Berlin-Hannover Algebraic Geometry Workshop
| Simon Brandhorst (Saarbrücken) | Thomas Dedieu (Toulouse) | |
| Soheyla Feyzbakhsh (Imperial College London) | Alice Garbagnati (Milano) | |
| Christian Gleissner (Bayreuth) | Yoav Len (St. Andrews) | |
| Andrés Rojas (HU Berlin) | Sara Torelli (Hannover) |
| Angela Ortega (HU Berlin) | Matthias Schütt (Hannover) |
Schedule
Abstractsclick to show or hideSimon Brandhorst K3 surfaces of zero entropy (Joint work with Giacomo Mezzedimi) Automorphisms of complex surfaces come in 3 flavours:1) The orbit of every point is finite. 2) There exists a point with an infinite orbit, but no orbit is Zariski dense. 3) There is a Zariski dense orbit. In the first and second case the automorphism has zero topological entropy while in the last case it is of positive entropy. We say that a surface has zero entropy if every of its automorphisms has zero entropy. In this talk we classify K3 surfaces of zero entropy yet with infinite automorphism group, equivalently, K3 surfaces which have a unique elliptic fibration whose Jacobian has infinite Mordell-Weil group. Thomas Dedieu Moduli of curves on Enriques surfaces and Enriques-Fano varieties I will give the respective dimensions of the general fibers of the forgetful map from the moduli space of pairs (S,C) where C is a smooth curve on an Enriques surface S to the moduli space of curves. The irreducible components of the source moduli space correspond to those of the moduli space of polarized Enriques surfaces, which may be identified by means of the so-called simple isotropic decomposition of the polarization. I will focus on the relations between these dimensions and the existence of higher dimensional varieties having Enriques surfaces as linear sections, and pose a number of open problems. This is joint work with C. Ciliberto, C. Galati, and A. L. Knutsen.Soheyla Feyzbakhsh Brill-Noether-type problems on curves through Bridgeland stability conditions In this talk, we will discuss two different methods for studying classical Brill-Noether-type problems on curves using Bridgeland stability conditions on triangulated categories. One approach involves embedding the curve into special types of surfaces, such as K3 surfaces. The other approach uses the bounded derived category of coherent systems. The latter is joint work in progress with Angela Ortega.Alice Garbagnati Bidouble covers of rational surfaces and Hodge structures of K3 type A bidouble coveris a Galois cover whose Galois group is the Klein group. If \(f:X\rightarrow Y\) is a bidouble cover there exist 3 intermediate double covers of \(Y\). We require that these intermediate double covers are either K3 surfaces or surfaces with \(h^{2,0}=0\). In this case the Hodge structure on \(H^2(X,\mathbb{Z})\) splits into the direct sum of Hodge substructures of K3 type (this has strong consequences, for example on the Mumford Tate conjecture and on the infinitesimal Torelli property for these surfaces).Under the condition that \(X\) is smooth and \(Y\) is minimal and rational, we classified the surfaces obtained as above. We will describe these surfaces and we will present generalizations of the previous construction, obtained considering iterated bidouble covers or allowing some singularities. The talk is based on a joint work with M. Penegini. Christian Gleissner Rigid four-dimensional torus quotients In this talk, we provide a fine classification of rigid and free torus quotients in dimension four up to biholomorphism and diffeomorphism. The proof is based on Bieberbach's structure theorems for crystallographic groups. It turns out that all examples arise as quotients of a product of Fermat elliptic curves. This is a joint work with Andreas Demleitner.Yoav Len The Geometry of Prym Varieties The talk will revolve around Prym varieties, a class of Abelian varieties that shows up in the presence of double covers of curves. Pryms have deep connections with torsion points of Jacobians, bitangent lines on quartics, and spin structures. As I will explain, problems concerning Pryms may be reduced, via tropical geometry, to chip-firing games on graphs. Consequently we obtain new results in the geometry of special algebraic curves and a generalization of Kirchhoff's matrix-tree theorem.Andrés Rojas Cohomological rank functions on abelian surfaces via Bridgeland stability In the context of abelian varieties, Z. Jiang and G. Pareschi have introduced interesting invariants called cohomological rank functions, associated to \(\mathbb{Q}\)-twisted (complexes of) coherent sheaves. We will show that, in the case of abelian surfaces, Bridgeland stability provides an alternative description of these functions. This helps to understand their general structure, and allows to compute geometrically meaningful examples. As a main application, we will give new results on the syzygies of abelian surfaces. This is a joint work with Martí Lahoz.Sara Torelli Correspondences acting on constant cycle curves Constant cycle curves on K3 surfaces \(X\) have been introduced by Huybrechts as curves whose points all define the same class in the Chow group. In this talk we introduce correspondences \(Z \subseteq X\times X\) over \(\mathbb{C}\) that act on the group of cycles generated by constant cycle curves. We construct for any \(n\geq 2\) and any very ample line bundle \(L\) a locus \(Z_n(L)\subseteq X\times X\) of expected dimension 2, which yields a correspondence that acts on the group of cycles generated by constant cycle curves, when it has the expected dimension. We provide examples for low \(n\) and use them to produce non rational constant cycle curves.RegistrationFor the registration please write a short email to schuett at math.uni-hannover.de with your name and affiliation. Location
Due to construction, there will be no S-Bahn connection between Schöneweide and Adlershof stations. Instead, busses are running between these two stations to replace the journey. If you are coming from Mitte we suggest you to take the S-Bahn to Ostkreuz. Change there to the regional train and then get off in Schöneweide (which goes non-stop between Ostkreuz and Schöneweide) to take the bus. Hotel Recommendations
Hotels in Aldershof: AcknowledgementThe event is generously supported by: "MATH+ Distinguished Fellow" prize from Gavril Farkas. 
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