Seminar Algebra and Geometry - Spring 2019

September 17 Anna Bot
Automorphisms of rational surfaces The starting point of this talk is a finite number of distinct points lying on an irreducible cubic curve in 2 which we blow up to obtain a rational surface S. What conditions must the points satisfy such that a fixed element of the Weyl group associated to Pic(S) is realised as an automorphism of the surface S? I will discuss the conditions I worked on in my master thesis, present the relevant notions and results, and speak about the potential directions which could be further pursued.
September 23-27 Conference in Toulouse (Del Duca Workshop on Cremona Groups)
October 8 Christian Urech
On continuous automorphisms of Cremona groups Julie Déserti showed that every automorphism of the plane Cremona group is inner up to a field automorphism of the base-field. In this talk we generalize this result to Cremona groups of arbitrary rank, however, only under the additional restriction that the automorphisms are also homeomorphisms with respect to the Zariski or the Euclidean topology on the Cremona group. We will consider similar questions for groups of polynomial automorphisms and groups of birational diffeomorphisms. This is joint work with Susanna Zimmermann.
October 15 Tomasz Pełka
-homology planes satisfying the Negativity Conjecture A smooth complex normal algebraic surface S is a -homology plane if Hi(S,ℚ)=0 for i>0. This holds for example if S is a complement of a rational cuspidal curve in 2. The Negativity Conjecture of K. Palka asserts that for a smooth completion (X,D) of S, κ(KXD)=-∞, so the minimal model of (XD) is a log Mori fiber space. Assume that S is of log general type, otherwise the geometry is well understood. It turns out that, as expected by tom Dieck and Petrie, all such S can be arranged in finitely many discrete families, each obtainable in a uniform way from certain arrangements of lines and conics on 2. As a consequence, they all satisfy the Strong Rigidity Conjecture of Flenner and Zaidenberg; and all their automorphism groups are subgroups of S3. To illustrate this surprising rigidity, I will show how to construct all rational cuspidal curves (with complements of log general type, satisfying the Negativity Conjecture) inductively, by iterating quadratic Cremona maps.
October 22  
no talk
October 29 Stavros Papadakis
Graded Rings and Birational Geometry Un projection theory, initiated by Miles Reid, aims to construct and analyze complicated commutative rings in terms of simpler ones. It can also be considered as an algebraic language for birational geometry. The main purpose of the talk is to give a short introduction to the theory and describe some of its applications.
November 5 Shengyuan Zhao
Birational Kleinian groups and birational structures Let Y be a smooth complex projective surface. Let U be a connected Euclidean open set of Y. Let G be a subgroup of Bir(Y) which acts by holomorphic diffeomorphisms on U (i.e. preserves U and without indeterminacy points in U), in a free, properly discontinuous and cocompact way, so that the quotient X=U/G is a compact complex surface. Such a birational transformation group G, or more precisely such a quadruple (Y,U,G,X), will be called a birational Kleinian group. Once we have a birational Kleinian group, the quotient surface is equipped with a birational structure, i.e. an atlas of local charts with rational changes of coordinates. I will present some basic properties and subtleties of birational structures, compared to the classical geometric structures. Then I will begin by studying birational structures on a special type of non-algebraic surfaces, the Inoue surfaces, to reveal some of the general strategy. Using classification of solvable and abelian groups of the Cremona group, and by relating the foliations on Inoue surfaces with some birational dynamical systems via Ahlfors-Nevanlinna currents, I will show that the Inoue surfaces have one unique birational structure. Then I will move on to the general study of birational Kleinian groups with the additional hypothesis that the quotient surface is projective. I will explain how to use powerful results from Cremona groups, holomorphic foliations and non-abelian Hodge theory to get an almost complete classification of such birational Kleinian groups.
November 12 Enrica Mazzon
(Imperial College)
Dual complexes of degenerations and Berkovich geometry To a degeneration of varieties, we can associate the dual intersection complex, a topological space that encodes the combinatoric of the central fiber and reflects the geometry of the generic fiber. The points of the dual complex can be identified to valuations on the function field of the variety, hence the dual complex can be embedded in the Berkovich space of the variety. In this talk I will explain how this interpretation gives an insight in the study of the dual complexes. I will focus on some degenerations of hyper-Kähler varieties and show that we are able to determine the homeomorphism type of their dual complex using techniques of Berkovich geometry. The results are in accordance with the predictions of mirror symmetry, and the recent work about the rational homology of dual complexes of degenerations of hyper-Kähler varieties, due to Kollár, Laza, Saccà and Voisin. This is joint work with Morgan Brown.
November 18-20 Graduate colloquium in Geneva
November 26 Carlos Amendola
Maximum Likelihood Estimation of Toric Fano Varieties We study the maximum likelihood estimation problem for several classes of toric Fano models. We start by exploring the maximum likelihood degree for all 2-dimensional Gorenstein toric Fano varieties. We show that the ML degree is equal to the degree of the surface in every case except for the quintic del Pezzo surface with two singular points of type A1 and provide explicit expressions that allow to compute the maximum likelihood estimate in closed form whenever the ML degree is less than 5. We then explore the reasons for the ML degree drop using A-discriminants and intersection theory. Based on joint work with Dimitra Kosta and Kaie Kubjas.
December 3 Erik Paemurru
Birational models of terminal sextic double solids It is known that quasismooth 3-fold Fano hypersurfaces with index 1 in weighted projective spaces over ℂ are birationally rigid (not birational to any other Fano 3-folds, conic bundles or del Pezzo fibrations). But very little is known when they carry non-orbifold singularities. I consider sextic double solids, one of the simplest such 3-folds, which have an isolated cA_n singularity. I have shown that n is at most 8, and that rigidity fails for n > 3. In this talk, I will illustrate this phenomenon by giving some examples.
December 9
(15h )
Spiegelgasse 5, 05.002
Michel van Garrel
Prelog Chow rings and stable rationality in semistable degenerations In this joint work with Christian Böhning and Hans-Christian von Bothmer we apply Voisin's criterion of existence of a decomposition of the diagonal to semistable degenerations. In doing so, we obtain partial results towards proving that very general cubic threefolds are stably irrational.
December 10 Elisa Gorla
Blowup algebras of sparse determinantal varieties Let X be a sparse generic matrix, i.e. a matrix whose entries are either zeros or distinct variables. A sparse determinantal variety is the locus where X does not have full rank. While determinantal varieties, i.e. degeneracy loci of matrices whose entries are distinct variables with no zeros, are in many respects well-understood, this is not yet the case for sparse determinantal varieties. However, sparse determinantal varieties have recently received increased attention, as new approaches for studying them have been introduced by Boocher (2011) and by Conca, De Negri and myself in a series of works (since 2015).

Blowup algebras - such as the Rees algebra, the special fiber ring, and the associated graded ring - are an active area of study within commutative algebra. They are algebraic objects related to the concept of blowing up a variety along a subvariety. In this talk, I will present some new results on the blowup algebras of sparse determinantal varieties. Our approach makes an essential use of the theory of SAGBI bases, which I will introduce during the talk. The new results that I will present are part of an ongoing joint work with E. Celikbas, E. Dufresne, L. Fouli, K.-N. Lin, C. Polini, and I. Swanson.