Abstracts

It is a well known fact that a Jacobian variety has a 4−dimensional family of 3-secants lines to its Kummer Variety. Recently the problem of characterizing Jacobians by the existence of just one trisecant has been solved.

My work consist on generalizing this fact, studying 4-secant planes to the Kummer variety of an indecomposable principally polarized abelian variety X; particularly, if one has a curve of 4−secant planes to the Kummer variety of X. one has that X is a Prym Variety under the additional condition that a certain triple intersection of translates of the Theta divisor is a complete intersection. The origin of this technical issue that there is not an exact analog to the Matsusaka-Ran criterion (containing a curve that is the minimal class) for the study of Abelian Varieties that has a curve that is twice the minimal class, but thanks to the work of Welters this separares in 3 particular cases, which are being studying in the ongoing work.

The moduli space of K3 surfaces with a purely non-symplectic automorphism of order n ≥ 2 is one dimensional exactly when φ(n) = 8 or 10. We provide explicit equations for the very general members of the irreducible components of maximal dimension of such moduli spaces and we describe their geometry. In particular, we show that there is a unique one-dimensional component for n = 20,22,24, three irreducible components for n = 15 and two components in the remaining cases. This is joint work with Paola Comparin and María Elisa Valdés.

For any projective variety embedded into projective space, we study the question of when a linear projection restricted to said variety gives a Galois morphism. In particular, we will show that the set of linear projections that restrict to Galois morphisms has a natural scheme structure, and offers interesting insight into the geometry of the variety itself.

Brill-Noether theory on curves has been studied from different points of view. In this talk I will explain the twisted Brill-Noether theory and some results on non-emptiness obtained with Peter Newstead.

From the point of view of manifold topology, the moduli space BDiff(M) of a smooth manifold M is the space that classifies smooth M-bundles over some base space. The cohomology and homotopy groups of moduli spaces of smooth manifolds are some of the most basic invariants of manifold bundles: the former contain all the characteristic classes and the latter classify smooth bundles over spheres. Complete calculations of these groups are challenging, even for the simplest compact manifolds. It is then desirable to know, at least, some qualitative information, for example whether these groups are (degreewise) finitely generated. In this talk, I will discuss a method to attack this question which leads to the following theorem: if M is a closed smooth manifold of even dimension > 5 with finite fundamental group, then the cohomology and higher homotopy groups of BDiff(M) are finitely generated abelian groups.

In this talk we show a complete classification of homogeneous locally nilpotent derivation on ℂ[S] when S is a commutative cancellative monoid via a combinatorial element called Demazure root. This classification is a generalization of the classification when S is in correspondence with normal affine toric varieties, i.e., when S is a saturated affine semigroup. Finally, as a geometric application, we obtain a family of non-normal toric surfaces X veri- fying Aut(X) is isomorphic with Aut(Xn) where Xn is the normalization of X. These results are part of a joint work with Alvaro Liendo.

We describe all the schematic limits of divisors associated to any family of linear series on any one-dimensional family of projective varieties degenerating to any connected reduced projective scheme X defined over any field, under the assumption that the total space of the family is regular along X. More precisely, the degenerating family gives rise to a special quiver Q, called a Z^n-quiver, a special representation L of Q in the category of line bundles over X, called a maximal exact linked net, and a special subrepresentation V of the representation induced from L by taking global sections, called a pure exact finitely generated linked net. Given g=(Q, L, V) satisfying these properties, we prove that the quiver Grassmanian G of subrepresentations of V of pure dimension 1, called a linked projective space, is Cohen-Macaulay, reduced and of pure dimension. Furthermore, we prove that there is a morphism from G to the Hilbert scheme of X whose image parameterizes all the schematic limits of divisors along the degenerating family of linear series if g arises from one. Joint work with Eduardo Vital and Renan Santos.

We consider as ingredients a complex Grassmannian G(d,n) and a class L in its homology group. We are interested in the problem of finding subvarieties (integral closed subschemas) of G(d,n) with homology class L, but which are also invariant under the action of the maximal torus T of G(d,n). We will see that this problem is governed by combinatorial objects called matroids, and we will solve it completely for the case of T-orbits. Later we will study the same problem in the case of the symplectic Grassmannian of lines SpG(2,2n), which is now governed by combinatorial objects called symplectic matroids of rank 2. These results are part of a joint work with Pedro Luis Del Ángel, Javier Elizondo, Alex Fink and Felipe Zaldivar.

Limit canonical systems are degenerations of the canonical linear system along families of curves degenerating to nodal curves. They appeared prominently in works by Eisenbud and Harris for general curves of compact type and by Esteves, Medeiros and Salehyan for certain general nodal curves far from being of compact type. In this talk, we’ll present an approach to bridge these works, describing all the limit canonical systems along degenerations to all nodal curves that are general for their dual graphs, based on the theory of twisted differentials, introduced by Bainbridge, Chen, Gendron, Grushevsky and Möller to compactify certain strata of Abelian differentials. This is a joint work with Eduardo Esteves.

Let C be a smooth curve of positive genus and let C_2 be the symmetric product of C with itself. In this talk I will discuss the pseudo-effective cone of the blow-up of C_2 at a general point, showing that if C is very general, then the cone is non-polyhedral. This result generalizes previous results of J.F. García and G. McGrat about the genus 1 case. To prove the statement we first show that having a polyhedral pseudo-effective cone is a closed property for families of surfaces and then prove it in the hyperelliptic case. This is joint work with Luca Ugaglia.

In a paper of 2019 Boissière-Camere-Sarti prove that there exists an isomorphism between the moduli space of smooth cubic threefolds and the moduli space of IHS fourfolds of K3 [2]-type with a non-symplectic automorphism of order three, whose invariant lattice is generated by a class of square 6. They study then the degeneration of the auto- morphism along the locus of the generic nodal degeneration of a cubic threefold, showing the birationality of this locus with some moduli space of IHS fourfolds of K3 [2]-type with a nonsymplectic automorphism of order three belonging to a different family. I will present their results together with an implementation which consents to go further in the analysis of the degeneration and give a similar result also in the non-generic nodal case.

An old question steaming from Klein’s Erlangen Program can be phrased in modern terms as: Is a given geometric object uniquely determined by its group of symmetries? The first part of this talk consists of an introduction to the problem with some selected examples from outside algebraic geometry. In the second part of the talk we come to the setting of algebraic geometry, where we show that, in general, the answer to the above question is negative. After restricting the class of varieties we will show an instance where the answer is affirmative. Indeed, 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.

Lagrange interpolation is a classical and elementary tool with important connections across mathematics. One of its higher dimensional generalizations, involving higher-rank vector bundles, is a topic of current research with implications in birational geometry.Lagrange interpolation is a classical and elementary tool with important connections across mathematics. One of its higher dimensional generalizations, involving higher-rank vector bundles, is a topic of current research with implications in birational geometry. In this talk I will discuss two compelling examples where solving interpolation problems is crucial in understanding the birational geometry of certain moduli spaces.

This work is inspired by the results of Bourqui (2002) on the asymptotic behavior of the number of rational points of bounded height on Hirzebruch surfaces over function fields. A natural generalization of these Hirzebruch surfaces are the Hirzebruch-Kleinschmidt varieties, which encompass all projective toric varieties with Picard rank 2. We provide an asymptotic formula for the number of rational points of bounded height on Hirzebruch-Kleinschmidt varieties over function fields, and show that this coincides with the asymptotic formula predicted by Peyre (2012), which is a precise formulation of a function eld analogue of a conjecture of Manin originally stated only for number fields.

Del Pezzo varieties arise as a natural higher-dimensional generalization of the classical Del Pezzo surfaces. Over the complex numbers, they were extensively studied by T. Fujita in the 1980s, who classified them according to their degree. In degree 5, it follows from Fujita’s classification that all of these manifolds are obtained as linear sections of the 6-dimensional Grassmannian $\textup{Gr}(2,5)$ with respect to the Plücker embedding, whose points parametrize 2-dimensional linear subspaces of a vector space of dimension 5. In this talk, we will discuss the existence and uniqueness of $\mathbf{G}_a^n$-structures on these varieties, i.e., we will determine when and in how many ways one can obtain them as equivariant compactifications of the abelian unipotent group $\mathbf{G}_a^n$. To do so, we study the Hilbert schemes of certain linear subspaces on such varieties and we analyze some explicit equivariant Sarkisov links. As an application, we give some new results on k-forms of quintic del Pezzo varieties over an arbitrary field k of characteristic zero. This is a joint work with Adrien Dubouloz (Dijon, France) and Takashi Kishimoto (Saitama, Japan).

Due to Poincaré’s Reducibility Theorem, we know that for an Abelian variety A there exists an isogeny decomposition of A into simple factors. However, this decomposition hides many interesting properties of the set of Abelian subvarieties of A. For example, it is not clear from the decomposition ”how many” Abelian subvarieties A actually has. In this talk, for a principally polarized Abelian variety (A, L) of dimension g in the moduli space Ag and some positive integers t and n, we define: N:A(t, n) := #{S ≤ A : S is an abelian subvariety with dim S = n, χ(L) ≤ t}, which counts the number of Abelian subvarieties of dimension n and reduced degree bounded by t. After some reductions to the problem of obtaining a bound for this number, we characterize the set of abelian subvarieties of dimension n on Eg, for E an elliptic curve, in terms of the Grassmannian variety via the Stiefel variety. Then we use machinery from Diophantine Geometry to obtain an asymptotic estimation for NA(t, n) for any (A,L) ∈ Ag.

K3 surfaces are characterized by having a nowhere vanishing rational 2-form and irregularity equal to zero. Quartic surfaces in ℙ³ are such surfaces. Given a smooth quartic K3 surface S ⊂ ℙ³, Gizatullin was interested in which automorphisms of S are induced by Cremona transformations of ℙ³. Later on, Oguiso answered it for some interesting examples and he posed the following natural question: Is every automorphism of finite order of any smooth quartic surface S ⊂ ℙ³ induced by a Cremona transformation? In this talk, we will give a negative answer to this question by constructing a family of smooth quartic K3 surfaces Sₙ with Picard number two such that Aut(Sₙ) = D∞ together with an involution σn of Sₙ, that is not derived by any element of Bir(ℙ³). This is joint work with Ana Vitoria M Quedo.

The study of Weierstrass points has contributed significantly to the understanding of several aspects of the geometry of Riemann surfaces and algebraic curves during the last century. In this talk, after reviewing briefly the most important properties of Weierstrass points, we shall discuss whether or not the action of the automorphism group is transitive on them. This is a joint work with Pietro Speziali (Universidade Estadual de Campinas).

The moduli space of vector bundles on smooth projective varieties were constructed using GIT in the 70’s by Mumford, Newstead, Seshadri, Maruyama and by Gieseker. Since then, these moduli spaces have been studied extensively by several authors. However, there is still a huge number of questions left open. In this talk we determine a bound of the dimension of an irreducible component of the Hilbert scheme of the moduli space of torsion-free sheaves on surfaces. Let X be a non-singular irreducible complex surface and let E be a vector bundle of rank n on X. We use the m-elementary transformation of E at a point x ∈ X to show that there exists an embedding from the Grassmannian variety G(Ex, m) into the moduli space of torsion-free sheaves MX,H(n; c1, c2 + m) which induces an injective morphism from X × MX,H(n; c1, c2) to Hilb MX,H(n;c1,c2+m).

The period matrix for a polarized abelian variety A, defining the relation between the real and the complex coordinate functions of its lattice and of its vector space respectively, captures deep geometric information about A. As a consequence, period matrices are useful tools to describe loci of moduli spaces of abelian varieties with interesting geometric properties. However, finding a period matrix for a given polarized abelian variety A is not easy. In this talk we will first briefly discuss general facts and some open questions regarding abelian varieties, their moduli spaces and the Torelli locus, that we intend to approach through the study of period matrices. Then, we will present some new methods to compute the period matrix of a polarized abelian variety, depending on the given geometric information about it. Finally, we will show a couple of applications. This is a joint work with Rubí E. Rodríguez.

I will introduce N-resolutions, which are the negative analog of the Kollár–Shepherd-Barron (1988) P-resolutions of a 2-dimensional cyclic quotient singularity. (We will instead work with the corresponding M-resolutions of Benkhe-Christophersen (1994).) I will start by describing an algorithm to find them based on the explicit algorithm for P-resolutions (Park-Park-Shin-Urzúa (2018), which geometrically recovers Christophersen-Stevens’ zero continued fractions correspondence (1991)) that in turn is based on the explicit MMP described by Hacking-Tevelev-Urzúa (HTU 2017). I will also describe another way to find N-resolutions via antiflips (HTU 2017) starting with an M-resolution, showing an action of the braid group on all its associated Wahl resolutions. This will bring us to Hacking exceptional collections (2013-2016) on surfaces that are Q-Gorenstein smoothings of particular singular surfaces, where Karmazyn-Kuznetsov-Shinder (2022) have expressed their derived categories via derived categories of Kalck-Karmazyn algebras (2017). This can be put together through Kawamata’s bundles (2018-2022), and I will describe our main theorem on semi-orthogonal decompositions that shows up from the M- and N-resolutions. Applications to all simply-connected Dolgachev surfaces will be shown. This is based on the joint work with Jenia Tevelev https://arxiv.org/abs/2204.13225.

In this talk, I will present a report of my thesis on the geometry of some families of curves in projective 3-space; in particular, curves of degree six and genus three. The main motivation to study these families stems from Liaison Theory. I will talk briefly about some aspects of this theory and its possible relation with the birational geometry of the Hilbert schemes that parametrize families of curves in the projective 3-space.