Forschungsseminar Algebraische Geometrie

Wintersemester 2009/10

Dienstag

13.10.2009 Motohico Mulase (Univ. of California at Davis)

Topological recursions and integrable equations

Abstract: The Virasoro conjecture is trivially true for Calabi-Yau spaces, and the Virasoro constraint condition produces no information about their Gromov-Witten invariants.
Then what should be the correct formulation that generalizes the Witten-Kontsevich theory to Calabi-Yau spaces?
Recently a surprising answer has been proposed by physicists Bouchard, Klemm, Marino, and Pasquetti, based on the work of Eynard-Orantin and Dijkgraaf-Vafa.
There are three important ingredients in Witten-Kontsevich theory: (1) the Virasoro constraint; (2) an effective recursion; and (3) the KdV/KP equations. The physicists' recent proposal seems to contain all these ingredients for toric Calabi-Yau 3-folds.
In this talk I will give an algebro-geometric introduction to these recent developments.

21.10.2009 Michele Bolognesi (HU Berlin)

Forgetful linear system on the projective space and rational normal curves over M_{0,2n}^{GIT}

Abstract: Let M_{0,n} the moduli space of n-pointed rational curves. The aim of my talk is to give a new, geometric construction of M_{0,2n}^{GIT}, the GIT compactification of M_{0,2n}, in terms of linear systems on P^{2n-2} that contract all the rational normal curves passing by the points of a projective base. These linear systems are a projective analogue of the forgetful maps between \bar{M}_{0,2n+1} and \bar{M}_{0,2n}. The construction is performed via a study of the so-called contraction maps from the Knudsen-Mumford compactication \bar{M}_{0,2n} to M_{0,2n}^{GIT} and of the canonical forgetful maps.

28.10.2009 Cristina Manolache (HU Berlin)

Gromov-Witten Invariants: Definition and functoriality properties

Abstract: One typical enumerative problem is the following. We are given a smooth projective variety X and we would like to know how many genus-g curves in X, of fixed homology type intersect some given general cycles of X. These numbers are closely related to the Gromov-Witten invariants of X. I will define Gromov-Witten invariants of a smooth projective variety and I will try to answer the following question: "Suppose we have a morphism of varieties f:X--> Y and let us suppose that we know the Gromov-Witten invariants of Y. Can we say anything about the Gromov-Witten invariants of X?"

04.11.2009 Herbert Lange (Univ. Erlangen)

Generalization of the tetragonal construction

Abstract: A tetragonal curve is by definition a smooth projective curve admitting a 4-fold covering of the projective line. Donagi showed in 1981 that to any etale double covering of a simply ramified tetragonal covering one can associate 2 other such coverings such that the corresponding Prym varieties are isomorphic. The aim of the talk is to generalize this construction to simply ramified 4-fold coverings Y of an arbitrary curve X. This is joint work with Elham Izadi.

11.11.2009 Eduardo Esteves (IMPA Rio de Janeiro)

Abel maps for singular curves

Abstract: Abel maps have been at the center of the study of compact Riemann surfaces, or complex curves, since its roots. I will sketch the history of Abel maps from Abel, passing through Riemann, and will expose on recent progress in extending the construction of such maps to degenerations of Riemann surfaces, more specifically to Deligne-Mumford stable curves.

18.11.2009 Fabrizio Catanese (Univ. Bayreuth)

Explicit descriptions of connected components of the moduli space of surfaces with p_g =0: Burniat's and Keum Naie surfaces

(joint work with Ingrid Bauer)

Abstract: The classification and the moduli space of surfaces with p_g = 0 were on the one side motivated in the 70's by the problem of pluricanonical maps of surfaces of general type, which was finally solved by Bombieri, with final touches by Miyaoka, Benveniste and the present author. The classical Godeaux and Campedelli surfaces, and the more recent Burniat surfaces were then rediscovered. The work by Miyaoka and Reid motivated several interesting and important researches in the field. The discovery of new surfaces with p_g = 0 went on in the 80's , motivated on the one side by the Bloch conjecture, and on the other side by differential topology: for instance a big achievement was the result that the Barlow surface is homeomorphic but not diffeomorphic to the rational surface X obtained by blowing up the plane in 8 points. This in turn had an important byproduct concerning the disproof of the Besse conjecture: together with Claude Le Brun, we showed that the self product of X with itself admits Kaehler Einstein metrics. In the last 5 years there has been an explosion of new results , constructions and partial classification of surfaces with p_g = 0. Due to the celebrated Bogomolov-Miyaoka-Yau inequality the minimal models of these surfaces have K^2 at most 9. Due to Yau's uniformization theorem, the case where K^2 = 9 corresponds to ball quotients, and using a result of Klingler stating the arithmeticity of the fundamental groups a complete classification seems to have been found by Prasad and Yeung, with the help of Steger, Cartwright and a lot of computer calculations. While Park and Lee have been working on the construction of surfaces with p_g = 0 which are simply connected, in the last 3-4 years I have been working with Ingrid Bauer and Fritz Grunewald (later also with Roberto Pignatelli) in order to construct new such surfaces, classifying all such surfaces which can be constructed as the quotient of a product of curves by the action of a finite group. This project also needed computer assistance. We succeed in describing the fundamental groups of such surfaces, and in this way we could construct more than 40 new connected components of the moduli space of surfaces with p_g=0, distinguished by the fundamental group. Looking at the case where the fundamental groups were not new has motivated an investigation of the moduli spaces of certain surfaces with p_g = 0. This is joint work with Ingrid Bauer, and I am going to report on this. We solve the moduli problem for the primary Keum-Naie surfaces and the primary Burniat surfaces, showing that we get in both cases an irreducible connected component of the moduli space to which any surface with the same homotopy type belongs. In the case of primary Burniat surfaces the Inoue description plays a crucial role. More delicate and interesting is the case of secondary Burniat surfaces, having K^2 = 5,4. We show that they form three connected components of the moduli space. In the case of K^2=4, there are two components, the nodal ones yield a Gieseker moduli space (i.e., moduli space of the canonical models) which is everywhere non reduced (a fortiori, the order of nilpotency for the moduli space of canonical models is even higher). While the Burniat surface with K^2 = 2 was already recognized as being a very special Campedelli surface, we find that tertiary Burniat surfaces, i.e. those with K^2 = 3, do not form an irreducible component of the moduli space. We are able to describe explicitly the other surfaces which appear in this component. This work is still in progress and we have not yet decided whether this is also a connected component of the moduli space.

25.11.2009 Alex Abreu (IMPA Rio de Janeiro)

Wroskian classes in the moduli space of curves

Abstract: The purpose of this talk is to compute the class of a divisor in the moduli space of stable curves of genus 2n, defined as the closure of the locus of smooth curves C possessing a pair of points (P,Q), such that P is a special ramification point of the linear system K_C(-nQ) and Q is a special ramification point of K_C(-nP).

02.12.2009 Marian Aprodu (IMAR Bucharest)

Koszul cohomology of canonical curves

Abstract: We discuss recent developments on syzygies of canonical curves, with special emphasis on curves on K3 surfaces. The talk is mainly based on a joint work in progress with G. Farkas.

09.12.2009 Claire Voisin (IHES Paris)

Hyper-Kähler fourfolds and Grassmann geometry

Abstract: This is joint work with Olivier Debarre. The second punctual Hilbert scheme of an algebraic K3 surface is an hyper-Kaehler fourfold with Picard number 2. General deformation theory tells us that given any polarization on them, there is a 20-dimensional family of deformations of this polarized variety. The corresponding moduli spaces can be studied by the period map, and recent work of Gritsenko-Hulek-Sankaran shows that very few of them have $-\infty$ Kodaira dimension. I will describe in this talk a construction of one of these families of deformations, which is unirational. This is the fourth explicitely described such family. It also has a curious numerical property: while it is unirational, it follows from work of Gritsenko-Hulek-Sankaran that there is another family of deformations of a pair as above, where the polarization is also primitive with the same degree, but which has nonnegative Kodaira dimension.

06.01.2010 Samuel Grushevsky (Stony Brook)

Integrable discrete Schrodinger equations and a characterization of Prym varieties by a pair of quadrisecants

Alessandro Chiodo (Univ. Grenoble)

The geometric interpretation of limit roots of line bundles

Abstract: For r a positive integer, given two line bundles L and N on a curve C, we say that L is an rth root of N if we have an isomorphism between the tensor power L^r and N. Let us recall that the moduli functor of rth roots is proper over the moduli of smooth curves of genus g>1. In other words, consider a family of smooth curves f : C -> B and a line bundle N -> C whose degree on the fibres of f is a multiple of r. Then, the rth roots of the restriction of N to the fibers of f form a proper (and etale) covering S_B of B. When f : C -> B is a stable (nodal) curve with smooth fibres over a nonempty open subscheme of B, we still get an etale covering S_B, but the properness is lost in general. The issue is how to compactify S_B over B. One could think of the points in the compactification as representing "limit roots". We provide two useful interpretations of these points: via semi-stable curves and via stack-theoretic curves. This will allow us to further illustrate the LG/CY correspondence introduced in http://www.raumzeitmaterie.de/veranstaltungen.php?evt=select&sqn=6826& lang=en (NB the material discussed there will not be taken for granted).

27.01.2010 Norbert Hoffmann (FU Berlin)

Moduli of special instanton bundles

Abstract: Mathematical instanton bundles are certain algebraic vector bundles on the complex projective space P^{2n+1} with n at least one. I will explain the notion of special instanton bundle and their moduli space, following Spindler and Trautmann. Then I will discuss the birational type of this moduli space. This generalizes work of Hirschowitz and Narasimhan about special t'Hooft bundles on P^3.

03.02.2010 Marcello Bernardara (Univ. Duisburg-Essen)

A categorical invariant for cubic threefolds

Abstract: Let X be a smooth cubic threefold. The derived category of X contains a nontrivial triangulated subcategory T as the orthogonal complement of an exceptional sequence. In this talk, I will show how the category T characterizes the isomorphism class of X. This result is a joint work with Macri, Mehrotra and Stellari and is based on the construction of a stability condition on T. More generally, one can consider a conic bundle Y over a rational surface S. I will give at least one more example in which the birational properties of Y are encoded in a triangulated subcategory of the derived category.

10.02.2010 Sukmoon Huh (Korea Institute for Advanced Study KIAS)

Restriction of vector bundles on surfaces to curves

Abstract: There are many ways to study the moduli space of stable vector bundles on curves, for example, extension families, theta maps and so on. In this talk, we consider the moduli space of stable sheaves on a surface containing the given curve and investigate the rational map between moduli spaces given by the restriction. This method is effective in studying global properties and also describing higher Brill-Noether loci explicitly. As examples, we will look into the cases of non-hyperelliptic curves of genus 3 and 4.

13.10.2009	Motohico Mulase (Univ. of California at Davis)
	Topological recursions and integrable equations
	Abstract: The Virasoro conjecture is trivially true for Calabi-Yau spaces, and the Virasoro constraint condition produces no information about their Gromov-Witten invariants. Then what should be the correct formulation that generalizes the Witten-Kontsevich theory to Calabi-Yau spaces? Recently a surprising answer has been proposed by physicists Bouchard, Klemm, Marino, and Pasquetti, based on the work of Eynard-Orantin and Dijkgraaf-Vafa. There are three important ingredients in Witten-Kontsevich theory: (1) the Virasoro constraint; (2) an effective recursion; and (3) the KdV/KP equations. The physicists' recent proposal seems to contain all these ingredients for toric Calabi-Yau 3-folds. In this talk I will give an algebro-geometric introduction to these recent developments.
21.10.2009	Michele Bolognesi (HU Berlin)
	Forgetful linear system on the projective space and rational normal curves over M_{0,2n}^{GIT}
	Abstract: Let M_{0,n} the moduli space of n-pointed rational curves. The aim of my talk is to give a new, geometric construction of M_{0,2n}^{GIT}, the GIT compactification of M_{0,2n}, in terms of linear systems on P^{2n-2} that contract all the rational normal curves passing by the points of a projective base. These linear systems are a projective analogue of the forgetful maps between \bar{M}_{0,2n+1} and \bar{M}_{0,2n}. The construction is performed via a study of the so-called contraction maps from the Knudsen-Mumford compactication \bar{M}_{0,2n} to M_{0,2n}^{GIT} and of the canonical forgetful maps.
28.10.2009	Cristina Manolache (HU Berlin)
	Gromov-Witten Invariants: Definition and functoriality properties
	Abstract: One typical enumerative problem is the following. We are given a smooth projective variety X and we would like to know how many genus-g curves in X, of fixed homology type intersect some given general cycles of X. These numbers are closely related to the Gromov-Witten invariants of X. I will define Gromov-Witten invariants of a smooth projective variety and I will try to answer the following question: "Suppose we have a morphism of varieties f:X--> Y and let us suppose that we know the Gromov-Witten invariants of Y. Can we say anything about the Gromov-Witten invariants of X?"
04.11.2009	Herbert Lange (Univ. Erlangen)
	Generalization of the tetragonal construction
	Abstract: A tetragonal curve is by definition a smooth projective curve admitting a 4-fold covering of the projective line. Donagi showed in 1981 that to any etale double covering of a simply ramified tetragonal covering one can associate 2 other such coverings such that the corresponding Prym varieties are isomorphic. The aim of the talk is to generalize this construction to simply ramified 4-fold coverings Y of an arbitrary curve X. This is joint work with Elham Izadi.
11.11.2009	Eduardo Esteves (IMPA Rio de Janeiro)
	Abel maps for singular curves
	Abstract: Abel maps have been at the center of the study of compact Riemann surfaces, or complex curves, since its roots. I will sketch the history of Abel maps from Abel, passing through Riemann, and will expose on recent progress in extending the construction of such maps to degenerations of Riemann surfaces, more specifically to Deligne-Mumford stable curves.
18.11.2009	Fabrizio Catanese (Univ. Bayreuth)
	Explicit descriptions of connected components of the moduli space of surfaces with p_g =0: Burniat's and Keum Naie surfaces
	(joint work with Ingrid Bauer)
	Abstract: The classification and the moduli space of surfaces with p_g = 0 were on the one side motivated in the 70's by the problem of pluricanonical maps of surfaces of general type, which was finally solved by Bombieri, with final touches by Miyaoka, Benveniste and the present author. The classical Godeaux and Campedelli surfaces, and the more recent Burniat surfaces were then rediscovered. The work by Miyaoka and Reid motivated several interesting and important researches in the field. The discovery of new surfaces with p_g = 0 went on in the 80's , motivated on the one side by the Bloch conjecture, and on the other side by differential topology: for instance a big achievement was the result that the Barlow surface is homeomorphic but not diffeomorphic to the rational surface X obtained by blowing up the plane in 8 points. This in turn had an important byproduct concerning the disproof of the Besse conjecture: together with Claude Le Brun, we showed that the self product of X with itself admits Kaehler Einstein metrics. In the last 5 years there has been an explosion of new results , constructions and partial classification of surfaces with p_g = 0. Due to the celebrated Bogomolov-Miyaoka-Yau inequality the minimal models of these surfaces have K^2 at most 9. Due to Yau's uniformization theorem, the case where K^2 = 9 corresponds to ball quotients, and using a result of Klingler stating the arithmeticity of the fundamental groups a complete classification seems to have been found by Prasad and Yeung, with the help of Steger, Cartwright and a lot of computer calculations. While Park and Lee have been working on the construction of surfaces with p_g = 0 which are simply connected, in the last 3-4 years I have been working with Ingrid Bauer and Fritz Grunewald (later also with Roberto Pignatelli) in order to construct new such surfaces, classifying all such surfaces which can be constructed as the quotient of a product of curves by the action of a finite group. This project also needed computer assistance. We succeed in describing the fundamental groups of such surfaces, and in this way we could construct more than 40 new connected components of the moduli space of surfaces with p_g=0, distinguished by the fundamental group. Looking at the case where the fundamental groups were not new has motivated an investigation of the moduli spaces of certain surfaces with p_g = 0. This is joint work with Ingrid Bauer, and I am going to report on this. We solve the moduli problem for the primary Keum-Naie surfaces and the primary Burniat surfaces, showing that we get in both cases an irreducible connected component of the moduli space to which any surface with the same homotopy type belongs. In the case of primary Burniat surfaces the Inoue description plays a crucial role. More delicate and interesting is the case of secondary Burniat surfaces, having K^2 = 5,4. We show that they form three connected components of the moduli space. In the case of K^2=4, there are two components, the nodal ones yield a Gieseker moduli space (i.e., moduli space of the canonical models) which is everywhere non reduced (a fortiori, the order of nilpotency for the moduli space of canonical models is even higher). While the Burniat surface with K^2 = 2 was already recognized as being a very special Campedelli surface, we find that tertiary Burniat surfaces, i.e. those with K^2 = 3, do not form an irreducible component of the moduli space. We are able to describe explicitly the other surfaces which appear in this component. This work is still in progress and we have not yet decided whether this is also a connected component of the moduli space.
25.11.2009	Alex Abreu (IMPA Rio de Janeiro)
	Wroskian classes in the moduli space of curves
	Abstract: The purpose of this talk is to compute the class of a divisor in the moduli space of stable curves of genus 2n, defined as the closure of the locus of smooth curves C possessing a pair of points (P,Q), such that P is a special ramification point of the linear system K_C(-nQ) and Q is a special ramification point of K_C(-nP).
02.12.2009	Marian Aprodu (IMAR Bucharest)
	Koszul cohomology of canonical curves
	Abstract: We discuss recent developments on syzygies of canonical curves, with special emphasis on curves on K3 surfaces. The talk is mainly based on a joint work in progress with G. Farkas.
09.12.2009	Claire Voisin (IHES Paris)
	Hyper-Kähler fourfolds and Grassmann geometry
	Abstract: This is joint work with Olivier Debarre. The second punctual Hilbert scheme of an algebraic K3 surface is an hyper-Kaehler fourfold with Picard number 2. General deformation theory tells us that given any polarization on them, there is a 20-dimensional family of deformations of this polarized variety. The corresponding moduli spaces can be studied by the period map, and recent work of Gritsenko-Hulek-Sankaran shows that very few of them have $-\infty$ Kodaira dimension. I will describe in this talk a construction of one of these families of deformations, which is unirational. This is the fourth explicitely described such family. It also has a curious numerical property: while it is unirational, it follows from work of Gritsenko-Hulek-Sankaran that there is another family of deformations of a pair as above, where the polarization is also primitive with the same degree, but which has nonnegative Kodaira dimension.
06.01.2010	Samuel Grushevsky (Stony Brook)
	Integrable discrete Schrodinger equations and a characterization of Prym varieties by a pair of quadrisecants
	Alessandro Chiodo (Univ. Grenoble)
	The geometric interpretation of limit roots of line bundles
	Abstract: For r a positive integer, given two line bundles L and N on a curve C, we say that L is an rth root of N if we have an isomorphism between the tensor power L^r and N. Let us recall that the moduli functor of rth roots is proper over the moduli of smooth curves of genus g>1. In other words, consider a family of smooth curves f : C -> B and a line bundle N -> C whose degree on the fibres of f is a multiple of r. Then, the rth roots of the restriction of N to the fibers of f form a proper (and etale) covering S_B of B. When f : C -> B is a stable (nodal) curve with smooth fibres over a nonempty open subscheme of B, we still get an etale covering S_B, but the properness is lost in general. The issue is how to compactify S_B over B. One could think of the points in the compactification as representing "limit roots". We provide two useful interpretations of these points: via semi-stable curves and via stack-theoretic curves. This will allow us to further illustrate the LG/CY correspondence introduced in http://www.raumzeitmaterie.de/veranstaltungen.php?evt=select&sqn=6826& lang=en (NB the material discussed there will not be taken for granted).
27.01.2010	Norbert Hoffmann (FU Berlin)
	Moduli of special instanton bundles
	Abstract: Mathematical instanton bundles are certain algebraic vector bundles on the complex projective space P^{2n+1} with n at least one. I will explain the notion of special instanton bundle and their moduli space, following Spindler and Trautmann. Then I will discuss the birational type of this moduli space. This generalizes work of Hirschowitz and Narasimhan about special t'Hooft bundles on P^3.
03.02.2010	Marcello Bernardara (Univ. Duisburg-Essen)
	A categorical invariant for cubic threefolds
	Abstract: Let X be a smooth cubic threefold. The derived category of X contains a nontrivial triangulated subcategory T as the orthogonal complement of an exceptional sequence. In this talk, I will show how the category T characterizes the isomorphism class of X. This result is a joint work with Macri, Mehrotra and Stellari and is based on the construction of a stability condition on T. More generally, one can consider a conic bundle Y over a rational surface S. I will give at least one more example in which the birational properties of Y are encoded in a triangulated subcategory of the derived category.
10.02.2010	Sukmoon Huh (Korea Institute for Advanced Study KIAS)
	Restriction of vector bundles on surfaces to curves
	Abstract: There are many ways to study the moduli space of stable vector bundles on curves, for example, extension families, theta maps and so on. In this talk, we consider the moduli space of stable sheaves on a surface containing the given curve and investigate the rational map between moduli spaces given by the restriction. This method is effective in studying global properties and also describing higher Brill-Noether loci explicitly. As examples, we will look into the cases of non-hyperelliptic curves of genus 3 and 4.