Summer Semester 2019
Unless otherwise specified talks take place in room 119 (Arnimallee 3) at 16:15
!! Tuesday, 23.04.2019 at 12: 15 in room 210 (Arnimallee 3) !!
Duco van Straten (Mainz)
Continuous algebraic geometry: exploring fractional dimension.
Abstract: The concept of interpolation to a continuous variable is very old, and has turned out to be very useful, the Gamma-function being an example in case. In this talk I will report on joint work with V. Golyshev on projective spaces and Grassmanians of fractional dimension and the algebraic geometry related to it.
Jarek Wisniewski (Warsaw)
Combinatorics of C* action and varieties of small bandwidth.
Abstract: Motivated by LeBrun-Salamon conjecture about quaternion-Kahler manifolds we study polarized pairs (X,L) with an action of C* on the complex manifold X such that its general orbit has small degree with respect to the ample line bundle L. For this purpose we describe adjunction morphism for (X,L) using combinatorics arising from the action of C*. In passing we get a reinterpretation of classical objects in projective and birational geometry like Severi varieties and special Cremona tranformations. I will report on a joint work with Romano and Ochetta, Sola Conde.
Luca Battistella (MPI, Bonn)
Reduced Gromov-Witten invariants and singularities of genus one
Abstract: Classical enumerative geometry produced beautiful theorems such as: every smooth cubic surface contains exactly 27 lines. The subject has been reinvigorated since the introduction of the moduli space of stable maps, which, for example, allowed Kontsevich to solve a long-standing problem: the number of rational plane curves of degree d passing through 3d-1 general points. More generally, it made it possible to answer many questions on rational curves in projective complete intersections.
Dealing with curves of higher genus is more difficult, because the moduli spaces are not so well behaved. I will explain this with an example in genus one. We will then see two different approaches to the problem: a desingularisation of the moduli space, which led to the definition of reduced invariants, due to Li-Vakil-Zinger, and alternative compactifications obtained by Viscardi by allowing the source curve to acquire a singularity of genus one. For the quintic threefold, the two approaches are put in relation in joint work with Carocci and Manolache.
If time permits, I will discuss how log geometry - a far-fetching generalisation of toric geometry which is often useful to single out and improve the main component of moduli spaces - enters the picture, both by producing natural contractions to singular curves, and allowing us to study the relative problem, i.e. a count of curves tangent to a hyperplane section. This is joint work with Nabijou and Ranganathan, based on a ground-breaking paper of Ranganathan, Santos-Parker, and Wise.
Matej Filip (Mainz)
The versal deformation for toric varieties in special lattice degrees.
Abstract: I will describe the versal deformation for non-solated Gorenstein toric varieties in the Gorenstein lattice degree.
Klaus Altmann (FU-Berlin)
Deformation of toric singularities by universal extensions of semigroups
Abstract: We show how extensions of semigroups lead to deformations of toric singularities. On the level of semigroups, universal extensions exist, and they lead to versal deformations of the toric singularities in a prescribed multidegree. This is joint work with Alexandru Constantinescu (FUB) and Matej Filip (Mainz).
Christian Sevenheck (Chemnitz)
Hodge ideals for certain free divisors
Abstract: In recent years, there has been renewed interest in the theory of mixed Hodge modules, mainly motivated by questions from birational geometry. In particular, Mustata and Popa have defined the so-called Hodge ideals, which describes to a certain extend to Hodge filtration on the complement of a singular divisor. In this talk, I will explain what Hodge ideals are and then discuss a particular case, namely that of certain free divisors, where quite explicit statements about the Hodge filtration can be made. This is joint work with Luis Narvaez Macarro (Sevilla) and Alberto Castaño Domínguez (Santiago de Compostella).