Humboldt Universität zu Berlin
Mathem.-Naturwissenschaftliche Fakultät
Institut für Mathematik

Forschungsseminar "Algebraische Geometrie"

Sommersemester 2017

Das Forschungsseminar findet mittwochs in der Zeit von 13:00 - 15:00 Uhr in der Rudower Chaussee 25, 12489 Berlin-Adlershof, Raum 2.006, statt.

Seminar: Algebraic Geometry an der FU

19.04.2017 Alessandro Verra (U Roma 3)
Title: On quasi étale coverings of K3 surfaces and their moduli

Abstract: Let X be a complex K3 surface endowed with a cyclic group G of order n of symplectic automorphisms. Then Y := X/G is a normal K3 surface and the quotient map from X to Y is a quasi étale cover. The talk will survey about the well known classification of the families of these surfaces Y and on some recent advances in the study of their geometric properties and of their moduli. The case n = 3, of triple covers, will be specially considered, together with some applications to the study of the moduli spaces of étale triple covers of curves of genus g, g < 6.
26.04.2017 Rahul Pandharipande (ETH Zürich)
Title: Cohomological field theories from local curves

Abstract: I will explain work in progress with H.H. Tseng on the higher genus Gromov-Witten theory of the Hilbert scheme of points of the plane and its relationship to local curve theories.
03.05.2017 kein Seminar
10.05.2017 kein Seminar
17.05.2017 Juan Carlos Naranjo (U Barcelona)
Title: On the Xiao conjecture for fibred surfaces

Abstract: Xiao's conjecture deals with the relation between the natural invariants present on a fibred surface ƒ : SB: the irregularity q of S, the genus b of the base curve B and of the genus g of the fibre of ƒ. In a paper with M. A. Barja and V. González-Alonso we have proved the inequality qbgc, where c is the Clifford index of the generic fibre. This gives in particular a proof of the (modified) Xiao's conjecture, qbg / 2+1, for fibrations whose general fibres have maximal Clifford index. In this talk we will report on that paper and also on a recent progress on the Xiao's conjecture for fibrations whose generic fibre is a plane curve. More precisely in a joint work with F. Favale an G. P. Pirola we have proved the conjecture for fibrations of quintic plane curves, we also have proved the inequality qbgc − 1 for fibrations of plane curves of any degree ≥ 5.
24.05.2017 Marco Ramponi (U Poitiers)
Title: Curves on K3 double covers of Enriques surfaces

Abstract: Recent works by Farkas and Kemeny on the Green-Lazarsfeld and Prym-Green conjectures rely on the possibility of computing Clifford indices and gonalities of curves on special K3 surfaces. In this talk we consider curves lying on K3 surfaces X carrying a fixed-point free involution. Such an automorphism is also called an Enriques involution, since the quotient of X by it is an Enriques surface. Building on work by Knutsen and Lopez around the Brill-Noether theory for curves on Enriques surfaces, we show that the gonality and the Clifford index of all curves on X is governed by the genus 1 fibrations carried by the K3 surface X and coming from the Enriques quotient.
31.05.2017 Patrick Graf (U Bayreuth)
Title: Finite quotients of complex tori

Abstract: Let X be a compact Kähler threefold with canonical singularities and vanishing first Chern class. I will show that if the second orbifold Chern class of X intersects some Kähler form trivially, then X admits a quasi-étale (i.e. étale in codimension one) cover by a complex torus. This result generalizes a theorem of Shepherd-Barron and Wilson for projective varieties. It should be seen as complementing the structure theory of Kähler threefolds (aka Minimal Model Program).
Part of the talk is devoted to explaining the notion of second orbifold Chern class for complex spaces with canonical singularities, since this topic has not been treated in the literature up to now. If time permits, I will also discuss possible generalizations to klt singularities and to higher dimensions. Joint with Tim Kirschner (Essen).
07.06.2017 András Némethi (Rényi Institute Budapest)
Title: Links of surface singularities

Abstract: We prove that a complex normal surface singularity is rational if and only if its link is an L-space (in the sense of Heegaard Floer theory of Ozsvath and Szabo) and if and only if the fundamental group of this link is not left-ordarable. In the proof we will use the lattice cohomology associated with surface singularities as well.
14.06.2017 kein Seminar
21.06.2017 Peter Bürgisser (TU Berlin)
Title: No occurrence obstructions in geometric complexity theory

Abstract: The permanent versus determinant conjecture is a major problem in complexity theory that is equivalent to the separation of the complexity classes VPs and VNP. Mulmuley and Sohoni suggested to study a strengthened version of this conjecture over the complex numbers that amounts to separating the orbit closures of the determinant and padded permanent polynomials. In that paper it was also proposed to separate these orbit closures by exhibiting occurrence obstructions, which are irreducible representations of GL(n2,C), which occur in one coordinate ring of the orbit closure, but not in the other. We prove that this approach is impossible. However, we do not rule out the general approach to the permanent versus determinant problem via multiplicity obstructions. Joint work with Christian Ikenmeyer and Greta Panova (arXiv:1604.06431)
28.06.2017 Jérémy Guéré (HU Berlin)
Title: Enumerative geometry of curves from GIT quotients

Abstract: Gromov-Witten theory of toric varieties is known in any genus. It is not the case for projective hypersurfaces in positive genus, and even in genus zero, it is not known for hypersurfaces in weighted projective spaces. To try to answer this difficult question, Fan, Jarvis, and Ruan have moved to another point of view: they see the polynomial defining the hypersurface as an orbifold singularity and they defined an analogue of Gromov-Witten theory for it. In this talk, I will describe a work in progress with Ciocan-Fontanine, Favero, Kim, and Shoemaker whose goal is to define an analogue of Gromov-Witten theory of complete intersections in toric varieties. The construction is based on matrix factorizations and gives a new point of view on Gromov-Witten theory.
05.07.2017 13:15 - 14:15 Samuel Grushevsky (Stony Brook University)
Title: Vanishing and relations in the tautological ring of M_g, via the theta divisor

Abstract: We consider various Abel-Jacobi maps from the moduli space of curves with marked points to the universal abelian variety, by taking weighted sums of points on the Jacobian. On the universal abelian variety it is known by applying Fourier-Mukai transform that the (g+1)'st power of the universal theta divisor, trivialized along the zero section, vanishes in the Chow ring. By pulling back this vanishing relation under all the Abel-Jacobi maps, we reprove in an elementary combinatorial way the vanishing part of Faber's conjecture on tautological rings of M_{g,n}. Furthermore, pulling back the expressions for the double ramification cycle, we provide an algorithm for explicitly expressing vanishing tautological classes as being supported on the boundary of the Deligne-Mumford compactification. Based on joint work with E. Clader, F. Janda, D. Zakharov
14:30 - 15:30 John Ottem (U Oslo)
Title: Positivity of the diagonal

Abstract: A natural approach to classifying varieties X is via the positivity properties of the diagonal in the self-product X x X. For instance, if the tangent bundle of X is nef, then the diagonal is nef (as a cycle). We analyze when the diagonal is big, movable or nef, with special emphasis on the case of surfaces. We also give several criteria for establishing rigidity of the diagonal. This is joint work with Brian Lehmann.
19.07.2017 Keiji Oguiso (U Tokio)
Title: On complex dynamics of inertia groups on surfaces - a question of Professor Igor Dolgachev

Abstract: I give a few explicit examples which answer an open minded question of Professor Igor Dolgachev on complex dynamics of the inertia group of a smooth rational curve on a projective K3 surface:
"It is challenging to find (further) pairs (S, C) of a projective K3 surface and a smooth rational curve C ⊂ S such that Ine(C) contains an element with positive topological entropy"
and its variants for rational surfaces. I also would like to explain why K3 surfaces and rational surfaces with big inertia groups of smooth rational curves are interesting from both a birational algebraic geometric and a complex dynamics point of view. Some part of my talk will be a joint work with Professor Xun Yu.