Abstracts and Schedule

edtcremona

Minicourses:


  • Yves Cornulier: Partial actions, commensurating actions and applications to birational groups.

The purpose of the lectures is to explain the result of https://arxiv.org/abs/1910.07802. It gives restrictions on birational actions of groups satisfying a certain combinatorial rigidity property called Property FW. The outline of the lectures (3 hours) is roughly (1) Introduction to partial actions and examples (2) Introduction to commensurating actions and Property FW (3) Construction of a partial action for the group of birational self-transformations of an algebraic variety (4) The regularization theorem and its proof.


  • Stéphane Lamy: Introduction to the Cremona group.

In these lectures I will give an introduction to the classical Cremona group, which is the group of birational transformations of the complex plane.

In the first two lectures I will discuss the classical point of view: basic notions such as the base locus and the resolution of a map by sequences of blow-ups, lots of examples, and a description of generators following Noether and Castelnuovo (but with a modern twist to give an introduction to the Sarkisov program).

In the last two lectures I will discuss the action of the Cremona group on an infinite dimensional hyperbolic space, and I will state with some ideas of proof some 21th century results about the Cremona group, such as the Tits alternative or the non-simplicity.

Notes for the lectures can be found here: https://www.math.univ-toulouse.fr/~slamy/stock/cuernavaca_lectures.pdf


  • Nick Salter: Higher spin mapping class groups in algebraic and flat geometry

Abstract: An r-spin structure is a choice of r’th root of the canonical bundle of a Riemann surface. Such structures arise in a variety of settings in geometry; in this lecture series, we will focus on their role in two places at the interface of algebraic geometry and topology: linear systems on algebraic surfaces (especially toric surfaces), and translation surfaces (also known as abelian differentials). In both these settings, there are “topological monodromy groups” valued in the mapping class group that encode important information about these families of Riemann surfaces and their degenerations, and the presence of r-spin structures is reflected in the underlying group theory. We will outline some recent developments in the theory of these “higher spin mapping class groups” that allow us to understand monodromy in the above problems, and ultimately to gain new insights into the behavior of these families. No specialized knowledge of topology or the mapping class group will be assumed. Portions of this work are joint with Aaron Calderon.


Lecture notes can be found here: http://math.columbia.edu/~nks/expository-notes/cuernavacalectures.pdf


  • Susanna Zimmermann: Homomorphisms from Cremona groups.

Abstract: We will look at how to construct non-trivial group homomorphisms from a Cremona group to a finite group. After seeing that such a homomorphism can only exist for Cremona groups in higher dimension or for the plane Cremona group over a non-closed field, we will attack the construction in the latter case, and we will stay over the rational numbers. I will introduce so-called Sarkisov links and we will study relations between them. This will bring us to the construction of the group homomorphisms. At the end of the lecture, I will give an overview of the construction in higher dimensions.

Talks:

  • Corey Bregman: Surface bundles and complex projective varieties.

Abstract: Kodaira, and independently Atiyah, gave the first examples of surface bundles over surfaces whose signature does not vanish, demonstrating that the signature of fiber bundles need not be multiplicative. These examples, called Kodaira fibrations, are in fact complex projective surfaces admitting a holomorphic submersion onto a complex curve, whose fibers have nonconstant moduli. After reviewing the Atiyah-Kodaira construction, we consider Kodaira fibrations with nontrivial holomorphic invariants in degree one. When the dimension of the invariants is at most two, we show that the total space admits a branched covering over a product of curves.


  • Alexander Duncan: Finite subgroups of Cremona groups. (2 Talks)

I will overview how to study finite subgroups of Cremona groups using ideas from the minimal model program. In this case, one can reduce to considering actual automorphism groups of varieties rather than birational automorphism groups. However, the tradeoff is that one must now consider many different varieties instead of only projective spaces. Nevertheless, this perspective has been very fruitful. I will focus on the case of the complex plane Cremona group, where an almost complete classification is known, but results in higher dimensions and over other fields will also be discussed.


  • Rita Jiménez Rolland: Powers of the Euler class for pure mapping class groups.

In this talk we will consider a discrete group acting on the circle by orientation-preserving homeomorphisms and review some of the properties of the Euler class associated to this action. In particular, we will be interested in the Euler class associated to the Nielsen action on the circle of the pure mapping class group of an orientable surface with one marked point. We will describe some partial results, in ongoing work with Solomon Jekel, on the vanishing and non-vanishing behaviour of the powers of this class.


  • Anne Lonjou: Elements generating a proper normal subgroup of the Cremona group.

Abstract: The group of birational transformations (isomorphisms between two dense open subsets) of the projective space is called the Cremona group. An important tool for the study of this group is its isometric action on a hyperbolic space that Stéphane Lamy will introduce in its mini-course. Until now, the known elements generating a proper normal subgroup of the Cremona group are loxodromic elements (for this action). Hence, it is natural to ask if there exist other types of isometries having this property. We will answer this question in this talk, which is based on a joint work with Serge Cantat and Vincent Guirardel.


  • César Lozano Huerta: The Lefschetz principle in birational geometry.

Solomon Lefschetz was an American mathematician with a significant impact on Mexican mathematics; including mathematics in Cuernavaca. In this talk I will report on research that can be traced back to Solomon Lefschetz; namely the Lefschetz hyperplane theorem. This theorem tells us that some topological invariants of an algebraic variety are often fully determined by those of its ambient space. It appears that this phenomenon is so robust that it also occurs in the context of birational geometry. In this talk I will make this precise and present examples of subvarieties for which their birational information is fully determined by the ambient space.