Title: Fundamental groups of low-coregularity Calabi-Yau type pairs.
Abstract: Our object of study will be the orbifold fundamental groups of the smooth locus of Calabi-Yau type pairs. The absolute coregularity of a pair, is a measure of combinatorial complexity of a variety. We will show that the fundamental groups of klt pairs with low-coregularity and arbitrary dimension behave similarly to the fundamental groups of low-dimensional Calabi-Yau pairs. More explicitly, we will show that klt pairs with coregularity 0 have finite fundamental groups, and klt pairs with coregularity 1 or 2 have virtually abelian fundamental groups of rank at most 2 or 4. The indices of the abelian subgroups cannot be uniformly bound, even if we control the dimension. In this direction, we will end with a statement that gives effective bounds on the index of solvable groups in the case of pairs of fixed dimension and low-coregularity. This talk is based on joint work with Lukas Braun.
Abstract: I will report on joint work with Valentino Tosatti and Simion Filip in which we show by example that for a pseudo-effective divisor D and ample A, the volume function vol(D+tA) for small values of t can exhibit various pathological behaviors.
Abstract: Esnault-Viehweg (resp. S. Ishii) proved that two-dimensional klt (resp. lc) singularities are stable under small deformations. Unfortunately, an analogous statement fails in higher dimensions, because the generic fiber is not necessarily Q-Gorenstein if the special fiber is klt. In this talk, I present a generalization of the results of Esnault-Viehweg and Ishii under the assumption that the generic fiber is Q-Gorenstein (but the total space is not necessarily Q-Gorenstein). This talk is based on joint work with Kenta Sato.
Title: Irrationality of degenerations of Fano varieties
Abstract:In this talk, I will introduce a recent result about bounding degrees of irrationality of degenerations of klt Fano varieties of arbitrary dimensions. This proves the generically bounded case of a conjecture proposed by C. Birkar and K. Loginov for log Fano fibrations of dimensions greater than three. Our approach depends on a method to modify the klt Fano fibration to a toroidal morphism of toroidal embeddings with bounded general fibres. This is a joint work with Prof. C. Birkar.
Title: Fano 4-folds with b_2 > 12 are products of surfaces.
Abstract: Let X be a smooth, complex Fano 4-fold, and b_2 its second Betti number. We will discuss the following result: if b_2 > 12, then X is a product of del Pezzo surfaces. The proof relies on a careful study of divisorial elementary contractions f : X -> Y such that the image S of the exceptional divisor is a surface, together with my previous work on Fano 4-folds. In particular, given f : X -> Y as above, under suitable assumptions we show that S is a smooth del Pezzo surface with -K_S given by the restriction of -K_Y.
Title: Polarized endomorphisms of Fano varieties with complements.
Abstract: An endomorphism on a normal projective variety X is said to be polarized if the pullback of an ample divisor A is linearly equivalent to a qA, for some integer q>1. Examples of these endomorphisms are naturally found in toric varieties and abelian varieties. Indeed, it is conjectured that if X admits a polarized endomorphism, then X is a finite quotient of a toric fibration over an abelian variety. In this talk, I will restrict to the case when X is a Fano type variety admitting a 1-complement, meaning that there exists an effective divisor B such that (X,B) is log Calabi-Yau, and K_X + B ~ 0. I will prove that if (X,B) has a polarized endomorphism that preserves the complement structure, then (X,B) is a finite quotient of a toric log Calabi-Yau pair. This is joint work with Joaquin Moraga and Wern Yeong.
Title: Birational complexity of log Calabi-Yau pairs.
Abstract: The complexity of a log Calabi-Yau pair is a rather simple invariant that measures how close the pair is from being toric.
The birational complexity, as the name suggests, is a birational variant of the aforementioned invariant.
The birational complexity measures how close is a given log Calabi-Yau pair from having a toric birational model.
In this talk, I will introduce this new invariant, and show how it reflects on the topology of dual complexes
and the existence of conic bundles on birational models of a log Calabi-Yau pair.
Abstract: We study equivariant geometry and rationality of moduli spaces of points on the projective line, for twists associated with
permutations of the points. (joint with Yuri Tschinkel and Zhijia Zhang)
Title: Umemura quadric fibrations and maximal subgroups of Cremona groups.
Abstract: Maximal connected algebraic subgroups of groups of birational transformations Bir(X) of a projective variety X appear as automorphism groups of Mori fiber spaces birational to X. When X = P^n, for n = 2, 3, we have a classification of such groups and the
models they act on, while in dimension 4 and on, this is an open problem. In this talk, after explaining the main machinery and known results, we will explore the equivariant geometry of a class of quadric fibrations over P^1. As a result, we will see how their automorphism
groups gives us infinite families of maximal subgroups of Bir(P^n), for any n at least 3. This is joint work with Enrica Floris.
Title:Rationality questions for Fano schemes of intersections of two quadrics.
Abstract:We study rationality questions for the Fano schemes of non-maximal linear spaces on a smooth complete intersection X of two quadrics, especially over non-closed fields. We start by showing that they are all geometrically rational. We then ask their rationality over k and analyze in details the case of second maximal linear spaces. In particular, we generalize results of Hassett-Tschinkel and Benoist-Wittenberg when X has odd dimension, and extend work of Hassett-Kollár-Tschinkel when X has even dimension and k = R. This is joint work with Lena Ji.
Abstract: Recall that an algebraic variety is rational if it is birational to a projective space. In the past decade it was established that several classes of complex varieties are not (stably) rational: this includes very general hypersurfaces in a certain degree range, cyclic covers, and other varieties.
These results were obtained using the specialization method: to summarize, X is not stably rational if it degenerates to a well chosen mildly singular "reference" variety with some particular nontrivial invariants.
Constructing such reference varieties could be a difficult task in general. In this talk, we will give an overview of the recent progress and we will discuss one specific example of a reference variety, namely, a fibration in cubic surfaces: we will describe a general formula for the invariant coming from the unramified Brauer group.