K-stability, Geometry and Group Actions

Motivic enrichment of the count of lines in a Del Pezzo surface of degree 3 and 4

Victor Chachay

The Del Pezzo surfaces are projective varieties that have different characterizations as blow-ups of points in $\mathbb{P}^2$ or zero loci of polynomials in $\mathbb{P}^n$. We can use them to highlight how to recover the famous count of 27 lines in a cubic surface (over $\mathbb{C}$) and the similar result for a degree 4 surface. From this result as a degree of Chern class over the Grassmannian of lines in $\mathbb{P}^n$, we'll see the link that can be done to justify why we call the new quadratic invariant "enrichment" (and what new information they provide).

Around the K-stability of Fano-Mukai fourfolds of genus 9

Sebastian Fuentes

We review the known results on the K-stability of del Pezzo and Fano-Mukai manifolds of dimension 4 and Picard rank one. We focus on the construction and geometry of Fano-Mukai fourfolds of genus 9, presenting some evidence towards their K-stability. We conclude by discussing certain properties of their automorphism groups.

Singularities of Toric Varieties from the Topological Point of View

Shahryar Ghaed Sharaf

In this talk, I will study the topological properties of toric varieties. From the topological point of view, toric varieties are pseudomanifolds with a stratification induced by the moment map. I will briefly introduce the notion of a pseudomanifold. We will study the link bundle of complex 2-dimensional toric varieties and show that, in this case, the topological singularities are the same as the algebraic ones.

Forms of cylindrical Fano threefolds

Takashi Kishimoto

The existence of cylinders on Fano varieties, which is often reduced to the cylindricity of Mori fiber spaces birational to the initial Fano, has some connection with a study of K-stability stuff. To some extent, the cylindricity of the generic fiber of Mori fiber space, that is a rank one Fano variety defined over the function field of the base variety, plays an important role to detect that of the total space. From this viewpoint, in the talk we will focus on the cylindricity of forms of Fano threefolds defined over a field of characteristic zero. This is based on the joint work with Adrien Dubouloz and Kento Fujita.

MMP and horospherical varieties

André Lapuyade

The goal of this talk is to present my Phd subject. We propose an introduction to the MMP and to horospherical varieties, with the goal to study divisorial contractions.

Nash blowup fails to resolve singularities in dimensions four and higher

Alvaro Liendo

In this talk, we present a (very) recent joint work with Federico Castillo, Daniel Duarte, and Maximiliano Leyton-Álvarez, where we show that iterating Nash blowups, or normalized Nash blowups, does not resolve the singularities of algebraic varieties of dimension four or higher over an algebraically closed field of arbitrary characteristic. This result disproves a long-standing conjecture in the theory of resolution of singularities.

Normal varieties with torus action

Gary Martinez Nunez

The most famous varieties of this kind are toric varieties. These varieties have been studied since Demazure’s work in the seventies and are very useful to test theories. Toric varieties have a very nice combinatorial description in terms of cones for the affine ones and fans for the general ones. In 2006, Altmann and Hausen introduce the concept of proper polyhedral divisors, which is a generalisation of the cones in the affine toric case for any affine normal variety endowed with an effective torus action. Two years later, Altmann, Hausen and Süss developed the divisorial fans as a counterpart of a normal variety endowed with an effective torus action. Both works are studied over a field of characteristic 0 and algebraically closed. In this talk we will present this objects and share the recent works towards a description over any field.

On the geometry of equivariant compactifications of the vector group

Pedro Montero

In this talk, we will summarize results on the geometry of equivariant compactifications of commutative unipotent groups, and their consequences in affine geometry, arithmetic geometry, and positivity in algebraic geometry. We will conclude with some specific questions and possible perspectives for future research.

Gromov ellipticity and torus actions

Charlie Petitjean

The aim of this presentation is to provide a quick overview of the notion of ellipticity in Gromov's sense, and to introduce examples of varieties possessing this property. We will focus on so-called uniformly rational varieties, that is, varieties of dimension n covered by open sets isomorphic to open sets in the affine $n$-space. Group actions will allow us, in a second part, to "rigidify" the definitions and thus provide additional examples.

Group theoretical characterization of rationality and Borel subgroups

Andriy Regeta

In this talk we will discuss two questions about the group of birational transformations $\mathrm{Bir}(X)$ of an irreducible variety $X$:
(1) If $\mathrm{Bir}(X)$ is isomorphic to $\mathrm{Bir}(\mathbb{P}^n)$ does this imply that $X$ is rational?
(2) What are the Borel subgroups of $\mathrm{Bir}(X)$?
In 2014 Serge Cantat proved that the first question has an affirmative answer if the dimension of $X$ is less than or equal to n. We show that (1) has a positive answer without this extra assumption. Regarding the second question: it is well-known that Borel subgroups of an algebraic group (over algebraically closed fields) are conjugate. This is not the case for $\mathrm{Bir}(\mathbb{P}^2)$. Nevertheless, all Borel subgroups of $\mathrm{Bir}(\mathbb{P}^2)$ were classified by Jean-Philippe Furter and Isac Héden.In this talk we will present a result concerning Borel subgroups of $\mathrm{Bir}(X)$ for arbitrary (irreducible) variety $X$: we prove that a Borel subgroup of $\mathrm{Bir}(X)$ has derived length at most twice the dimension of $X$, and in the case of equality $X$ is rational, and the Borel subgroup is conjugate to the standard Borel subgroup in $\mathrm{Bir}(\mathbb{P}^n)$. Moreover, we provide examples of Borel subgroups in $\mathrm{Bir}(\mathbb{P}^n)$ of derived length strictly smaller than $2n$ for any $n\geq 2$. This confirms the conjectures by Popov and by Furter-Héden. We also show that the automorphism group of the affine space $\mathbb{A}^n$ contains non-conjugate Borel subgroups if $n\geq 2$. This is joint work with Christian Urech and Immanuel van Santen.

Local volume of singularities and the Mahler conjecture for convex bodies

Hendrik Suess