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

Forschungsseminar "Algebraische Geometrie"

Wintersemester 2014/15

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

15.10.2014 Mike Roth (Queen's University, zzt. HU Berlin)
Title: Roth’s theorem for arbitrary varieties

Abstract: If X is a variety of general type defined over a number field k, then the Bombieri-Lang conjecture predicts that the k-rational points of X are not Zariski dense. The conjecture is a prediction that a global condition on the canonical bundle (that it is ''generically positive'') implies a global condition about rational points. By the local-global philosophy in geometry we should look for local influence of positivity on the accumulation of rational points. To do that we need measures of both these local phenomena. Let L be an ample line bundle on X, and x an algebraic point. The central theme of the talk is the interrelations between αx(L), an invariant measuring the accumulation of rational points around x as gauged by L, and the Seshadri constant εx(L), measuring the local positivity of L near x. In particular, the classic approximation theorem of K. F. Roth on P1 generalizes as an inequality between αx and εx valid for all projective varieties.
22.10.2014 Thomas Krämer (Universität Heidelberg)
Title: Generic vanishing theory, cubic threefolds and the monodromy of the Gauss map

Abstract: To any closed subvariety of an abelian variety one may attach a reductive algebraic group in a natural way, using the Tannakian formalism. The arising groups are new invariants with interesting applications to the moduli of abelian varieties and the Schottky problem, but their proper geometric interpretation remains mysterious. After a motivated introduction to the Tannakian framework via a generalization of the Green-Lazarsfeld vanishing theorems, I will show that for the theta divisor on the intermediate Jacobian of a smooth cubic threefold the Tannaka group is an exceptional group of type E6. This is the first known exceptional case, and it suggests a surprising connection with the monodromy of the Gauss map and the Fourier-Mukai transform for Higgs bundles.
29.10.2014 Jarod Alper (Australian National University, zzt. HU Berlin)
Title: A Luna etale slice theorem for algebraic stacks

Abstract: Quotient stacks are a distinguished class of algebraic stacks which provide key intuition for studying the geometry of general algebraic stacks. It has long been believed that certain algebraic stacks are in some sense "locally" quotient stacks. In this talk, we will prove that this expectation holds by providing a description of the etale local structure of algebraic stacks near points with linearly reductive stabilizer. We will then discuss a number of striking applications of this result. This is joint work with Jack Hall and David Rydh.
05.11.2014 kein Seminar
12.11.2014 kein Seminar
19.11.2014 Michael Joswig (TU Berlin)
Title: Moduli of Tropical Plane Curves

Abstract: We study the moduli space of metric graphs that arise from tropical plane curves. There are far fewer such graphs than tropicalizations of classical plane curves. For fixed genus g, our moduli space is a stacky fan whose cones are indexed by regular unimodular triangulations of Newton polygons with g interior lattice points. It has dimension 2g+1 unless g≤3 or g=7. We compute these spaces explicitly for g≤5. Joint work with Sarah Brodsky, Ralph Morrison and Bernd Sturmfels.
21.11.2014 Friday Complex Algebraic Geometry - A workshop on the occasion of Herbert Kurke's 75th birthday
26.11.2014 Benjamin Bakker (HU Berlin)
Title: Bounding torsion in geometric families of abelian varieties

Abstract: A celebrated theorem of Mazur asserts that the order of the torsion part of the group of rational points of an elliptic curve over Q is absolutely bounded; it is conjectured that the same is true for abelian varieties over number fields K, though very little progress has been made in proving it. The natural geometric analog where K is replaced by the function field of a complex curve - known as the geometric torsion conjecture - is equivalent to the nonexistence of low genus curves in congruence towers of the moduli space of abelian varieties. In joint work with J. Tsimerman, we prove this conjecture for abelian varieties with real multiplication. We will discuss a general method for bounding the genera of curves in locally symmetric varieties using hyperbolic geometry to bound Seshadri constants and apply it to some related problems.
03.12.2014 kein Seminar
10.12.2014 Damiano Testa (University of Warwick)
Title: K3 surfaces arising from arithmetic problems

Abstract: In my talk I will present three different arithmetic problems. In all these problems, the easiest unknown case asks for rational points on K3 surfaces. I will show how to use the common features of these K3 surfaces to obtain geometric information about these surfaces and how in turn to use this information to obtain some insight on the initial problems.
17.12.2014 Andrew Kresch (Universität Zürich)
Title: Relative stable maps, logarithmic stable maps and virtual localization

Abstract: This largely expository talk will begin with a brief introduction to logarithmic geometry and proceed with a survey of some developments related to the two notions of stable maps mentioned in the title and the virtual localization formulas in each setting.
07.01.2015 Remke Nanne Kloosterman (HU Berlin)
Title: Ciliberto-Di Gennaro conjecture on non-factorial hypersurfaces

Abstract: In 2004 Ciliberto and Di Gennaro conjectured that a nodal threefold of degree d in P4 with at most 2(d-2)(d-1) nodes is either factorial, contains a plane or contains a quadric surface. In this talk we present a proof for this conjecture if d is at least 7. We use techniques similar to the ones Voisin used in her proof for the fact that the two largest components of the Noether-Lefschetz locus of degree d surfaces in P3 are the components parametrizing surfaces containing a line or a conic.
14.01.2015 Alessandro Verra (Università Roma Tre)
Title: On Nikulin surfaces of genus 10 and related topics

Abstract: In the talk the moduli space N10 of complex Nikulin surfaces with a genus 10 polarization is considered. A standard construction provides an embedding of a general Nikulin surface S of genus 10 in the Grassmannian G(2,6). Such a realization implies that a general S can be recovered from the data (A, E, L), where E is a stable rank 2 vector bundle of degree 10 on a general curve A of genus 2 and L is a rank one subbundle of E of degree 4. Building on these data various properties of the birational geometry of N10 are deduced, in particular its unirationality. The final part of the talk is dedicated to survey the moduli spaces of K3 surfaces with a symplectic involution and their birational geometry.
21.01.2015 kein Seminar
28.01.2015 Daniele Agostini (HU Berlin)
Title: Asymptotic behaviour of graded Betti numbers

Abstract: A recent conjecture of L. Ein, D. Erman and R. Lazarsfeld says that the graded Betti numbers of a projective variety have an asymptotic Gaussian behaviour as the positivity of the embedding grows. They have proved this for curves and in this talk I will present an approach for the next simplest cases, namely P2 and P1 x P1.
04.02.2015 Workshop jointly organized with Rahul Pandharipande and Einstein Stiftung Berlin
Motivic invariants related to K3 and abelian geometries, Berlin, February 3-6, 2015

11.02.2015 Seminar Einstein Stiftung: Rahul Pandharipande und Felix Janda (ETH Zürich)
Title: Pixton's formula for double ramification cycles I and II

Abstract: We will explain Pixton's conjectural formula, the geometric background and a strategy for proof.


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