ABSTRACTS
Michel Brion - Projective homogeneous varieties over an arbitrary field
The objects of the talk are the projective algebraic varieties over a field that are homogeneous under a smooth algebraic group. Over the complex numbers, every such variety is the product of an abelian variety and a generalized flag variety G/P, where G is semi-simple and P parabolic (Borel-Remmert). This was generalized to any algebraically closed field by Sancho de Salas. In characteristic p > 0, P may be non-smooth; its structure was described by Wenzel for p > 3, and Maccan for p = 2, 3. The talk will first present these results, and then their extensions to an arbitrary field based on work in progress with Matilde Maccan and Srimathy Srinivasan.
Cinzia Casagrande - Fano varieties with Lefschetz defect 2
We will start by discussing the definition of the Lefschetz defect delta(X) for a (smooth, complex) Fano variety X, and by recalling its properties: when delta(X)>3 the variety X is a product, while when delta(X)=3 there is a structure theorem for X. Then we will present a work in progress on the structure of Fano varieties with Lefschetz defect 2.
Thomas Dedieu - TBA
Enrico Fatighenti - Fano 4-folds of type c5 and even nodal surfaces
A classical construction in projective geometry realizes the blow-up of a cubic fourfold along a line as a conic bundle whose discriminant locus is a quintic surface with 16 nodes. A similar construction can be carried out for Gushel–Mukai fourfolds, as shown in joint work with M. Bernardara, G. Kapustka, M. Kapustka, L. Manivel, G. Mongardi, and G. Tanturri. In this talk, we explain how to obtain an analogous result via a degeneration of certain Küchle fourfolds of type (c5), which are of K3 type. This is joint work with Federico Tufo.
Evgeny Ferapontov - Differential equations for modular forms
It is well known that every classical modular form f on a discrete subgroup of SL(2, R) satisfies a third-order nonlinear ODE that expresses algebraic dependence of the functions f, f’, f’’ and f'''. Here, we clarify the structure and properties of these ODEs from the viewpoint of symmetry analysis of differential equations.
Thus, ODEs for classical modular forms are automatically invariant under the Lie group SL(2, R), which acts on the solution spaces thereof with an open orbit (and a discrete stabiliser of a generic solution). Similarly, every modular form satisfies a fourth-order nonlinear ODE that is invariant under the Lie group GL(2, R), acting on its solution space with an open orbit. ODEs for modular forms can be compactly expressed in terms of differential invariants of the corresponding actions. The invariant forms of both ODEs define plane algebraic curves naturally associated with every modular form; the corresponding ODEs can be seen as modular parametrisations of the associated curves.
After reviewing examples of ODEs satisfied by classical modular forms (such as Eisenstein series, modular forms on congruence subgroups of level two and three, theta constants, and some newforms of weight two), we generalise these results to Jacobi forms; these satisfy involutive nonlinear third-order PDE systems that are invariant under the semidirect product of SL(2, R) with the Heisenberg group. All results on Jacobi forms are essentially new.
It is precisely via the corresponding differential equations that modular forms feature in various problems of mathematical physics and applied mathematics. The present paper emphasises the point of view of modular forms as special functions of mathematical physics.
Sara Angela Filippini - Residual intersections and Schubert varieties
The notion of residual intersections was introduced by Artin and Nagata. Roughly speaking, given an algebraic variety X and a closed subscheme Y in X, which is contained in another closed subscheme Z, then a closed subscheme W such that W \cup Y = Z is a residual intersection of Y in Z.
This idea can be formalized as follows: Let I be an ideal in a local Cohen-Macaulay ring R, and A = (a_1, \ldots, a_s) \subsetneq I. Then J = A:I is called an s-residual intersection of I if ht(J) \geq s \geq ht(I). Residual intersections provide a generalization of linkage: if J = A:I and I = A:J for A a regular sequence, I and J are said to be linked.
I will show how results of Huneke and of Kustin and Ulrich on residual intersections for standard deteminantal ideals and Pfaffian ideals respectively arise in the context of ideals of Schubert varieties in the big opposite cell of homogeneous spaces. This is joint work with X. Ni, J. Torres and J. Weyman.
Baohua Fu - Symplectic singularities and Hamiltonian reductions
I'll report a joint work with Jie Liu (AMSS, China) showing that many Hamiltonian reductions are affine closures of some cotangent bundles of smooth quasi-projective varieties. As an example, we discover a novel connection between minimal nilpotent orbits of type A and those of type D.
Frederic Han - Geometry of genus sixteen K3 surfaces
Polarized K3 surfaces of genus sixteen have a Mukai vector bundle of rank two. We study the geometry of the projectivization of this bundle. We prove that it has an embedding in P9 with an ideal generated by quadrics. We give an effective method to compute these quadrics from a general choice in Mukai’s unirationalization of the moduli space. This linear system gives a double cover of P9 ramified on a degree 10 hypersurface. It gives relative Weddle/Kummer surfaces over a Peskine variety associated to an explicit trivector. This work is also motivated by hyperkähler geometry and Debarre-Voisin varieties. Oberdieck showed that the Hilbert square of a general K3-surface of genus 16 is a Debarre-Voisin variety for some trivector. We start to investigate the relationship between these two trivectors.
Alexander Kuznetsov - Odd biisotropic symplectic Grassmannians
In the talk I will discuss the geometry of the subvariety of Gr(k,2n+1) parameterizing subspaces of dimension k in a vector space of dimension 2n+1 isotropic with respect to a general pencil of skew-symmetric forms, with an emphasis on the case where k = n = 3.
Joseph M. Landsberg - Tensors of minimal border rank
A common geometric question is as follows: given an algebraic group G, a G-module V, and a vector v in V, describe the G-orbit closure of v. Many incidences of this problem arise naturally in theoretical computer science. I will discuss one such which immediately leads to an intractable problem in algebraic geometry (characterizing the smoothable component of the Hilbert scheme of points). Nevertheless, one can prove useful and interesting qualitative results that I will discuss. This is joint work with J. Jelisiejew and A. Pal as well as work with F. Gesmundo, J. Jelisiejew, and T. Mandziuk.
Michalek Mateusz - Beyond Linear Flattenings of Tensors
Equations of secant varieties play an important role in several branches of pure and applied mathematics. Landsberg and Manivel were among the pioneers in studying such equations using modern representation theory and algebraic geometry. Among the most useful equations are Koszul and Young flattenings, developed by Landsberg and Ottaviani, with ideas going back to Strassen. However, it was known that this approach in fact always produces equations for a larger cactus variety.
I will report on recent joint work with Dolezalek in which we provide determinantal equations that distinguish secant varieties from cactus varieties. As an application, we obtain a simple, computer-free proof that the border rank of the 2×2 matrix multiplication tensor is seven. I will also discuss generalizations of these equations, based on work in progress with Dolezalek, Misinová, and Stastny.
Rosa Maria Miró-Roig - Uniform and homogeneous vector bundles on projective spaces
In my talk I will deal with uniform and homogeneous vector bundles on projective spaces and I will explain recent contributions to the longstanding open conjecture: Any uniform vector bundle of rank r on the projective space P^n with 2r<n is homogeneous
Giovanni Mongardi - The classification of Enriques manifolds of K3^[n] type
We show that an Enriques manifold whose universal covering is a hyper-Kähler manifold of K3^[n] type is always associated to an Enriques surface. More precisely, we show that a group of automorphisms of prime order on a hyper-Kähler manifold X of K3^[n]-type has always fixed points, unless the order is 2, X is isomorphic to a moduli space of stable twisted sheaves on a K3 surface S, and the involution is induced from a fixed point free involution on S preserving all numerical data. This is joint work in progress with Emanuele Macrì.
Giorgio Ottaviani - vInvariants of points in computer vision
We report on a joint work with Rekha Thomas. Given two sets of k general points in P^n, we call the centers-variety the locus of pairs (a,b) in (P^n x P^n) such that their projections from a and b are projectively equivalent. We describe all the centers-varieties for n=3; the relevant cases correspond to k=5, 6, 7 points. As a by-product, we find an alternative description of the Weddle surface in P^3, which is a Kummer variety, and describes the locus of points a in P^3 such that the projections of six given points from a lie on a conic.
Nicolas Perrin - Quantum cohomology and irrationality of Gushel-Mukai fourfolds
I will report on a recent joint work with Vladimiro Benedetti and Laurent Manivel. We compute a presentation of the quantum cohomology ring of Gushel-Mukai fourfold and in particular the multiplication by the Euler field. Using recent work of Katzarkov, Kontsevich, Pantev, Yu and of Guéré, we deduce that the very general Gushel-Mukai fourfold is irrational.
Francesco Russo - On the tangent degree of a projective variety
The tangent degree tau(X) of a projective variety X^n \subset P^N is the number of tangent spaces to X at smooth points passing through a general point of the tangent variety Tan(X)\subset P^N if dim(Tan(X))=2n; it is equal to zero if dim(Tan(X))<2n. After showing that tau(X) \neq 1 when N=2n we shall describe some classification results for N=2n and tau(X)=2 either in small dimension and/or under the smoothness assumption. Finally for N \geq 2n+1 we shall consider varieties
X^n subset P^N having tau(X)>1 (unexpected behaviour), provide their classification for n=2 and discuss the case n \geq 3. This is joint work with Jordi Hernandez Gomez.
Alessandra Sarti - On quotients of generalized Fermat manifolds and Log Enriques varieties
Generalized Fermat manifolds (GFM) are defined by a system of Fermat-type equations and extend the classical Fermat manifolds, which are given by a single equation. The aim of this talk is to study certain quotients of these manifolds. In particular, we focus on GFMs that are Calabi–Yau manifolds and investigate specific automorphisms of finite order acting on them. We show that the resulting quotients are either singular Calabi–Yau
manifolds or Log Enriques varieties of index 2. Log Enriques varieties have been introduced only recently by several authors; they can be viewed as singular analogues of Enriques manifolds, which themselves generalize Enriques surfaces.
Moreover, we prove that, with only a few exceptions, these quotients have terminal singularities. The results are contained in a joint work in progress with A. Palomino.
Jieao Song - Coble type hypersurfaces and hyperkähler fourfolds
If a hypersurface in a projective space is singular exactly along a variety X, we call it a Coble type hypersurface for X. Hyperkähler manifolds are natural generalizations of K3 surfaces in higher dimensions, and geometric descriptions for locally complete families are only known in very few cases. In this talk, we present two new examples of K3[2]-type, in terms of Coble type hypersurfaces. Based on joint work with Benedetta Piroddi, Ángel Ríos Ortiz, and Andrés Rojas.