Our seminar is partially supported by Sonata Bis 3 Narodowe Centrum Nauki Poland Grant No NCN 2014/10/E/ST1/00688.
19 - 22 May 2018, Motives of Calabi-Yau Manifolds
Here is the list of talks given during our non-regular local seminar at Polish Academy of Sciences in Cracow. We usually meet together at chosen Wednesdays around 12.45. Each talk is about 1h 30 min.
In this talk, I will discuss the notion of lattice-polarized irreducible holomorphic symplectic manifolds and the construction of their moduli spaces. Then I will explain a generalization of a lattice-theoretical mirror symmetry, as known for K3 surfaces, to higher dimensions. Finally we will see some examples in the case of fourfolds of K3^-type.
In this talk, we review the notion of Mori Dream Spaces in birational terms, and we speak about their main geometrical properties, such as the bijection between contractions and faces on movable cones. Then we focus on some known results about blow-ups of Projective Spaces which are Mori Dream Spaces, by discussing the geometrical information which arise from the structure of their effective cones.
Donaldson-Thomas invariants enumerate sheaves on Calabi-Yau 3-folds. The rank one case corresponds to counting curves in a given homology class.
We will explain the idea of motivic refinements of these numbers. We provide a construction of motivic DT invariants in the rank one case:
these invariants are “local”, in that they refine the contribution of a single (smooth) curve to the DT invariants attached to its homology class.
Given a subset of points Z in the projective plane over the complex numbers, we are interested in cases in which a general fat point P does not impose independent conditions on the linear system [I(Z)]_d. When this occurs, we say that Z admits an unexpected curve of degree d. Recently, an example of a set of nine points in P^2 admitting an unexpected quartic was given by R. Di Gennaro, G. Ilardi and J. Valles, and studied by D. Cook II, B. Harbourne, J. Migliore and U. Nagel. We discuss this example, and study which subsets of points in the projective plane admit an unexpected curve of degree 4. This is joint work with F. Galuppi, L. Sodomaco and B. Trok.
I am going to provide a short introduction into two interesting problems on surfaces of general type, namely the BNC and Vojta's conjecture. These conjectures are devoted to reduced and irreducible algebraic curves. I will present a short outline and at the end I will present some steps towards the theory of abelian covers of surfaces branched along curve configurations. The ultimate goal is to pass towards Fermat covers of P^3 branched along arrangements of eight hyperplanes. This will be useful in the context of Chadmark Schoen's paper on Albanese exotic 3-folds.
I will speak about finite groups acting by birational automorphisms of surfaces over algebraically non-closed fields, mostly function fields. One of important observations here is that a smooth geometrically rational surface S is either birational to a product of a projective line and a conic (in particular, S is rational provided that it has a point), or finite subgroups of its birational automorphism group are bounded. I will also discuss some particular types of surfaces with interesting automorphism groups, including Severi-Brauer surfaces. The talk is based on a joint work with Vadim Vologodsky.