Forschungsseminar Algebraische Geometrie

Sommersemester 2012

11.04.2012 kein Seminar

18.04.2012 Balazs Szendroi (Oxford)

Nekrasov's formula and refined sheaf counting

Abstract: I revisit the identification of Nekrasov's K-theoretic partition function, counting instantons on R⁴, and the (refined) Donaldson-Thomas partition function of the associated local Calabi-Yau threefold. The main example will be the case of the resolved conifold, corresponding to the gauge group U(1). I will show how recent mathematical results about refined DT theory confirm this identification, and speculate on how one could lift the equality of partition functions to an isomorphism of vector spaces.

25.04.2012 Yuri Tschinkel (New York University)

Almost abelian anabelian geometry

Beginn: 15.00 Uhr

Carel Faber (KTH Stockholm)

Detecting and constructing Teichmüller modular forms

Abstract: A classical Teichmüller modular form is a section on the moduli space M_g of a power of the determinant of the Hodge bundle. Vector-valued Teichmüller modular forms are sections on M_g of the vector bundles obtained by applying Schur functors for irreducible representations of GL(g) to the Hodge bundle. Siegel modular forms, sections on A_g of the corresponding bundles, give Teichmüller modular forms via pull-back to M_g, but for g at least 3 there exist ‘genuine’ Teichmüller modular forms, not obtained via pull-back. Classical Teichmüller modular forms have been studied in detail by Ichikawa. I will try to explain how we have found, in a rather roundabout way, strong indications for the existence of genuine Teichmüller modular forms, in four cases. If time permits, I will discuss how a natural construction appears to produce many Teichmüller modular forms. The construction covers in particular the four cases mentioned. This is work in progress, joint with Jonas Bergström and Gerard van der Geer.

Beginn: 16.15 Uhr

02.05.2012 Angela Ortega (HU Berlin)

The Brill-Noether curve and Prym-Tyurin varieties

Abstract: Prym-Tyurin varieties are principally polarized abelian subvarieties of a Jacobian, which generalize the classical Prym varieties associated to a étale double covers between curves. In this talk we will show that the general curve C of genus g=2a+1 can be realized as a Prym-Tyurin variety for the Brill-Noether curve W¹_a+2(C). As a consequence of this result we are able to compute the class of the sum of secant divisors of the curve C, embedded with a complete linear series g^a-1_3a-2.

Beginn: 15.15 Uhr

Ronnie Sebastian (HU Berlin)

Smash nilpotence on abelian varieties

Abstract: To have a good theory of algebraic cycles on smooth projective varieties, one has to go modulo equivalence relations on the group of cycles. One such equivalence relation is smash nilpotence, which was introduced by Voevodsky, who conjectured it to be same as numerical equivalence. After defining an adequate equivalence relation and giving some examples, I will state some results from Kimura's theory of finite dimensionality of motives, and as an application we will see some non trivial examples of smash nilpotent cycles.

Beginn: 16.30 Uhr

09.05.2012 Alessandro Verra (Univ. Roma Tre)

Prym moduli spaces in low genus and nodal conic bundles

Abstract: The talk reports some results and conjectures on the global geometry of the Prym moduli space R_g for g = 6, 7, 8. Parametrizations via some Severi varieties of nodal conic bundles are described. Related question on the uniruledness of R₈ and new proofs of the unirationality of R₇ and R₆ are proposed.

16.05.2012 Gian Pietro Pirola (Univ. Pavia)

The Fano surface of the cubic 3-fold and its normal function

Abstract: The Fano surface of the cubic 3-fold has been used by Clemens and Griffiths to prove the nonrationality of a smooth cubic 3-fold. The Fano surfaces also give very interesting examples of irregular surfaces. We discuss their geometry in connection with classification problems on surfaces and the theory of cycles on abelian varieties. In particular we study a natural normal function defined by the embeddings of the Fano surfaces in their Albanese varieties. The new results have been obtained in a joint work with Alberto Collino and Juan Carlos Naranjo.

23.05.2012 kein Seminar

30.05.2012 Noah Giansiracusa (Univ. Zürich)

GIT compactifications of M_0,n and flips

Abstract: Despite intensive investigation over the years and its innocuously classical appearance, there remain some tantalizing open questions regarding the birational geometry of the moduli space of marked rational curves. One such question, dating from a 2000 paper of Hu and Keel, is to determine whether M_0,n is a Mori dream space. Roughly speaking, this would say that its Mori-theoretic information is completely determined by variational GIT in a natural way. In this talk I will discuss joint work with Dave Jensen and Han-Bom Moon in which we construct a wide range of birational models using a GIT construction inspired by two constructions of Kapranov from 1993. These models include M_0,n itself as well as all the Hassett weighted models. A consequence is that we exhibit explicit flips between models and give a glimpse of the Mori dream behavior envisioned by Hu and Keel.

06.06.2012 Arnaud Beauville (Univ. Nice)

Abelian varieties associated to Gaussian lattices

Abstract: Let Γ be a self-dual lattice, endowed with an automorphism of square -1. Then A_Γ =Γ_R/Γ is a principally polarized abelian variety, with an automorphism i of square -1. I will show that the configuration of i-invariant theta divisors of A_Γ follows a pattern very similar to the classical theory of theta characteristics; as a consequence A_Γ has a high number of vanishing thetanulls. When Γ =E₈ we recover the 10 vanishing thetanulls of the abelian fourfold discovered by R. Varley.

13.06.2012 kein Seminar

20.06.2012 kein Seminar

27.06.2012 kein Seminar

04.07.2012 Gerard van der Geer (Univ. Amsterdam)

Vector-valued Picard modular forms and curves of genus three

Abstract: The talk deals with vector-valued Picard modular forms on a unitary group of signature (2,1) over the ring of Eisenstein numbers. We construct modular forms and discuss the structure of modules of such modular forms. It is related to the cohomology of local systems on a moduli space of curves of genus three. This is joint work with Fabien Clery and Jonas Bergstroem.

11.07.2012 Frank Gounelas (HU Berlin)

Free curves on varieties

Abstract: This talk will be about various ways in which a variety can be "connected by curves of a fixed genus", mimicking the notion of rational connectedness. At least in characteristic zero, in the specific case of the existence of a single curve with a large deformation space of morphisms to a variety implies that the variety is in fact rationally connected. Time permitting I will discuss attempts to show this result in positive characteristic.

11.04.2012	kein Seminar

18.04.2012	Balazs Szendroi (Oxford)
	Nekrasov's formula and refined sheaf counting
	Abstract: I revisit the identification of Nekrasov's K-theoretic partition function, counting instantons on R⁴, and the (refined) Donaldson-Thomas partition function of the associated local Calabi-Yau threefold. The main example will be the case of the resolved conifold, corresponding to the gauge group U(1). I will show how recent mathematical results about refined DT theory confirm this identification, and speculate on how one could lift the equality of partition functions to an isomorphism of vector spaces.
25.04.2012	Yuri Tschinkel (New York University)
	Almost abelian anabelian geometry
	Beginn: 15.00 Uhr
	Carel Faber (KTH Stockholm)
	Detecting and constructing Teichmüller modular forms
	Abstract: A classical Teichmüller modular form is a section on the moduli space M_g of a power of the determinant of the Hodge bundle. Vector-valued Teichmüller modular forms are sections on M_g of the vector bundles obtained by applying Schur functors for irreducible representations of GL(g) to the Hodge bundle. Siegel modular forms, sections on A_g of the corresponding bundles, give Teichmüller modular forms via pull-back to M_g, but for g at least 3 there exist ‘genuine’ Teichmüller modular forms, not obtained via pull-back. Classical Teichmüller modular forms have been studied in detail by Ichikawa. I will try to explain how we have found, in a rather roundabout way, strong indications for the existence of genuine Teichmüller modular forms, in four cases. If time permits, I will discuss how a natural construction appears to produce many Teichmüller modular forms. The construction covers in particular the four cases mentioned. This is work in progress, joint with Jonas Bergström and Gerard van der Geer.
	Beginn: 16.15 Uhr

02.05.2012	Angela Ortega (HU Berlin)
	The Brill-Noether curve and Prym-Tyurin varieties
	Abstract: Prym-Tyurin varieties are principally polarized abelian subvarieties of a Jacobian, which generalize the classical Prym varieties associated to a étale double covers between curves. In this talk we will show that the general curve C of genus g=2a+1 can be realized as a Prym-Tyurin variety for the Brill-Noether curve W¹_a+2(C). As a consequence of this result we are able to compute the class of the sum of secant divisors of the curve C, embedded with a complete linear series g^a-1_3a-2.
	Beginn: 15.15 Uhr

	Ronnie Sebastian (HU Berlin)
	Smash nilpotence on abelian varieties
	Abstract: To have a good theory of algebraic cycles on smooth projective varieties, one has to go modulo equivalence relations on the group of cycles. One such equivalence relation is smash nilpotence, which was introduced by Voevodsky, who conjectured it to be same as numerical equivalence. After defining an adequate equivalence relation and giving some examples, I will state some results from Kimura's theory of finite dimensionality of motives, and as an application we will see some non trivial examples of smash nilpotent cycles.
	Beginn: 16.30 Uhr

09.05.2012	Alessandro Verra (Univ. Roma Tre)
	Prym moduli spaces in low genus and nodal conic bundles
	Abstract: The talk reports some results and conjectures on the global geometry of the Prym moduli space R_g for g = 6, 7, 8. Parametrizations via some Severi varieties of nodal conic bundles are described. Related question on the uniruledness of R₈ and new proofs of the unirationality of R₇ and R₆ are proposed.
16.05.2012	Gian Pietro Pirola (Univ. Pavia)
	The Fano surface of the cubic 3-fold and its normal function
	Abstract: The Fano surface of the cubic 3-fold has been used by Clemens and Griffiths to prove the nonrationality of a smooth cubic 3-fold. The Fano surfaces also give very interesting examples of irregular surfaces. We discuss their geometry in connection with classification problems on surfaces and the theory of cycles on abelian varieties. In particular we study a natural normal function defined by the embeddings of the Fano surfaces in their Albanese varieties. The new results have been obtained in a joint work with Alberto Collino and Juan Carlos Naranjo.
23.05.2012	kein Seminar

30.05.2012	Noah Giansiracusa (Univ. Zürich)
	GIT compactifications of M_0,n and flips
	Abstract: Despite intensive investigation over the years and its innocuously classical appearance, there remain some tantalizing open questions regarding the birational geometry of the moduli space of marked rational curves. One such question, dating from a 2000 paper of Hu and Keel, is to determine whether M_0,n is a Mori dream space. Roughly speaking, this would say that its Mori-theoretic information is completely determined by variational GIT in a natural way. In this talk I will discuss joint work with Dave Jensen and Han-Bom Moon in which we construct a wide range of birational models using a GIT construction inspired by two constructions of Kapranov from 1993. These models include M_0,n itself as well as all the Hassett weighted models. A consequence is that we exhibit explicit flips between models and give a glimpse of the Mori dream behavior envisioned by Hu and Keel.
06.06.2012	Arnaud Beauville (Univ. Nice)
	Abelian varieties associated to Gaussian lattices
	Abstract: Let Γ be a self-dual lattice, endowed with an automorphism of square -1. Then A_Γ =Γ_R/Γ is a principally polarized abelian variety, with an automorphism i of square -1. I will show that the configuration of i-invariant theta divisors of A_Γ follows a pattern very similar to the classical theory of theta characteristics; as a consequence A_Γ has a high number of vanishing thetanulls. When Γ =E₈ we recover the 10 vanishing thetanulls of the abelian fourfold discovered by R. Varley.
13.06.2012	kein Seminar

20.06.2012	kein Seminar

27.06.2012	kein Seminar

04.07.2012	Gerard van der Geer (Univ. Amsterdam)
	Vector-valued Picard modular forms and curves of genus three
	Abstract: The talk deals with vector-valued Picard modular forms on a unitary group of signature (2,1) over the ring of Eisenstein numbers. We construct modular forms and discuss the structure of modules of such modular forms. It is related to the cohomology of local systems on a moduli space of curves of genus three. This is joint work with Fabien Clery and Jonas Bergstroem.
11.07.2012	Frank Gounelas (HU Berlin)
	Free curves on varieties
	Abstract: This talk will be about various ways in which a variety can be "connected by curves of a fixed genus", mimicking the notion of rational connectedness. At least in characteristic zero, in the specific case of the existence of a single curve with a large deformation space of morphisms to a variety implies that the variety is in fact rationally connected. Time permitting I will discuss attempts to show this result in positive characteristic.