptional components never meet). The notion of balanced line bundle on an \(n\)-pointed quasistable curve extends the notion of balanced line bundle on a quasistable curve introduced by \textit{L. Caporaso} [J. Am. Math. Soc. 7, No. 3, 589--660 (1994; Zbl 0827.14014)] inspired by the ``Basic Inequalities'' of \textit{D. Gieseker} [Lectures on moduli of curves (Lectures on Mathematics and Physics. Mathematics, 69. Tata Institute of Fundamental Resarch, Bombay. Springer, Berlin-Heidelberg-New York) (1982; Zbl 0534.14012)].
The main result of the paper asserts that \({\overline {\mathcal P}}_{d,g,n}\) is a smooth and irreducible (Artin) stack of dimension \(4g - 3 + n\) endowed with a universally closed morphism onto \({\overline{\mathcal M}}_{g,n}\). The author reduces the proof of this result to the following statement: for all \(d \in {\mathbb Z}\) and \(n > 0\) with \(2g -2 + n > 1\), \({\overline {\mathcal P}}_{d,g,n}\) is isomorphic to the universal family over \({\overline {\mathcal P}}_{d,g,n-1}\). The main technical ingredients in the proof of the last statement are a criterion of normal generation for line bundles over \(n\)-pointed semistable curves and a generalization of Knudsen's notion of \textit{contraction} of families of nodal curves. compactified Picard stacks; stable curve with marked points; balanced line bundle; quasistable curve Artin, M.: Algebraization of formal moduli: I. Global analysis, papers in honor of K. Kodaira, pp. 21-71. Princeton University Press, Princeton (1969) Stacks and moduli problems, Picard groups, Families, moduli of curves (algebraic), Picard schemes, higher Jacobians Compactified Picard stacks over the moduli stack of stable curves with marked points | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians In the 1980's \textit{M. Cornalba} and \textit{J. Harris} [Ann. Sci. Éc. Norm. Supér., IV. Sér. 21, No. 3, 455--475 (1988; Zbl 0674.14006)] discovered a relation among the Hodge class and the boundary classes in the Picard group with rational coefficients of the moduli space of stable, hyperelliptic curves. They proved the relation by computing degrees of the classes involved for suitable one-parameter families. In the present article we show that their relation can be obtained as the class of an appropriate, geometrically meaningful empty set, thus conforming with \textit{C. Faber}'s general philosophy [in: Moduli of curves and abelian varieties. Aspects Math. E 33, 109--129 (1999; Zbl 0978.14029)] of finding relations among tautological classes in the Chow ring of the moduli space of curves. The empty set we consider is the closure of the locus of smooth, hyperelliptic curves having a special ramification point. Families, moduli of curves (algebraic), Picard groups A geometric interpretation and a new proof of a relation by Cornalba and Harris | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is 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 \(\overline{{\mathcal M}_{g,n}}\) be the moduli space of \(n\)-pointed genus \(g\) stable curves, and let \(\overline{S_{g,n}}\) be the moduli space of \(n\)-pointed quasi-stable curves \(C\) of genus \(g\) with a line bundle \({\mathcal L}\) such that \({\mathcal L}^{\otimes 2}\simeq\omega_c\), the dualizing sheaf.
The authors apply an elementary double induction argument to prove various results on the rational cohomology of these spaces, which had been previously known via other methods. They derive some vanishing of Hodge numbers of type \((p,0)\) for \(\overline{{\mathcal M}_{g,n}}\), describe the rational Picard group of \(\overline{S_{g,n}}\) and compute the Kodaira dimension of spin moduli spaces in several cases. Bini G., Fontanari C.: Moduli of curves and spin structures via algebraic geometry. Trans. Am. Math. Soc. 358, 3207--3217 (2006) Families, moduli of curves (algebraic), Rationality questions in algebraic geometry, Picard groups, Classical real and complex (co)homology in algebraic geometry Moduli of curves and spin structures via algebraic geometry | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians The author studies the moduli stack \(\mathcal{H}_g\) of hyperelliptic curves of genus \(g\geq 1\) over the algebraic closure of \(\mathbb F_2\). The main theorem of this paper describes this stack as a quotient of a smooth variety by a non-reductive group. Secondly, the author describes the structure of the Picard group of this stack: \(\text{Pic}(\mathcal{H}_g)\) equals \(\mathbb Z/(8g+4)\mathbb Z\) or \(\mathbb Z/(4g+2)\mathbb Z\), depending on whether \(g\) is odd or even. These two results are extensions to characteristic \(2\) of results (for any characteristic not dividing \(2g+2\)) by \textit{A. Arsie} and \textit{A. Vistoli} [Compos. Math. 140, No. 3, 647--666 (2004; Zbl 1169.14301)]. However, the methods differ. This is mainly due to the fact that the Galois group (associated with the double cover map) equals \(\mathbb Z/2\mathbb Z\). In characteristic 2 this group is not isomorphic to \(\mu_2\). The author continues by studying the stratification of \(\mathcal{H}_g\) by means of higher ramification data.
Reviewer's remark: Some of the internal references are incorrect, e.g., the references to Remarque 4.3 should actually be Remarque 4.5. Deligne-Mumford stacks; Moduli of hyperelliptic curves; Picard group Families, moduli of curves (algebraic), Generalizations (algebraic spaces, stacks), Modular and Shimura varieties, Picard groups, Arithmetic ground fields for curves The stack of smooth hyperelliptic curves in characteristic two. (Le champ des courbes hyperelliptiques lisses en caractéristique deux) | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians Let X be a smooth projective curve of genus \(g\geq 2\) over \({\mathbb{C}}\). Let U(r,d) (resp. \(U_ s(r,d))\) be the moduli space of algebraic semistable vector bundles (resp. the open subset corresponding to the stable bundles) of rank \(r\geq 2\) and degree d over X. It is known that \(U(r,d)\) is a normal, irreducible, projective variety. If \(gcd(g,r)\neq 1\) and one excludes also the case \(g=r=2\), d even then \(U(r,d)\) is not smooth, \(Sing(U(r,d))=U(r,d)\setminus U_ s(r,d)\) and \(co\dim_{U(r,d)}U(r,d)\setminus U_ s(r,d)\geq 2\). For \(L\in Pic(X)\), \(\deg (L)=d\) let denote by U(r,L) (resp. \(U_ s(r,L))\) the closed subvariety of \(U(r,d)\) (resp. \(U_ s(r,d))\) corresponding to the vector bundles with determinant isomorphic to L. The aim of this paper is to give a complete description of \(Pic(U(r,d))\) and \(Pic(U(r,L))\) when \(gcd(g,r)\neq 1\) and \((g,r)\neq (2,2)\), d even.
The first result is that even they are singular, \(U(r,d)\) and \(U(r,L)\) are locally factorial. Let now \(\gcd (r,d)=n\) and let \({\mathcal F}\) be a vector bundle on X such that \(\deg({\mathcal F})=(-d+r(g-1))/n\) and \(rk({\mathcal F})=r/n\). Then \(\chi({\mathcal E}\otimes {\mathcal F})=0\) for all vector bundles \({\mathcal E}\) on X of rank r and degree d. One can show that \({\mathcal F}\) above can be chosen such that there exists \({\mathcal E}\in U_ s(r,d)\) with \(H^ 0(X,{\mathcal E}\otimes {\mathcal F})=H^ 1(X,{\mathcal E}\otimes {\mathcal F})=0\). Then for such an \({\mathcal F}\) denote by \(\Theta^ s_{{\mathcal F}}\) (respectively \(\Theta^ s_{{\mathcal F},L})\) the set of points of \(U_ s(r,d)\) (resp. \(U_ s(r,L))\) which correspond to stable bundles \({\mathcal E}\) with \(H^ 0(X,{\mathcal E}\otimes {\mathcal F})\neq 0\). These are showed to be hypersurfaces in \(U_ s(r,d)\) respectively in \(U_ s(r,L)\). Their closure in \(U(r,d)\) (respectively \(U(r,L)\)) are denoted by \(\Theta_{{\mathcal F}}\) (resp. \(\Theta_{{\mathcal F},L})\) and called theta divisors.
The line bundle \({\mathcal O}(\Theta_{{\mathcal F},L})\) is independent of the choice of \({\mathcal F}\) and \(Pic(U(r,L))\) is isomorphic to \({\mathbb{Z}}\) having \({\mathcal O}(\Theta_{{\mathcal F},L})\) as generator. Let \(I^{(d)}\) be the Jacobian of the line bundles of degree d on X. Then, through the canonical morphism \(\det: U(r,d)\to I^{(d)},\) \(Pic(I^{(d)})\) is seen as a subgroup of Pic(U(r,d)) and one has the isomorphism \(Pic(U(r,d))\cong Pic(I^{(d)})\oplus {\mathbb{Z}}{\mathcal O}(\Theta_{{\mathcal F}})\). Here \({\mathcal O}(\Theta_{{\mathcal F}})\) is dependent on the choice of \({\mathcal F}:\) \({\mathcal O}(\Theta_{{\mathcal F}'})\cong {\mathcal O}(\Theta_{{\mathcal F}})\otimes \det^*(\det {\mathcal F}'\otimes (\det {\mathcal F})^{-1}).\)
The paper also contains a complete description of the dualizing sheaves of \(U(r,L)\) and \(U(r,d)\) and a proof of the nonexistence of Poincaré bundles on open subsets of the moduli space \(M_ s({\mathbb{P}}_ 2({\mathbb{C}}),r,c_ 1,c_ 2)\) in case r, \(c_ 1\) and \(\chi\) are not prime to each other. factoriality of moduli space of algebraic semistable vector bundles; Picard group; smooth projective curve; determinant; theta divisors; Jacobian Drézet, J.-M.; Narasimhan, M. S., Groupe de Picard des variétés de modules de faisceaux semi-stables sur les courbes algébriques, Invent. Math., 97, 53-94, (1989) Picard groups, Families, moduli of curves (algebraic), Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20] Groupe de Picard des variétés de modules de fibrés semi-stable sur les courbes algébriques. (Picard groups of moduli varieties of semi- stable bundles on 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 C be a smooth complete curve of genus g over an algebraically closed field k. Consider the d-th Picard variety \(Pic_ d\) of C[1] and a Poincaré sheaf \({\mathcal P}_ d\) on \(C\times Pic_ d\). If \(d>2g-2\) and \(\pi\) is the projection from \(C\times Pic_ d\) to \(Pic_ d\), then \(W_ d=\pi ({\mathcal P}_ d)\) is locally free. In a not too difficult fashion, the author proves that \(W_{2g-1}\) is a stable bundle with respect to polarization by the theta divisor. moduli space of curves of genus g; Picard variety; Poincaré sheaf; stable bundle; polarization by the theta divisor Kempf, G., Rank \(g\) Picard bundles are stable, Am. J. Math., 112, 397-401, (1990) Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20], Picard groups, Families, moduli of curves (algebraic) Rank g Picard bundles are stable | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is 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, we prove that the \(\mathbb{Q}\)-Picard group of the moduli space of \(n\)-pointed stable curves of genus \(g\) over an algebraically closed field is generated by the tautological classes. We also prove that the cycle map to the second étale cohomology group is bijective. Atsushi Moriwaki, The \Bbb Q-Picard group of the moduli space of curves in positive characteristic, Internat. J. Math. 12 (2001), no. 5, 519 -- 534. Families, moduli of curves (algebraic), Arithmetic ground fields for curves, Picard groups The \(\mathbb Q\)-Picard group of the moduli space of curves in positive characteristic. | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians The Jacobian of a non-singular, compact Riemann surface is the group \(\mathrm{Pic}^{0}\) of divisors of degree zero factored out by principal divisors. As such, \(g\) being the genus of \(C,\) the Jacobian is \({\mathbb{C}}^{g}/\Lambda,\) \(\Lambda\) being the \(g-\)dimensional lattice of periods. In this case of the genus 2 (hyperelliptic) surface, there are coordinates \(x: C \rightarrow {\mathbb{P}}^{1}\) and \(y: C \rightarrow {\mathbb{P}}^{1}\) with poles of orders 2 and 5 respectively which satisfy a relation of the form
\[
y^{2} = 4x^{5} + \lambda_{4} x^{4} + \lambda_{3} x^{3} + \lambda_{2} x^{2} + \lambda_{1} x + \lambda_{0},
\]
the \(\lambda_{i}\) being constants in the ground field. Functions associated with more general special divisors provide us with other models. Such models are related by birational transformations. Thus we will be concerned with (singular) models of the genus 2 curve in the form
\[
y^{2} = g_{6} x^{6} + 6 g_{5} x^{5} + 15 g_{4} x^{4} + 20 g_{3} x^{3} + 15 g_{2} x^{2} + 6 g_{1} x + g_{0},
\]
which are related amongst themselves and to the quintic by simple Moebius maps.
If \(\mathrm{Pic}^{0}\) is identified with \(\mathrm{Pic}^{2}\) and \(\mathrm{Jac}(C)\) is constructed as a quadric variety in \({\mathbb{P}}^{15},\) the locus of seventy two linearly independent quadratic identities. Sixteen homogeneous coordinates on \({\mathbb{P}}^{15}\) are chosen to be symmetric functions in two points on the curve. The purpose of the current paper is to use a little representation theory to oil the wheels of this machinery and to uncover some structure intrinsic to the collection of quadratic identities. The idea is that the coordinates on \(\mathrm{Jac}(C)\) can be chosen to belong to irreducible \(G-\)modules where \(G\) is a group of birational transformations. Quadratic functions arise by tensoring up these modules and decomposing into irreducibles. Next he is presenting the Lie algebraic action of the coordinate transformations on the variables and the coefficients of the curve and define the construction of a highest weight element that use for a component of the decomposition. Thus author present a treatment of the algebraic description of the Jacobian of a general genus two plane curve which exploits an \(\mathrm{SL}_{2}(k)\) equivariance and clarifies the structure of Flynn's 72 defining quadratic relations. The treatment is also applied to the Kummer variety. Jacobian of a non-singular; compact Riemann surface; hyperelliptic surface; special divisors; birational transformations; homogeneous coordinates; Kummer variety Athorne C., On the equivariant algebraic Jacobian for curves of genus two, J. Geom. Phys., 2012, 62(4), 724--730 Picard groups, Riemann-Roch theorems, Rational and birational maps, Families, moduli of curves (algebraic), Jacobians, Prym varieties On the equivariant algebraic Jacobian for curves of genus two | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians This article provides two different, but closely related, moduli problems, which in characteristic zero provide a type of compactification of the universal Picard over the moduli of stable curves. Although neither is of finite type, both are limits of a sequence of stacks, each of which is a separated algebraic stack of finite type. We discuss relations to previous compactifications and partial compactifications, give a number of examples related to this compactification, and work out the structure of its fibres over certain fixed curves. Some applications are also discussed. moduli problems; compactification of the universal Picard; moduli of stable curves; algebraic stack Coelho, J., Esteves, E. and Pacini, M., Degree-2 Abel maps for nodal curves, to appear in \textit{Int. Math. Res. Not.}, available at the webpage: http://arxiv.org/abs/1212.1123. Families, moduli of curves (algebraic), Picard groups, Supervarieties Compactification of the universal Picard over the moduli of stable curves | 0 |
Applying a method of Weil, \textit{J.-P. Serre} constructed in his book ``Groupes algébriques et corps de classes'' (Paris 1959; Zbl 0097.35604) a generalized Jacobian for a curve with a singular point. For this he used the language of \textit{A. Weil}'s book: ``Foundations of algebraic geometry'' (Providence 1962; Zbl 0168.18701). The present paper works out the same proof in the language of schemes. The main tool is Grothendieck's base change theorem. Moreover, it is proved that the generalized Jacobian coincides with the Picard variety of the curve. In an appendix it is shown that the functor of relative Cartier divisors of the curve is representable. Picard variety of a curve; generalized Jacobian; relative Cartier divisors Families, moduli of curves (algebraic), Picard groups On the construction of generalized Jacobians The purpose of this paper is to study the geometry of the Harris-Mumford compactification of the Hurwitz scheme. The Hurwitz scheme parametrizes certain ramified coverings \(f:C \to\mathbb{P}^1\) of the projective line by smooth curves. Thus, from the very outset, one sees that there are essentially two ways to approach the Hurwitz scheme:
(1) We start with \(\mathbb{P}^1\) and regard the objects of interest as coverings of \(\mathbb{P}^1\).
(2) We start with \(C\) and regard the objects of interest as morphisms from \(C\) to \(\mathbb{P}^1\).
One finds that one can obtain the most information about the Hurwitz scheme and its compactification by exploiting interchangeably these two points of view. Our first main result is the following theorem.
Let \(b,d\), and \(g\) be integers such that \(b=2d+ 2g-2\), \(g\geq 5\) and \(d>2g+4\). Let \({\mathcal H}\) be the Hurwitz scheme over \(\mathbb{Z} [{1\over b!}]\) parametrizing coverings of the projective line of degree \(d\) with \(b\) points of ramification. Then \(\text{Pic} ({\mathcal H})\) is finite.
The number \(g\) is the genus of the ``curve \(C\) upstairs'' of the coverings in question. Note, however, that the Hurwitz scheme \({\mathcal H}\), and hence also the genus \(g\), are completely determined by \(b\) and \(d\). -- Note that although in the statement of the theorem we spoke of ``the'' Hurwitz ``scheme,'' there are in fact several different Hurwitz schemes used in the literature, some of which are, in fact, not schemes, but stacks. The main idea of the proof is that by combinatorially analyzing the boundary of the compactification of the Hurwitz scheme, one realizes that there are essentially three kinds of divisors in the boundary, which we call excess divisors, which are ``more important'' than the other divisors in the boundary in the sense that the other divisors map to sets of codimension \(\geq 2\) under various natural morphisms. On the other hand, we can also consider the moduli stack \({\mathcal G}\) of pairs consisting of a smooth curve of genus \(g\), together with a linear system of degree \(d\) and dimension 1. The subset of \({\mathcal G}\) consisting of those pairs that arise from Hurwitz coverings is open in \({\mathcal G}\), and its complement consists of three divisors, which correspond precisely to the excess divisors. Using results of Harer on the Picard group of \({\mathcal M}_g\), we show that these three divisors on \({\mathcal G}\) form a basis of \(\text{Pic} ({\mathcal G}) \otimes_\mathbb{Z} \mathbb{Q}\), and in fact, we even compute explicitly the matrix relating these three divisors on \({\mathcal G}\) to a certain standard basis of \(\text{Pic} ({\mathcal G}) \otimes_\mathbb{Z} \mathbb{Q}\). The above theorem then follows formally.
Crucial to our study of the Hurwitz scheme is its compactification by means of admissible coverings and we prove a rather general theorem concerning the existence of a canonical logarithmic algebraic stack \(({\mathcal A}, M)\) parametrizing such coverings:
Fix non-negative integers \(g,r,q,s,d\) such that \(2g-2+r =d(2q-2+s) \geq 1\). Let \({\mathcal A}\) be the stack over \(\mathbb{Z}\) defined as follows: For a scheme \(S\), the objects of \({\mathcal A}(S)\) are admissible coverings \(\pi:C\to D\) of degree \(d\) from a symmetrically \(r\)-pointed stable curve \((f:C\to S\); \(\mu_f \subseteq C)\) of genus \(g\) to a symmetrically \(s\)-pointed stable curve \((h:D \to S\); \(\mu_h \subseteq D)\) of genus \(q\); and the morphisms of \({\mathcal A} (S)\) are pairs of \(S\)-isomorphisms \(\alpha: C\to C\) and \(\beta: D\to D\) that stabilize the divisors of marked points such that \(\pi\circ \alpha= \beta\circ \pi\). Then \({\mathcal A}\) is a separated algebraic stack of finite type over \(\mathbb{Z}\). Moreover, \({\mathcal A}\) is equipped with a canonical log structure \(M_{\mathcal A} \to {\mathcal O}_{\mathcal A}\), together with a logarithmic morphism \(({\mathcal A}, M_{\mathcal A}) \to \overline {{\mathcal M} {\mathcal S}}^{\log}_{q,s}\) (obtained by mapping \((C;D;\pi) \mapsto D)\) which is log étale (always) and proper over \(\mathbb{Z} [{1\over d!}]\). finite Picard group; Hurwitz scheme; Hurwitz coverings; admissible coverings S. Mochizuki, ''The geometry of the compactification of the Hurwitz scheme,'' Publ. Res. Inst. Math. Sci., vol. 31, iss. 3, pp. 355-441, 1995. Families, moduli of curves (algebraic), Picard groups, Coverings in algebraic geometry The geometry of the compactification of the Hurwitz scheme | 0 |
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