Schedule (CEST time) and abstracts

9:00 - 14:00. Registration of the participants.

14:00 - 14:50. Alessio Corti. The potential of the cubic surface.

Abstract: I speak about mirror symmetry for cubic surfaces and write down the LG potential explicitly. This is an excercise in Gross–Siebert technology. I discuss some of the things that we learn along the way.

15:00 - 15:50. Eleonora Romano. C*-actions and Mori Dream Spaces.

Abstract: In this talk we give an overview on recent results about C*-actions on complex smooth projective varieties, most of which are Fano. We then focus on small modifications of Mori Dream Spaces arising from C*-actions, by relating them to their GIT quotients. We review these results by discussing the cases of actions of small criticality. Joint work with L. Barban, G. Occhetta, L. Sola Conde and J. Wisniewski.

16:00 - 16:50. Lara Bossinger. Newton--Okounkov bodies for cluster varieties.

Abstract: In this talk I will explain how the theory of Newton--Okounkov bodies can be elegantly generalized for partial compactifications of cluster varieties. Throughout the talk I will elaborate the theory in the example of the Grassmannian. Cluster varieties are schemes glued from (eventually infinitely many copies) of an algebraic torus whose gluing morphisms are determined by the combinatorics that govern Fomin--Zelevinsky's cluster algebra mutation. To each copy of the torus we associate a valuation that determines a toric degeneration of the Grassmannian. Behind all the associated Newton--Okounkov bodies lies a single "convex polytope" living inside a tropical space. This point of view allows for example, to deduce wall-crossing formulas and to compute the Newton--Okounkov bodies as certain "broken-line" convex sets. (based on joint work in progress with M.Cheung, T.Magee and A.Nájera Chávez).

9:10 - 10:00. Cinzia Casagrande. Fano 4-folds with a small contraction.

Abstract: Let X be a smooth, complex Fano 4-fold. We will talk about the following result: if X has a small elementary contraction, then the Picard number of X is at most 12. This result is based on a careful study of the geometry of X, from the point of view of birational geometry and of families of rational curves.

I will give an idea of the context and of the main tools: the study of families of lines in Fano 4-folds and the construction of divisors covered by lines; the classification of fixed prime divisors; the properties of the faces of the effective cone; the detailed study of rational contractions of fiber type.

10:10 - 11:00. Sara Angela Filippini. Finite free resolutions, root systems and Schubert varieties.

Abstract: I will discuss the link between finite free resolutions of length 3 and Kac-Moody Lie algebras related to the T-shaped graph T_{p,q,r} following work of Weyman. When the graph T_{p,q,r} is Dynkin, the resolutions are related to certain intersections of opposite Schubert varieties with the big cell of the homogeneous spaces G(T_{p,q,r})/P, where P is a maximal parabolic subgroup, as shown by Sam and Weyman. A similar link also holds between Gorenstein ideals of codimension 4 with n generators and the root system of type E_n. I will explain this relation and describe several examples as well as recent developments obtained in collaboration with J. Torres and J. Weyman.

11:30 - 12:20. Alex Küronya. Functions on Newton-Okounkov bodies

Abstract: We will consider simple functions (more concretely, concave transforms of filtrations) supported on Newton-Okounkov bodies. This class of functions possesses good formal properties and are intimately related with the geometry of the polarized variety in question. After reviewing their construction and formal properties, we will provide a general structure theorem for them.

LUNCH

17:00. Online poster session.

Lorenzo Barban. Toric non-equalized flips associated to C*-actions

Abstract: Starting from C*-actions on complex projective varieties, we construct and investigate birational maps among the corresponding extremal fixed point components. We study the case in which such birational maps are locally described by toric flips, either of Atiyah type or so called non-equalized. We relate this notion of toric flip with the property of the action being non-equalized. Finally, we present explicit examples of C*-varieties whose weighted blow-up at the extremal fixed point components gives a birational map among two projective varieties which is locally a toric non-equalized flip.

Francesco Denisi. Boucksom-Zariski chambers on irreducible holomorphic symplectic manifolds

Abstract: We provide for the big cone of a projective irreducible holomorphic symplectic manifold a decomposition into chambers, called Boucksom‐Zariski chambers, in each of which the support of the negative part of the Boucksom‐Zariski decomposition is constant. We show how the obtained decomposition allows to describe the volume function and we determine when the Boucksom‐Zariski chambers are “numerically determined

Wendelin Lutz. Towards a geometric proof of the classification of T-polygons.

Abstract: One formulation of mirror symmetry predicts (omitting a few adjectives) a 1-1 correspondence between equivalence classes of certain lattice polygons and deformation families of certain del Pezzo surfaces.

Lattice polygons corresponding to smooth Del Pezzo surfaces are called T-polygons, and these have been classified by Kasprzyk-Nill-Prince using combinatorial methods. I will sketch a new geometric proof of their classification result.

Erik Paemurru. All divisorial contractions are valuation-blowups.

Abstract: We give a coordinate-free, intrinsic definition of weighted blowups which we call valuation-blowups. We show that all divisorial contractions and the morphisms in all flips are valuation-blowups.

Qaasim Shafi. Quasimaps & Accordions

Abstract: Quasimaps provide an alternate curve counting system to Gromov-Witten theory, which are related by wall-crossing formulae. Relative Gromov-Witten theory has proved useful for constructions in mirror symmetry, as well as for determining ordinary Gromov-Witten invariants via the degeneration formula. One version of this theory relies on a construction known as accordions. This poster discusses how to adapt this approach to produce a proper moduli space parametrising quasimaps relative a smooth divisor in any genus.

9:10 - 10:00. Giulia Gugiatti. Hypergeometric families and new mirrors of Fano varieties.

Abstract: I will discuss a few aspects of hypergeometric motives in geometry and arithmetic. Building upon these, I will outline a method to construct mirrors of Fano anticanonical weighted complete intersections beyond the known constructions. I will show how to apply the method in a couple of explicit examples in dimension 2.

10:10 - 11:00. Stefano Urbinati. Quasi-monomial valuations and Newton Okounkov bodies.

Abstract: In this talk I will discuss a work in progress with Joaquim Roé.

The idea of this work comes from an unexpected phenomena in the construction of Newton-Okounkov bodies in dimension 2. In fact, in this case, to any monomial valuation, there is a naturally associated flag. In particular it has been noticed that the convex bodies don't change continuously as the valuation moves along the Berkovich tree. We give an explanation of this fact with the language of b-divisors.

11:30 - 12:20. Thomas Eckl. Seshadri constants and Explicit Kähler packings.

Abstract: We explain the connection between (multi-point) Seshadri constants and (multi-ball) Kähler packings, and then introduce a method to construct such Kähler packings explicitly, using moment maps and guided by Newton-Okounkov bodies. We use the method to construct multi-ball Kähler packings on the complex projective plane $\mathbb{C}\mathbb{P}^2$.

LUNCH

14:00 - 14:50. Andrea Petracci. On deformations of toric varieties: affine and Fano.

Abstract: Studying deformation spaces (i.e. the bases of the Kuranishi family) of algebraic varieties is important to understand the local properties of moduli spaces. Vakil has shown that deformation spaces can be arbitrarily bad (Murphy's law).

In this talk, thanks to the seminal work by Klaus Altmann, I will consider deformation of isolated Q-Gorenstein toric singularities and I will show that their deformation spaces are quite restricted and do not satisfy Murphy's law in the sense of Vakil.

Moreover, I will give some applications to the local properties of K-moduli of Fano varieties. In particular, I will show that the number of local branches of the K-moduli stack of K-semistable Fano varieties and of the K-moduli space of K-polystable Fano varieties is unbounded in each dimension greater than 2.

15:00 - 15:50. Johannes Hofscheier. Log canonical pairs on spherical varieties.

Abstract: This talk will report on joint work in progress with Giuliano Gagliardi where it is shown that related statements on spherical varieties, such as the generalised Mukai conjecture or a purely combinatorial smoothness criterion, follow from the existence of certain log canonical pairs. The approach relies on a geometric characterisation of toric varieties using log pairs conjectured by Shokurov and proven by Brown, McKernan, Svaldi, and Zong. The talk will conclude with an outline of the construction of such pairs for the special case of horospherical varieties. The construction relies on the existence of dual canonical bases (or dual essential bases) of the coordinate ring of the affine variety G/U with "good" combinatorial properties and is inspired by ideas coming from the theory of Newton–Okounkov bodies.

16:10 - 17:00. Fatemeh Mohammadi. Combinatorial mutations of matching field polytopes with an application to toric degenerations of Grassmannians.

Abstract: Many toric degenerations of the Grassmannians Gr(2, n) are described by trees, or equivalently subdivisions of polygons. These degenerations can be also seen to arise from the cones of the tropicalization of the Grassmannian. In this talk, I focus on particular combinatorial types of cones in tropical Grassmannians indexed by matching fields, whose corresponding degenerations are toric. Moreover, I will show how their associated polytopes are connected by combinatorial mutations. I will present several combinatorial conjectures and computational challenges around this problem.

17:10 - 18:00. Elana Kalashnikov. Undoing toric degenerations: an analogue of Greene—Plesser mirror symmetry for the Grassmannian.

Abstract: The most basic construction of mirror symmetry compares the Calabi–Yau hypersurfaces of projective space and projective space quotient a finite group G. There is a natural analogue of this finite group action on the Grassmannian Gr(n, r). In this talk, I'll explain how toric degenerations, blow-ups, variation of GIT and mirror symmetry relate the Calabi–Yau hypersurfaces of Gr(n,r) and Gr(n,r)/G. This is joint work with Tom Coates and Charles Doran.

9:00 - 9:50. Catriona Maclean. Characterisation of approximable algebras.

Abstract: We use Newton Okounkov bodies to give a geometric characterisation of approximable algebras, ie. those for which a Fujita type approximation result holds.

9:50 - 10:40. Leonid Monin. Gorenstein algebras and computations of cohomology rings.

Abstract: It was observed by Pukhlikov and Khovanskii that the BKK theorem implies that the volume polynomial on the space of polytopes is the Macaulay generator of the cohomology ring of a smooth projective toric variety. This provides a way to express the cohomology ring of toric variety as a quotient of the ring of differential operators with constant coefficients by the annihilator of an explicit polynomial. The crucial ingredient of this observation is an explicit expression for the Macaulay generator of graded Gorenstein algebras generated in degree 1.

11:10 - 12:00. Anne-Sophie Kaloghiros. The Calabi problem for Fano 3-folds

Abstract: I will discuss progress on the Calabi problem for Fano 3-folds. The 105 deformation families of smooth Fano 3-folds, were classified by Iskovskikh, Mori and Mukai. We determine whether or not the general member of each of these 105 families admits a Kähler-Einstein metric. In some cases, it is known that while the general member of the family admits a Kähler-Einstein metric, some other member does not. This leads to the problem of determining which members of a deformation family admit a Kähler-Einstein metric when the general member does. This is accomplished for most of the families, and I will present a conjectural picture for some of the remaining families. This is a joint project with Carolina Araujo, Ana-Maria Castravet, Ivan Cheltsov, Kento Fujita, Jesus Martinez-Garcia, Constantin Shramov, Hendrik Süss and Nivedita Viswanathan.

LUNCH