Abstracts:
Klaus Altmann, Fujita's freeness conjecture for T-varieties of complexity one
Fujita made the conjecture that, for projective varieties X with
sufficiently mild singularities, all divisors (mH + K) are
basepoint free for all ample divisors H and all m ≥ dim X + 1.
This conjecture has been proven for varieties of dimension up
to five, on the one hand, and for toric varieties, on the other.
In this talk, we prove Fujita's freeness conjecture for projective
Gorenstein varieties X with at worst rational singularities being
equipped with an effective action of an algebraic torus of
dimension dim X − 1.
The main tool of the proof is the representation of X by a
divisorial polytope giving rise to a true poytope, the so-called
realization polytope. We will explain all these notions and show
how this helps to shed light on the base locus of torus
invariant divisors. This is joint work with Nathan Ilten.
Alessandra Bernardi, Skew-symmetric tensor decomposition
I will introduce the skew-symmetric apolarity and I will use it to propose an algorithm for the decomposition in the tri-vector case.
Serge Cantat, Halphen Twists
Halphen twists are examples of birational transformations of
the plane preserving a pencil of curves of genus 1. They form a distinguished
set of elements in the Cremona group of all birational transformations of the
plane. I will describe recent results concerning the algebraic properties of
these elements of the Cremona group.
Cinzia Casagrande, Fano 4-folds with rational fibrations
Smooth, complex Fano 4-folds are not classified, and we still lack a good understanding of their general properties. In the talk we will focus on Fano 4-folds with large second Betti number $b_2$, studied via birational geometry and the detailed study of their contractions and rational contractions. We recall that a contraction is a morphism with connected fibers onto a normal projective variety, and a rational contraction is given by a sequence of flips followed by a (regular) contraction.
The main result that we present is the following: let X be a Fano 4-fold having a rational contraction $X \dashrightarrow Y$ of fiber type (with dim Y > 0). Then either X is a product of surfaces, or $b_2(X)$ is at most 17, or Y is $\mathbb{P}^1$ or $\mathbb{P}^2$.
Milena Hering, Where can toric syzygies live?
In this talk I will give an introduction to syzygies and
will discuss some open questions on them. I will then talk about the
toric case, where defining equations and higher syzygies have a
natural grading by the character lattice of the torus, and give some
results on the the regions in the character lattice in which these
equations and syzygies can be supported.
Jun-Muk Hwang, Recognizing G/P by its varieties of minimal rational tangents
Let C be a general line on a rational homogeneous space G/P of Picard number 1. We are interested in the question of recognizing the germ of C in G/P. In other words, given a rational curve C' in a complex manifold X, how do we check that the germ of C' in X is biholomorphic to the germ of C in G/P? VMRT (variety of minimal rational tangents) is an invariant of such a germ. Is VMRT strong enough to determine the germ? An affirmative answer was given for G/P associated with a long root in the works of Mok and Hong-Hwang in 2008. But when G/P is associated with a short root, there are certain degenerations of G/P with the same VMRT.
We report on a recent joint work with Qifeng Li and Dmitry Timashev which gives a complete classification of such degenerations and shows that G/P is the only Fano manifold of Picard number 1 with the given VMRT.
Antonio Laface, On blowing up the weighted projective plane.
It is an open problem to decide if the blowing up
of a toric variety at a general point has finitely
generated Cox ring. In this talk I will focus on the
case of weighted projective planes. After reformulating
in this context the criterion by Y. Hu and S. Keel for the
finite generation, I will discuss necessary and sufficient
conditions, on the semigroup generated by the weights,
for the Cox ring to be generated in multiplicity at most two.
This is joint work with J. Hausen and S. Keicher.
JM Landsberg, Paths to upper bounds on the complexity of matrix multiplication
via geometry
The exponent of matrix multiplication is a fundamental constant
governing the complexity of operations in linear algebra. Since 1968,
when Strassen showed its significance by proving the standard way of
multiplying matrices (which leads to an exponent of three) was not
optimal, until 1989, when Coppersmith and Winograd showed the exponent
is less than 2.39, there was steady progress in upper-bounding it. In
2014 Ambainis, Filimus, and LeGall showed that current methods will not
prove the exponent is less than 2.3, although many computer scientists
believe it to be 2. In this talk (joint work with A. Conner, F.
Gesmundo, Y. Wang and E. Ventura), I will discuss new paths to improving
upper bounds on the exponent via algebraic geometry and representation
theory. Independent of their use in complexity, our results on
previously unstudied varieties of tensors are of interest in their own
right.
Alvaro Liendo, Characterization of affine toric varieties by their automorphism groups
We show that complex affine toric surfaces are determined by the abstract group structure of their regular automorphism groups in the category of
complex normal affine surfaces using properties of the Cremona group. As a generalization to arbitrary dimensions, we show that complex affine toric varieties, with the exception of the algebraic torus, are uniquely determined in the category of complex affine normal varieties by their automorphism groups seen as ind-groups. This is a joint work with Andriy Regeta and Christian Urech.
Elisa Postinghel, Mutations of Newton-Okounkov polytopes and classification of Fano manifolds.
Mutations are certain operations of Laurent polynomials, or of their Newton polytopes, that induce deformations between the corresponding toric varieties, as showed by Ilten. Mutations of Fano polytopes are studied by Corti et al. in order to classify Fano manifolds up to deformation using the mirror symmetry approach. On the other hand, to a pair given by a projective algebraic variety and a line bundle, we can associate the so called Okounkov bodies, that are a generalisation of Newton polytopes. In this talk, I will show how to obtain mutations of Okounkov polytopes, and hence toric deformations, of certain Fanos. This is join work (in progress) with A. Laface and S. Urbinati.
Bernd Sturmfels, The Geometry of Gaussoids
Gaussoids offer a new link between combinatorics, statistics and algebraic
geometry. Introduced by Lnenicka and Matus in 2007, their axioms describe
conditional independence for Gaussian random variables. We explain this
theory and how it relates to matroids. The role of the Grassmannian for
matroids is now played by a projection of the Lagrangian Grassmannian.
We discuss the classification and realizability of gaussoids, and we explore
oriented gaussoids, valuated gaussoids, and the analogue to positroids.
This is joint work with Tobias Boege, Alessio D'Ali and
Thomas Kahle.
Hendrik Suess, On irregular Sasaki-Einstein metrics in dimension 5
Sasakian geometry can be seen as the odd-dimensional counterpart of Kaehler geometry.
Indeed, a /regular/ Sasakian manifold M is a circle bundle over some Kaehler manifold Z.
In this situation the Sasakian geometry of M and the Kaehler geometry of Z are closely related to each other.
For example the problem of finding a Sasaki-Einstein metric on M is equivalent to the problem of
finding a Kaehler-Einstein metric on Z. However, in the so-called /irregular/ case this approach breaks
down. On the other hand, one also obtains a new tool in this situation: a torus action of higher rank.
In this talk I will explain how to make use of this new tool in order to prove the the existence or non-existence of
irregular Sasaki-Einstein metrics on certain 5-manifolds.
Milena Wrobel, On iteration of Cox rings
Looking at the well understood case of log terminal surface singularities, one observes that each of them is the quotient of a factorial one by a finite solvable group. The derived series of this group reflects an iteration of Cox rings of surface singularities.
We extend this picture to log terminal singularities in arbitrary dimension coming with a torus action of complexity one. Moreover we give a characterization of all varieties with a torus action of complexity one admitting an iteration of Cox rings and thus can be regained as the quotient of a factorial affine variety by a solvable finite torus extension.