## Location

## Dates

## Presentation

The goal of this CIMPA School is to train young mathematicians working in Latin America in some of the most active areas of research in Algebraic Geometry, as well as to promote greater interaction among researchers and students, and to build a network of collaborations. The previous editions of the ELGA were:

I ELGA – Buenos Aires and Cordoba, Argentina (2011)

II ELGA – Cabo Frio, Brazil (2015)

III ELGA – Guanajuato, Mexico (2017)

IV ELGA – Talca, Chile (2019)

Scientific program is available on the local website of the School: https://impa.br/en_US/v-latin-american-school-of-algebraic-geometry-and…

Official language of the school: English

## Administrative and scientific coordinators

## Scientific program

Course 1: "advanced course - Wahl singularities are the two-dimensional quotient singularities that admit a smoothing with Milnor number equal to zero. Originaly these singularities appeared in the work of Wahl in the early 80ies, and soon after they were very relevant in the 1988 work of Kollár--Shepherd-Barron on deformations of quotient singularities and construction of a compactification of the moduli space of surfaces of general type, and on the other side (and from another point of view), in the 1997 rational blowdown surgery for 4-manifolds by Fintushel-Stern and Park. The global set-up is a projective surface with only Wahl singularities together with a one-parameter smoothing, which is locally the Milnor number zero smoothing at each singularity. We abbreviate this situation by the name of W-surface. It turns out that W-surfaces have a rich theory analog to the classical well-known theory for nonsingular projective surfaces: intersection relative to the base, explicit birational maps to find minimal models (i.e. an MMP for W-surfaces), explicit canonical models defining families in the Kollár--Shepherd-Barron-Alexeev compactification. We know of many applications: construction of exotic 4-manifolds (and the relation to the geography of rational configurations), geography of surfaces of general type, optimal boundedness of singularities in W-surfaces of general type, constructions of semi-orthogonal decompositions of the derived category of the varieties involved. We plan to briefly introduce the W-surface theory, including the explicit MMP, and to show some of these applications with the aim of describing current open questions.", Giancarlo URZÚA (Pontificia Universidad Católica de Chile, Chile)

Course 2: "advanced course - Let X be a smooth, complex Fano variety. The Lefschetz defect delta(X) is a recently defined invariant of X, which sheds a new light on the study and classification of Fano varieties of arbitrary dimension. More precisely, delta(X) is a non-negative integer, defined as follows. Consider a prime divisor D in X and the restriction r:H^2(X,R)->H^2(D,R). The Leschetz defect delta(X) is the maximal dimension of ker(r), where D varies among all prime divisors in X. When delta(X)>3, then X is isomorphic to a product SxT, where S is a del Pezzo surface, and T a Fano variety smaller dimension. When delta(X)=3, X does not need to be a product, but it has a flat fibration in del Pezzo surfaces f: X->T, where T is smooth and Fano. Moreover f has a very precise structure, so that X and f are determined by T and some data on T. This structure result allows to classify completely Fano 4-folds X with delta(X)>2. We will explain these results and their proofs, which are related to birational geometry and the properties of Fano varieties as Mori dream spaces. We will also discuss the case delta(X)=2.", Cinzia CASAGRANDE (Università di Torino, Italy)

Course 3: "advanced course - Bridgeland stability is a powerful tool for extracting geometry from homological algebra. In particular, it gives a framework for studying moduli spaces of objects in a triangulated category, such as the derived category of an algebraic variety. The subject was born as a mathematical interpretation of work in string theory, but has since impacted many areas, including classical algebraic geometry, derived categories of coherent sheaves, enumerative geometry, homological mirror symmetry, and symplectic geometry.The goal of this mini-course is to develop the theory of Bridgeland stability, and to explain some of its applications within algebraic geometry. More precisely, we plan to cover the following topics. 1. General theory: We will define stability conditions on triangulated categories and prove Bridgeland’s deformation theorem, which gives the space of stability conditions a natural complex manifold structure. 2. Examples: We will construct stability conditions on curves, surfaces, and (dependingon time) some higher-dimensional examples. 3. Moduli spaces: We will discuss moduli spaces of Bridgeland stable objects and their behavior under variation of stability. We will describe the geometry of these spaces in more detail in the particularly nice case of K3 surfaces, as well as some applications (e.g. to the birational geometry of moduli spaces of sheaves). Along the way, we will cover some background material on semistable sheaves and derived categories.", Alexander PERRY (University of Michigan, United States)

Course 4: "advanced course - 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 course, I will give an overview of the construction in higher dimensions.", Susanna ZIMMERMANN (Mathematical Institute of Orsay, University of Paris-Saclay, France)

## Website of the school

## How to participate

For registration and application to a CIMPA financial support, follow the instructions given here https://www.cimpa.info/en/node/40

**Deadline for registration and application:
May 1, 2024**