ScheduleThe talks will be held in Room 144, Huxley Building, 180 Queen's Gate according to the following time table.
TalksSpeaker: Francesco RussoTitle: Explicit n-connectedness by irreducible curves and applications. Abstract: When an irreducible projective variety X is embedded in some projective space P^{N} (or more generally when an arbitrary Cartier divisor D is fixed on X), one can consider the property of being generically n-connected by irreducible curves of a fixed degree δ (with respect to D in the abstract case), that is through n general points of X ⊆ P^{N} there passes an irreducible curve of degree δ contained in X. Natural constraints for the existence of such varieties immediately appear. We shall present results of joint work with Luc Pirio showing that there is a natural bound on N (or on dim |D|) depending on dim(X), n and δ ≥ n-1 such that the boundary examples are rational varieties which are n-connected by smooth rational curves of degree δ in such a way that there exists a unique curve of the family passing though n general points. We shall also present an application of the previous bound to the top intersection of nef divisors on a variety X as above, generalizing a result of Fano. Moreover we shall illustrate the proof for the classification of the boundary cases for n=2 to show a projective incarnation of the proof of S. Mori characterizing projective spaces as the unique projective manifolds with ample tangent bundle. Speakers: Diletta Martinelli and Jakub Witaszek Title: Base point free theorem for log canonical threefolds over the algebraic closure of a finite field Abstract: Our Pragmatic project was about the semiampleness of the anticanonical line bundle in the case of a log canonical threefold defined over the algebraic closure of a finite field. We were able to generalize our result and obtained a base point free theorem. After an introduction about results and techniques over fields of positive characteristic we will give the sketch of the proof of our theorem. The first step is to reduce the problem to the case of surfaces - the surfaces we obtain are actually non irreducible. We prove that the statement is true for each component and then the last part is essentially a gluing problem. This is a joint work with Yusuke Nakamura. Speaker: Andrew Macpherson Title: Skeleta in non-Archimedean and tropical geometry Abstract: Non-Archimedean geometry is a tool to study degenerations of algebraic varieties (or complex manifolds). Tropical geometry is a way to capture leading order information from algebraic degenerations in combinatorial terms. It is therefore natural to try to associate tropical varieties to non-Archimedean analytic spaces. I'll introduce an algebro-geometric theory of skeleta, based on taking "Spec" of a semiring, designed to address this situation. The punchline (of the talk) will be the construction of a "universal tropicalisation" of a rigid analytic space. The talk will end with a food fight. Speakers: Andrea Fanelli and Luca Tasin Title: On the fibres of Mori fibre spaces Abstract: We are interested in understanding when a given Fano variety can be realised as a fibre of a Mori fibre space. We are able to provide two criteria, one sufficient and one necessary, which turn into a characterisation in the rigid case. In this talk we will also show how our criteria can be used to give a complete answer in the case of surfaces, an almost complete picture for 3folds and a combinatorial characterisation on the polytope in the toric case. This talk is based on a joint work with Giulio Codogni and Roberto Svaldi. Speaker: Alessio Corti Title: Fano varieties and mirror symmetry Abstract: The motivating philosophy of this new field is that Fano manifolds can be classified by classifying their Laurent polynomial mirrors. More precisely: starting from a Laurent polynomial f, one may construct a Picard–Fuchs differential operator L_{f}. Similarly, a Fano manifold X determines an associated regularised quantum differential operator Q_{X}, which encodes its quantum cohomology. A Laurent polynomial f is said to be mirror-dual to the Fano manifold X if L_{f} = Q_{X}. Recent work of Corti, Coates et al. reconstructs the classification of smooth del Pezzo surfaces and the Iskovskikh-Mori-Mukai classification of smooth Fano 3-folds by studying Laurent polynomials supported on reflexive polytopes. The projects which fall in this area are studying the possibility of extending the methods of Corti, Coates et al. to singular del Pezzo surfaces (equivalently: Fano orbifold surfaces). Speaker: Alessandro Oneto Title: Examples of Fano-LG correspondence for Del Pezzo orbisurfaces Abstract: The aim of this talk is to show examples of Fano-Landau-Ginzburg correspondence between quantum periods of Del Pezzo orbisurfaces and classic periods of polygons associated to Laurent polynomials. I will focus on some specific toric examples. The talk is based on several papers by T. Coates, A. Corti et al. and it is an ongoing project together with Andrea Petracci. Speaker: Ketil Tveiten Title: Computing the monodromy of the Picard-Fuchs operator Abstract: Given a maximally mutable Laurent polynomial f with Newton polygon P, its classical period integral π_{f}(t) is annihilated by a certain differential operator L_{f}, called the Picard-Fuchs operator of f. The monodromy of this operator around t=0 can be computed from the combinatorial data of P and some analytic and topological considerations. I will show how to do this, and give examples. Speaker: Mohammad Akhtar Title: Mutations of polygons Abstract: tba Speaker: Liana Heuberger Title: Families of del Pezzo surfaces with singularity basket B = {n x 1/3(1,1)} Abstract: tba |