Humboldt Universität zu Berlin
Institut für Mathematik
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.
|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.
|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.