Forschungsseminar Algebraische Geometrie

Wintersemester 2008/09

14.10.2008 Matthias Schuett (Univ. of Copenhagen)

K3 surfaces and modular forms

Abstract: A classical construction of Shimura associates every Hecke eigenform of weight 2 with rational coefficents to an elliptic curve over Q. The converse statement that every elliptic curve over Q is modular, is the Taniyama-Shimura-Weil conjecture, proven by Wiles et al. For higher weight, however, the opposite situation applies: Nowadays we know the modularity for wide classes of varietes, but it is an open problem whether all newforms of fixed weight with rational coefficients can be realised in a single class of varieties. I will present joint work with N. Elkies that provides the first solution to the realisation problem in higher weight: We show that every newform of weight 3 with rational coefficients is associated to a singular K3 surface over Q.

28.10.2008 Filippo Viviani (HU Berlin)

Torelli theorem for stable curves

Abstract: The classical Torelli theorem asserts that a smooth (connected and projective) curve is determined by its Jacobian together with the principal polarization induced by the theta divisor. In modular terms, it asserts that the natural (Torelli) map from the moduli space of smooth curves of genus g into the moduli space of principally polarized abelian varieties of dimension g is injective. Quite recently, Alexeev has extended the Torelli map, in a modular way, to natural compactifications of the above moduli spaces, namely the moduli space of stable curves and the moduli space of principally polarized stable semi-abelic pairs. In a joint work with L. Caporaso, we study the fibers of the above compactified Torelli map.

04.11.2008 Evgeni Materov (Univ. of Massachusetts at Amherst)

Tate resolutions and Weyman complexes

Abstract: The Tate resolution of a coherent sheaf on projective space is a bi- infinite exact complex over an exterior algebra. The terms of the complex are known from the work of Eisenbud, Floystad and Schreyer, but the differentials are only partially known. For example, for sheaves arising from Veronese embedding the differentials are induced by the Bezoutian, for sheaves arising from Segre embedding the toric Jacobian gives the choice of differentials. In my talk I will explain how to use the maps in Tate resolution to construct the Weyman-style complexes. Weyman complexes are important tools in computation of multidimensional resultants, discriminants and hyperdeterminants. The lecture is based on joint work with David Cox.

11.11.2008 Gavril Farkas (HU Berlin)

Variety of secants of the generic curve

Abstract: For a smooth projective curve, the cycles of e-secant k-planes are among the most studied objects in classical enumerative geometry, and there are well-known formulas due to Castelnuovo, Cayley and MacDonald concerning them. Despite various attempts, surprisingly little is known about the enumerative validity of such formulas. We completely clarify this problem in the case of the generic curve C of given genus and determine precisely under which conditions the cycle of e-secant k-planes is non-empty and compute its dimension.

18.11.2008 Jun-Muk Hwang (KIAS Seoul)

Fibrations of projective irreducible symplectic manifolds

Abstract: Given a projective hyperkaehler manifold M of dimension 2n, a projective manifold X and a surjective holomorphic map f:M -> X with connected fibers of positive dimension, we prove that X is biholomorphic to the projective space of dimension n. The proof is obtained by exploiting two geometric structures at general points ofX: the affine structure arising from the action variables of the Lagrangian fibration f and the structure defined by the variety of minimal rational tangents on the Fano manifold X.

25.11.2008 Gavril Farkas (HU Berlin)

Degeneration techniques for algebraic curves I

Abstract: Degeneration techniques have been used in algebraic geometry since the 19th century. Today they are still ubiquitous especially in enumerative geometry and the study of different moduli spaces. One of the main reason people like to work with compact moduli spaces is precisely to be able to use degeneration methods. The purpose of these two lectures is mainly educational. I will discuss standard degeneration methods for curves and line/vector bundles on them by focusing on the theory of admissible coverings (Harris-Mumford), limit linear series (Eisenbud-Harris) and localization of moduli spaces (Graber-Pandharipande).

02.12.2008 Gavril Farkas (HU Berlin)

Degeneration techniques for algebraic curves II

09.12.2008 Jaroslaw Wisniewski (Univ. of Warsaw)

On Kummer construction

Abstract: Given an integral representation of a finite group G and a crepant resolution of related quotient singularities one can construct examples of complex varieties with trivial canonical class. Among examples of varieties obtained in such a way there are classical Kummer surfaces, Hilbert schemes of points on them, Beauville's generalized Kummer varieties and special Calabi-Yau threefolds. Cohomology of such varieties can be computed by using virtual G-Poincare polynomials and McKay correspondence for G. I will report on a joint paper with Marco Andreatta and a thesis of my student Marysia Donten.

11.12.2008 Ezra Getzler (Northwestern Univ.)

Teichmueller space and topological field theory in two dimensions

Abstract: A topological field theory in d dimensions associates to each (d-1)-dimensional closed manifold M an inner-product space V(M), and to each d-dimensional manifold W with boundary M a vector v(W) in V(M), satisfying certain natural axioms; for example, V(-) takes disjoint unions to tensor products, and behaves well under diffeomorphisms. There are many flavours of topological field theories - one may for example assume that all of the manifolds are oriented, or spin, or carry a free action of a finite group G. It turns out that the two-dimensional case is especially simple: two-dimensional topological field theories are equivalent to commutative algebras with inner product (also known as commutative Frobenius algebras). In this talk, we relate this to a result in topology. Harvey has introduced a manifold with boundary containing the (6g-6)-dimensional Teichmueller space of genus g closed Riemann surfaces as its interior, and we define a filtration F(i) of this space such that the inclusion of F(i) into F(i+1) is i-connected. (The proof is an application of a triangulation of Teichmueller space constructed by Harer.) This result and its generalizations explain many pheonomena in topological field theory, including theorems of Moore and Seiberg, Moore and Segal, and Turaev.

16.12.2008 Frank Schreyer (Univ. Saarbrücken)

Betti numbers of Graded Moduls and Cohomology tables of Coherent Sheaves

Abstract: In a recent paper Boij and Söderberg introduced a series of conjectures, which characterize all possible syzygy numbers of graded modules over the polynomial ring up to rational multiples. In the talk, I report on joined work with David Eisenbud, which proves these conjectures and an analogous statements for cohomology tables of coherent sheaves.

20.01.2009 Filippo Viviani (HU Berlin)

Torelli theorem for tropical curves and graphs

Abstract: Mikhalkin-Zharkov ('07) have defined the Jacobian of a tropical curve. In the case when the tropical curve comes from a graph, their definition agress with the Albanese torus of a graph introduced by Kotani-Sunada ('01). In this talk, based on joint work with L. Caporaso, we prove a Torelli type theorem for tropical curves and graphs.

27.01.2009 Paul Larsen (HU Berlin)

The Faber-Fulton conjecture on moduli space of pointed rational curves

Abstract: Fulton's conjecture for curves claims that the closed cone of curves in the moduli space of pointed rational curves, $\bar{M}_{0,n}$, is generated by a finite, distinguished set of curves called F-curves. A variant of this conjecture, which we call the Faber-Fulton conjecture, asks if every divisor intersecting all F-curves non-negatively is linearly equivalent to an effective sum of boundary divisors. The second conjecture implies the first, and, if true, would give the surprising result that the cone of curves of $\bar{M}_{0,n}$ is polyhedral and finitely generated, even though for $n$ bigger than 5 $\bar{M}_{0,n}$ is not Fano. The Faber-Fulton conjecture has been proven for $n \leq 6$ by Farkas and Gibney, and Faber. We present a new proof that can be extended to $n=7$. The proof utilizes a natural subspace of the Picard group of $\bar{M}_{0,n}$ for which the Faber-Fulton conjecture holds for all $n$.

04.02.2009 David Ben-Zvi (Univ. of Texas at Austin)

Linear Algebra of Derived Categories of Sheaves

Abstract: The derived categories of sheaves on an algebraic variety or stack can be considered as algebro-geometric analogs of the spaces of distributions or L^2 functions on a manifold. For example the Fourier-Mukai transform provides an analog of the Fourier transform on abelian varieties. Using the tools of derived algebraic geometry, developed by Lurie and Toen-Vezzosi, one can now push this analogy further and work with derived categories as one would with classical function spaces. I will describe some basic features of this theory, based on joint work with John Francis and David Nadler (Northwestern University).

10.02.2009 Elisabetta Colombo (Univ. Milano)

Curvature of the moduli space of curves with the Siegel metric and second Gaussian map

Abstract: We study the curvature of the moduli space $M_g$ of curves of genus g with the Siegel metric induced by the period map $j:M_g ---> A_g$ and give an explicit formula for the holomorphic sectional curvature of $M_g$ along a Schiffer variation in terms of the holomorphic sectional curvature of $A_g$ and the second Gaussian map. Moreover we study related rank properties of the second Gaussian map. These results are obtained in collaboration with Paola Frediani.

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14.10.2008	Matthias Schuett (Univ. of Copenhagen)
	K3 surfaces and modular forms
	Abstract: A classical construction of Shimura associates every Hecke eigenform of weight 2 with rational coefficents to an elliptic curve over Q. The converse statement that every elliptic curve over Q is modular, is the Taniyama-Shimura-Weil conjecture, proven by Wiles et al. For higher weight, however, the opposite situation applies: Nowadays we know the modularity for wide classes of varietes, but it is an open problem whether all newforms of fixed weight with rational coefficients can be realised in a single class of varieties. I will present joint work with N. Elkies that provides the first solution to the realisation problem in higher weight: We show that every newform of weight 3 with rational coefficients is associated to a singular K3 surface over Q.
28.10.2008	Filippo Viviani (HU Berlin)
	Torelli theorem for stable curves
	Abstract: The classical Torelli theorem asserts that a smooth (connected and projective) curve is determined by its Jacobian together with the principal polarization induced by the theta divisor. In modular terms, it asserts that the natural (Torelli) map from the moduli space of smooth curves of genus g into the moduli space of principally polarized abelian varieties of dimension g is injective. Quite recently, Alexeev has extended the Torelli map, in a modular way, to natural compactifications of the above moduli spaces, namely the moduli space of stable curves and the moduli space of principally polarized stable semi-abelic pairs. In a joint work with L. Caporaso, we study the fibers of the above compactified Torelli map.
04.11.2008	Evgeni Materov (Univ. of Massachusetts at Amherst)
	Tate resolutions and Weyman complexes
	Abstract: The Tate resolution of a coherent sheaf on projective space is a bi- infinite exact complex over an exterior algebra. The terms of the complex are known from the work of Eisenbud, Floystad and Schreyer, but the differentials are only partially known. For example, for sheaves arising from Veronese embedding the differentials are induced by the Bezoutian, for sheaves arising from Segre embedding the toric Jacobian gives the choice of differentials. In my talk I will explain how to use the maps in Tate resolution to construct the Weyman-style complexes. Weyman complexes are important tools in computation of multidimensional resultants, discriminants and hyperdeterminants. The lecture is based on joint work with David Cox.
11.11.2008	Gavril Farkas (HU Berlin)
	Variety of secants of the generic curve
	Abstract: For a smooth projective curve, the cycles of e-secant k-planes are among the most studied objects in classical enumerative geometry, and there are well-known formulas due to Castelnuovo, Cayley and MacDonald concerning them. Despite various attempts, surprisingly little is known about the enumerative validity of such formulas. We completely clarify this problem in the case of the generic curve C of given genus and determine precisely under which conditions the cycle of e-secant k-planes is non-empty and compute its dimension.
18.11.2008	Jun-Muk Hwang (KIAS Seoul)
	Fibrations of projective irreducible symplectic manifolds
	Abstract: Given a projective hyperkaehler manifold M of dimension 2n, a projective manifold X and a surjective holomorphic map f:M -> X with connected fibers of positive dimension, we prove that X is biholomorphic to the projective space of dimension n. The proof is obtained by exploiting two geometric structures at general points ofX: the affine structure arising from the action variables of the Lagrangian fibration f and the structure defined by the variety of minimal rational tangents on the Fano manifold X.
25.11.2008	Gavril Farkas (HU Berlin)
	Degeneration techniques for algebraic curves I
	Abstract: Degeneration techniques have been used in algebraic geometry since the 19th century. Today they are still ubiquitous especially in enumerative geometry and the study of different moduli spaces. One of the main reason people like to work with compact moduli spaces is precisely to be able to use degeneration methods. The purpose of these two lectures is mainly educational. I will discuss standard degeneration methods for curves and line/vector bundles on them by focusing on the theory of admissible coverings (Harris-Mumford), limit linear series (Eisenbud-Harris) and localization of moduli spaces (Graber-Pandharipande).
02.12.2008	Gavril Farkas (HU Berlin)
	Degeneration techniques for algebraic curves II
09.12.2008	Jaroslaw Wisniewski (Univ. of Warsaw)
	On Kummer construction
	Abstract: Given an integral representation of a finite group G and a crepant resolution of related quotient singularities one can construct examples of complex varieties with trivial canonical class. Among examples of varieties obtained in such a way there are classical Kummer surfaces, Hilbert schemes of points on them, Beauville's generalized Kummer varieties and special Calabi-Yau threefolds. Cohomology of such varieties can be computed by using virtual G-Poincare polynomials and McKay correspondence for G. I will report on a joint paper with Marco Andreatta and a thesis of my student Marysia Donten.
11.12.2008	Ezra Getzler (Northwestern Univ.)
	Teichmueller space and topological field theory in two dimensions
	Abstract: A topological field theory in d dimensions associates to each (d-1)-dimensional closed manifold M an inner-product space V(M), and to each d-dimensional manifold W with boundary M a vector v(W) in V(M), satisfying certain natural axioms; for example, V(-) takes disjoint unions to tensor products, and behaves well under diffeomorphisms. There are many flavours of topological field theories - one may for example assume that all of the manifolds are oriented, or spin, or carry a free action of a finite group G. It turns out that the two-dimensional case is especially simple: two-dimensional topological field theories are equivalent to commutative algebras with inner product (also known as commutative Frobenius algebras). In this talk, we relate this to a result in topology. Harvey has introduced a manifold with boundary containing the (6g-6)-dimensional Teichmueller space of genus g closed Riemann surfaces as its interior, and we define a filtration F(i) of this space such that the inclusion of F(i) into F(i+1) is i-connected. (The proof is an application of a triangulation of Teichmueller space constructed by Harer.) This result and its generalizations explain many pheonomena in topological field theory, including theorems of Moore and Seiberg, Moore and Segal, and Turaev.
16.12.2008	Frank Schreyer (Univ. Saarbrücken)
	Betti numbers of Graded Moduls and Cohomology tables of Coherent Sheaves
	Abstract: In a recent paper Boij and Söderberg introduced a series of conjectures, which characterize all possible syzygy numbers of graded modules over the polynomial ring up to rational multiples. In the talk, I report on joined work with David Eisenbud, which proves these conjectures and an analogous statements for cohomology tables of coherent sheaves.
20.01.2009	Filippo Viviani (HU Berlin)
	Torelli theorem for tropical curves and graphs
	Abstract: Mikhalkin-Zharkov ('07) have defined the Jacobian of a tropical curve. In the case when the tropical curve comes from a graph, their definition agress with the Albanese torus of a graph introduced by Kotani-Sunada ('01). In this talk, based on joint work with L. Caporaso, we prove a Torelli type theorem for tropical curves and graphs.
27.01.2009	Paul Larsen (HU Berlin)
	The Faber-Fulton conjecture on moduli space of pointed rational curves
	Abstract: Fulton's conjecture for curves claims that the closed cone of curves in the moduli space of pointed rational curves, $\bar{M}_{0,n}$, is generated by a finite, distinguished set of curves called F-curves. A variant of this conjecture, which we call the Faber-Fulton conjecture, asks if every divisor intersecting all F-curves non-negatively is linearly equivalent to an effective sum of boundary divisors. The second conjecture implies the first, and, if true, would give the surprising result that the cone of curves of $\bar{M}_{0,n}$ is polyhedral and finitely generated, even though for $n$ bigger than 5 $\bar{M}_{0,n}$ is not Fano. The Faber-Fulton conjecture has been proven for $n \leq 6$ by Farkas and Gibney, and Faber. We present a new proof that can be extended to $n=7$. The proof utilizes a natural subspace of the Picard group of $\bar{M}_{0,n}$ for which the Faber-Fulton conjecture holds for all $n$.
04.02.2009	David Ben-Zvi (Univ. of Texas at Austin)
	Linear Algebra of Derived Categories of Sheaves
	Abstract: The derived categories of sheaves on an algebraic variety or stack can be considered as algebro-geometric analogs of the spaces of distributions or L^2 functions on a manifold. For example the Fourier-Mukai transform provides an analog of the Fourier transform on abelian varieties. Using the tools of derived algebraic geometry, developed by Lurie and Toen-Vezzosi, one can now push this analogy further and work with derived categories as one would with classical function spaces. I will describe some basic features of this theory, based on joint work with John Francis and David Nadler (Northwestern University).
10.02.2009	Elisabetta Colombo (Univ. Milano)
	Curvature of the moduli space of curves with the Siegel metric and second Gaussian map
	Abstract: We study the curvature of the moduli space $M_g$ of curves of genus g with the Siegel metric induced by the period map $j:M_g ---> A_g$ and give an explicit formula for the holomorphic sectional curvature of $M_g$ along a Schiffer variation in terms of the holomorphic sectional curvature of $A_g$ and the second Gaussian map. Moreover we study related rank properties of the second Gaussian map. These results are obtained in collaboration with Paola Frediani.