**Second ECOS-ANID workshop in Algebraic Geometry**

**Monday 19 and Tuesday 20 of July 2021**

**University of Poitiers (France)-Universidad de la Frontera, Temuco (Chili), Virtual Meeting**

**Organizers:
Paola Comparin (Universidad de la Frontera, Temuco),
Alessandra Sarti (University of Poitiers)**

We organize the second workshop in Algebraic Geometry in the framework of our
project between the University of Poitiers, Laboratoire
de Mathématiques et Applications and the Universidad de la Frontera Temuco,
Departamento de Matemática y Estadística, Research Center Geometry
at the frontier, supported by the french ECOS-Sud and
the chilean ANID. The first workshop
was held on Thursday 12 and Thursday 19 of November 2020. This is also a
follow up activity of the workshop *Women
in Algebraic Geometry* held virtually at ICERM in july 2020 and it is supported by the ANR
project *Symmetries and moduli spaces in
algebraic geometry and physics* and AWM Association for Women in Mathematics.

**
Registration**
__Participants____Poster____Photo 1 Photo 2
____
Videos of the talks__

**Speakers**

Michela Artebani (Concepción, Chile), Grâce Bockondas (Brazzaville, Congo), Alice Garbagnati (Milan, Italy), Jennifer Li (Massachusetts, USA), Dimitri Markouchevitch (Lille, France), Nathan Priddis (Provo, USA), Alejandra Rincón-Hidalgo (Trieste, Italy), Aline Zanardini (Pennsylvania, USA)

__Schedule__

__Talks of monday 19 of july__

** Nathan Priddis**, 9.00-9.45 Chilean time (15.00-15.45 French time)

**Grâce Bockondas **, 9.50-10.20 Chilean time (15.50-16.20 French time)

**Break** : 10.20-10.40 Chilean time (16.20-16.40 French time)

**Jennifer Li**, 10.40-11.10 Chilean time (16.40-17.10 French time)

**Michela Artebani**, 11.15-12.00 Chilean time (17.15-18.00 French time)

__Talks of tuesday 20 of july__

**Dimitri Markouchevitch**, 9.00-9.45 Chilean time (15.00-15.45 French time)

**Alejandra Rincón-Hidalgo **, 9.50-10.20 Chilean time (15.50-16.20 French time)

**Break**: 10.20-10.40 Chilean time (16.20-16.40 French time)

**Aline Zanardini**, 10.40-11.10 Chilean time (16.40-17.10 French time)

**Alice Garbagnati**, 11.15-12.00 Chilean time (17.15-18.00 French time)

__Titles and Abstracts__

**Title:**About Cox rings of K3 surfaces

**Abstract:**The Cox ring of a normal complex projective variety X with finitely generated divisor class group is the Cl(X)-graded algebra R(X) whose homogeneous pieces are Riemann-Roch spaces of divisors of X. This object is particularly interesting when it is finitely generated, since in such case X can be obtained as a GIT quotient of an open subset of Spec R(X) by the action of a quasi-torus. Finding a presentation or even a minimal generating set for R(X) is in general a difficult problem, already in the case of surfaces.
This talk will be about Cox rings of complex projective K3 surfaces. First, we will provide a general description of the degrees of a generating set of R(X) for any such K3 surface. After this, we will concentrate on K3 surfaces with finitely generated Cox ring and we will show some explicit computations of R(X) for small Picard number.
This is joint work with C. Correa Deisler, A. Laface and X. Roulleau.

**Title:**Triple lines on cubic threefolds

**Abstract:** Triple lines on a cubic threefold define a finite set.
In 1972, Murre showed that the set of double lines on a cubic threefold is a
curve on its Fano surface. In this talk we will give the description of
triple lines related to that curve. __Slides__

**Title:**Projective models of Nikulin orbifolds

**Abstract:** Given a hyperkaehler fourfold of K3^{[2]}-type and a
symplectic involution on it, it is known that the quotient does not admit
a smooth model which is a hyperkaehler manifold; nevertheless there exists
a partial resolution of the quotient which is a hyperkähler orbifold,
called Nikulin orbifold. We call orbifold of Nikulin type an orbifold which is a
deformation equivalent to a Nikulin orbifold.
The aim of this talk is twofold: we first relate families of projective fourfolds
of K3^{[2]}-type admitting a symplectic involution with the families of Nikulin orbifolds
which are their quotients, and to do that we describe the Neron--Severi groups and the
transcendental lattices of both these families. Then we analyze the projective models
of both the objects studied, relating certain (not necessarily Cartier) divisors on
fourfolds of K3^{[2]}-type and on the associated Nikulin orbifolds; to do that we
study the Riemann-Roch formula on orbifolds of Nikulin type.
We also give an explicit geometric description of the general member of a specific family
of orbifolds of Nikulin type.
The talk is based on a joint work with C. Camere, G. Kapustka and M. Kapustka.

**Title:**A cone conjecture for log Calabi-Yau surfaces

**Abstract:** In 1993, Morrison conjectured that the automorphism group
of a Calabi-Yau 3-fold acts on its nef cone with a rational polyhedral fundamental domain.
In this talk, I will discuss a version of this conjecture for log Calabi-Yau surfaces
that we have proved. In particular, for a generic log Calabi-Yau surface with singular
boundary, the monodromy group acts on the nef effective cone with a rational polyhedral
fundamental domain. In addition, the automorphism group of the unique surface with a split mixed
Hodge structure in each deformation type acts on the nef effective cone with a rational
polyhedral fundamental domain. __Slides__

**Title:**On special cubic 4-folds and EPW sextics associated to Enriques surfaces

**Abstract:**Fano models of Enriques surfaces are degree-10 surfaces in P^5
that contain tens of mutually incident plane elliptic curves. This correspondence
provides a birational identification of a 10-dimensional moduli component of tens
of mutually incident planes in P^5 with the moduli space of Enriques surfaces, marked by
a choice of generators of the Picard group. The same 10-dimensional moduli space also parametrizes
certain lattice-polarized cubic 4-folds and EPW sextics. This is a joint work with I. Dolgachev.
__Slides__

**Title:**Non-symplectic automorphisms of K3 Surfaces:
Uniqueness and Invariant lattices.

**Abstract:**In this talk I will discuss certain non-symplectic automorphisms
on K3 surfaces. Under certain conditions, the K3 surface is uniquely determined by
the automorphism. We will also discuss their invariant lattices. __Slides__

**Title:** Moduli of Bridgeland semistable holomorphic triples on curves

**Abstract:** A holomorphic triple (E1,E2,f) on a smooth projective curve C with g(C)>=1
consists of a pair of coherent sheaves E1,E2 on C and a morphism f\in
Hom(E1,E2). Let TCoh(C) be the abelian category of holomorphic triples. In
this talk we study Bridgeland stability conditions on $D^b (TCoh(C))$ and
the moduli spaces of Bridgeland semistable
objects. In particular, we shall prove that the moduli stacks are algebraic
of finite type over $\mathbb C$. This is joint work with Dominic Bunnett. __Slides__

**Title:** Stability of Halphen pencils of index two

**Abstract:**In this talk I will present some results about the GIT stability
of Halphen pencils of index two -- classical geometric objects which were
first introduced by G. Halphen in 1882. Inspired by the work of Miranda on
pencils of plane cubics, I will explain how to obtain explicit stability
criteria in terms of the types of singular fibers appearing in the associated
rational elliptic surfaces. In particular, I will describe how the log canonical
threshold plays an important role. __Slides__