Stefano Canino (Politecnico di Torino): Complete intersections on Veronese surfaces. In “Commentationes Geometricae” Euler asked when a sets of points in the plane is the intersection of two curves, that is, using the modern terminology, when a set of points in the plane is a complete intersection. In the same period, Cramer asked similar questions so that this kind of questions is presently known as the Cramer-Euler problem. In this talk, we consider a generalization of the Cramer-Euler problem: characterize the possible complete intersections lying on a Veronese surface, and more generally on a Veronese variety. The main result describes all possible reduced complete intersections on Veronese surfaces. We formulate a conjecture for the general case of complete intersection subvarieties of any dimension and we prove it in the case of the quadratic Veronese threefold. Our main tool is an effective characterization of all possible Hilbert functions of reduced subvarieties of Veronese surfaces, which we call d-sequences as a generalization of the well known O-sequences.
Davide Frapporti (Universitå di Genova): Segre and Goryunov-Kalker cubic hypersurfaces and associated discriminants. In this talk I report on a joint work with F. Catanese, Y. Cho and S. Coughlan. The Segre and the Goryunov-Kalker cubics are two sequences of maximally nodal cubic hypersurfaces of odd resp. even dimension. We prove a local rigidity result for such hypersurfaces and determine when the nodes can be smoothed independently. We also study the GK cubic 4-fold to construct a new 31-nodal Togliatti quintic surface as a symmetric determinantal.
Luis Núñez-Betancourt (CIMAT, Mexico): Differential aspects of the F-pure thresholf. Given a local ring, the F-pure threshold of its maximal ideal is a numerical invariant that detects and measures the singularity of the ring. In this talk, we will discuss a formula for this invariant in terms of differential operators. We will also see several consequences of this formulation.
Saverio Andrea Secci (Universitå di Torino): Anticanonical base locus of Fano manifolds and extremal contractions. Fano manifolds are a very natural class of projective varieties: they are characterised by the ampleness of the anticanonical divisor. Although a sufficiently large multiple is very ample, the anticanonical divisor of a Fano manifold is not always globally generated. In fact, examples are known where the anticanonical linear system admits nontrivial base locus: these examples are very few and it is expected that this is not the general behaviour. In this talk we will deal with low-dimensional Fano manifolds, trying to understand the relationship between the anticanonical base locus and their birational geometry.
Paolo Sentinelli (Politecnico di Milano): Immanant varieties and χ-matroid. For any simple character of a finite group, we will define a projective variety by projecting the Segre embedding. Such varieties are described by parametric equations involving the immanant (of the corresponding simple character) of a matrix of parameters. For one-dimensional characters, the geometry of these varieties suggests the definition of a combinatorial structure, which we call χ-matroid. A matroid is obtained by considering the alternating character of a symmetric group. We will enounce some conjectures and problems concerning the combinatorics and the geometry of immanant varieties.
Francesco Strazzanti (Universitå di Genova): Cohen-Macaulay and S_2 binomial edge ideals. A binomial edge ideal is an ideal of a polynomial ring generated by some minors of a generic 2xn matrix. Therefore, it is possible to put its generators in correspondence with the edges of a finite simple graph.Giving a description of the algebraic properties of binomial edge ideals in terms of the combinatorics of the associated graphs is a challenging problem, and in this talk I will report on some recent results in this direction regarding Cohen-Macaulay property and Serre's condition S_2. This is joint work with D. Bolognini, A. Macchia, and G. Rinaldo.
Luca Tasin (Universitå Statale di Milano): Sasaki-Einstein metrics on spheres. It is a classical problem in geometry to construct interesting metrics on spheres. I will report on a joint work with Yuchen Liu and Taro Sano in which we construct infinitely many families of Sasaki-Einstein metrics on odd-dimensional spheres that bound parallelizable manifolds, proving in this way conjectures of Boyer-Galicki-Kollár and Collins-Székelyhidi. The construction is based on showing the K-stability of certain Fano weighted orbifold hypersurfaces.
