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

Forschungsseminar "Algebraische Geometrie"

Sommersemester 2016

Das Forschungsseminar findet mittwochs in der Zeit von 15:00 - 17:00 Uhr in der Rudower Chaussee 25, 12489 Berlin-Adlershof, Raum 2.006 (Haus 2, Erdgeschoss), statt.

Seminar: Algebraic Geometry an der FU

20.04.2016 Juan Carlos Naranjo (Universitat de Barcelona)
Title: Xiao's conjecture for generic fibred surfaces

Abstract: Xiao's conjecture deals with the relation between the natural invariants present on a fibred surface f:S --> B: the irregularity q of S, the genus b of the base curve B and of the genus g of the fibre of f. In this talk we will prove that the inequality q-b <= g-c, where c is the Clifford index of the generic fibre. This gives in particular a proof of the (modified) Xiao's conjecture, q-b <= g/2 +1, for fibrations whose general fibres have maximal Clifford index. In the first part of the lecture we will review the state of the art and we will give some motivations. This is a joint work with Miguel Ángel Barja and Víctor González-Alonso.
04.05.2016 Alice Rizzardo (Trieste)
Title: An example of a non-Fourier-Mukai functor between derived categories of coherent sheaves

Abstract: Orlov's famous representability theorem asserts that any fully faithful exact functor between the bounded derived categories of coherent sheaves on smooth projective varieties is a Fourier-Mukai functor. We will show that this result is false without the fully faithfulness hypothesis. This is joint work with Michel Van den Bergh.
11.05.2016 Enrico Arbarello (Universita Roma La Sapienza)
Title: Surfaces with canonical sections and Brill-Noether-Petri curves

Abstract: In a joint work with A. Bruno, G. Farkas and G. Saccà we use surfaces having one elliptic singularity, and canonical curves as hyperplane sections, to construct explicit examples of smooth Brill-Noether-Petri curves, of any given genus, defined over the rational numbers. This provides a negative answer to a question raised by Harris and Morrison. The work stems from a joint result with A. Bruno and E. Sernesi where we give a complete characterization of Brill-Noether-Petri curves that are hyperplane sections of K3 surfaces or of a limit thereof.
18.05.2016 Richard Rimanyi (UNC Chapel Hill)
Title: Global singularity theory

Abstract: The topology of the spaces A and B may force every map from A to B to have certain singularities. To a singularity one may associate a polynomial (its Thom polynomial) which measures how topology forces this particular singularity. In this lecture, we will explore the theory of Thom polynomials and their applications to enumerative geometry. Along the way, we will meet a wide spectrum of mathematical concepts such as geometric theorems of the ancient Greeks, quivers, curve-counting, and dilogarithm identities.
01.06.2016 Alexis Kouvidakis (University of Crete)
Title: Maroni Divisors on Hurwitz Spaces

Abstract: The moduli spaces H_{d,g} of covers of genus g and degree d of the projective line are known as Hurwitz spaces. We will discuss the Maroni stratification on such Hurwitz spaces. This stratification has a stratum that is a divisor only if d − 1 divides g and we will approximate the class of this divisor in the compactified Hurwitz space. We will also construct an analogue of the above dvivisors on the Hurwitz spaces H_{d,g} for all pairs (d,g) with d less or equal than g, with a few exceptions. This is joint work with Gerard van der Geer.
15.06.2016 Maryna Viazovska (HU Berlin)
Title: Strongly regular graphs and Prym-Tyurin varieties.

Abstract: A strongly regular graph with parameters (v,k,l,m) is a k-regular graph in which every pair of adjacent vertices has l common neighbors and every pair of non-adjacent vertices has m common neighbors. Surprisingly enough, these combinatorial objects appear in different branches of mathematics. For example, certain types of strongly regular graphs can be used to construct Prym-Tyurin varieties. In this talk we will give an overview of the theory of these graphs, explain the construction of associated Prym varieties, and also will report on new non-existence results for strongly regular graphs.
22.06.2016 Jeremy Guere (HU Berlin)
Title: Quantum invariants for polynomial singularities

Abstract: In this talk, I will describe an analogue of Gromov-Witten theory associated to polynomial singularities. It was first introduced by Fan, Jarvis, and Ruan, and then an algebraic version has been developed by Polishchuk and Vaintrob using matrix factorizations. I will explain how to rephrase it in the language of K-theory and how to obtain an explicit formula for the virtual fundamental cycle of the theory, using a classical result about Koszul cohomology.


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