Schedule

Schedule

9:30 – 10:00 Welcome coffee and registration

10:00 – 10:45 Federico Tufo: Smooth Fano fourfolds: an update

11:00 – 11:45 Saverio Secci: The blow-up of the cubic fourfold along eight planes is Fano

12:00 – 14:00 Lunch

14:00 – 14:45 Gianluca Grassi: Cremona quadratic transformations of the 4-dimensional projective space

14:45 – 15:15 Coffee break

15:15 – 16:00 Francesco Angeli: A Reverse Tropical Hurwitz Problem

16:15 – 17:00 Qike Li: C∗-Equivariant Cohomological Crepant Resolution Conjecture for ADE Orbifolds

Abstracts

Smooth Fano fourfolds: an update – Federico Tufo (Bologna, Ljubljana)

The classification of deformation families of Fano varieties is a long-standing problem in algebraic geometry. Thanks to the results of Kollár, Miyaoka, and Mori, we know that there are only finitely many deformation families of Fano varieties in each dimension. They have been classified up to dimension three, but from dimension four onwards, the situation becomes much more complicated. In this talk, we will discuss some recent developments in the classification of deformation families of Fano fourfolds. 

The blow-up of the cubic fourfold along eight planes is Fano – Saverio Secci (SISSA)

I will discuss the classification of smooth complex Fano fourfold with large Picard rank, and then I will focus on a new class of examples: the blow-up of the cubic fourfold along a special configuration of planes. This talk is based on a work in progress. 

Cremona quadratic transformations of the 4-dimensional projective space – Gianluca Grassi (Ferrara)

Cremona transformations of projective spaces P^n are extremely relevant in birational geometry. While Cremona transformations have been extensively studied for P^2, going in higher dimensions gives a proper challenge, especially in the realm of classification. Quadratic Cremona transformations are also the easiest example and, for P^3, they have been classified back in 2001 by Pan, Ronga and Vust.

In this talk we will present the approach used to try a generalization in higher dimensions, focusing on the case study of P^4, showing interesting properties of all components of the set of quadratic birational maps of P^4, understanding how, given a specific multidegree, we can build those maps.

A Reverse Tropical Hurwitz Problem – Francesco Angeli (Udine)

The Hurwitz existence problem asks whether prescribed combinatorial branching data can be realized by a branched covering between compact Riemann surfaces.   

In this talk we move from algebraic geometry to tropical geometry and we consider a reverse perspective. We fix a tropical curve $\Gamma$, and we ask for which other tropical curves $\Gamma'$ exists a Hurwitz cover $\varphi:\Gamma \to \Gamma'$. We derive combinatorial constraints that any realizable cover must satisfy and use them to obtain bounds on the genus of the target and on the degree of the morphism.  

The talk will start with a broad overview of tropical geometry, with the goal of motivating the use of tropical geometry and introducing the mathematical objects necessary to understand the aforementioned problem. If the time allows it, we will also try to introduce tropical moduli spaces and formalize the problem in this setting.

C∗-Equivariant Cohomological Crepant Resolution Conjecture for ADE Orbifolds – Qike Li (Trieste)

Let G⊂SL2(C) be a non-trivial finite subgroup and let X→C2/G be the minimal crepant resolution. We prove the C∗-equivariant Cohomological Crepant Resolution Conjecture for all ADE surface orbifolds [C2/G]. The main idea is to define the cup product of C∗-equivariant orbifold cohomology directly, and then apply the map defined by Bryan and Gholampour to compare the structure constants of the cohomolgy rings with respect to appropriate generators, which give rise to the trigonometric identities that need to be proved.

While the A and D cases have already been proven in earlier papers by using mirror symmetry, the method used in this proof is completely new. And the exceptional cases are proven for the first time. 

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