Sommersemester 2011

13.04.2011 Florian Geiß (Univ. des Saarlandes)
The unirationality of Hurwitz-spaces of 6-gonal covers of genus less or equal than 40
Abstract: It is well known that the Hurwitz-spaces H_(d,g) of d-gonal covers of genus g are unirational for d=<5 and g>d-1. With a liaison construction we can prove the unirationality of H_(6,g) for g=<40. Our approach also involves computer-algebra in a substantial way. It is also shown how these results contribute to the existence of stable Ulrich bundles on cubic threefolds recently proved by M. Casanellas and R. Hartshorne.
20.04.2011 kein Seminar
27.04.2011  kein Seminar
04.05.2011 Bernd Sturmfels (Univ. of California Berkeley)
Quartic Curves and their Bitangents
Abstract: A smooth quartic curve in the complex projective plane has 36 inequivalent representations as a symmetric determinant of linear forms and 63 representations as a sum of three squares. These correspond to Cayley octads and Steiner complexes respectively. We present exact algorithms for computing these objects from the 28 bitangents. This expresses Vinnikov quartics as spectrahedra and positive quartics as Gram matrices. We explore the geometry of Gram spectrahedra and we find equations for the variety of Cayley octads. Interwoven is an exposition of much of the 19th century theory of plane quartics.
11.05.2011 kein Seminar (NoGAGS 12.-13.05.)
18.05.2011Dawei Chen (Univ. of Illinois at Chicago)
Geometry of Teichmueller Curves
Abstract: Teichmueller curves are central objects in geometry and dynamics. They provide fertile connections between polygon billiards, flat surfaces and moduli spaces. A class of special Teichmueller curves come from a branched cover construction. Using them as examples, I will introduce an algebro-geometric technique to study their dynamical properties. As an application, joint with Martin Moeller we prove Kontsevich-Zorich's conjecture about the non-varying property of Siegel-Veech constants of Abelian differentials in low genus. If time allows, I will also talk about their behavior in large genus to study the geometry of moduli space of curves. This talk will be accessible to a general audience.
25.05.2011 kein Seminar
01.06.2011 Valery Alexeev (Univ. of Georgia)
Explicit compactifications of moduli of surfaces of general type
08.06.2011 kein Seminar
15.06.2011 kein Seminar
22.06.2011 kein Seminar
29.06.2011 kein Seminar
06.07.2011 James McKernan (Massachusetts Institute of Technology)
Automorphisms of Varieties
Abstract: We give a survey of what is known about how many symmetries an algebraic variety can possess. We emphasize two relatively divergent cases, one where the automorphism group is relatively small, a case which is characterised by a classical theorem due to Hurwitz and one where the automorphism group is relatively large, a case which is characterised by a classical theorem due to Noether.
13.07.2011 Elham Izadi (Univ. of Georgia)
Counter-examples of high Clifford index to Prym-Torelli
Abstract: For an etale double cover of smooth curves, the Prym variety is essentially the ``difference'' between the jacobians of the two curves. The Torelli problem for the Prym map asks when two double covers have the same Prym variety. It is known that the Prym map from the moduli space of double covers of curves of genus g at least 7 to principally polarized abelian varieties of dimension g-1 is generically injective. Counter-examples to the injectivity of the Prym map were, up to now, given by Donagi's tetragonal construction and by Verra's construction for plane sextics. It was asked by Lange and Sernesi whether all counter-examples are obtained from double covers of curves of Clifford index at most 3. I will discuss counter-examples to this constructed by Herbert Lange and myself.