Speakers and abstracts

Title: A Pascal's Theorem for rational normal curves
Speaker: Alessio Caminata (Universita’ di Genova)
Abstract: Pascal's Theorem gives a synthetic geometric condition for six points A,…,F in the projective plane to lie on a conic: Namely, that the intersection points of the lines AB and DE, AF and CD, EF and BC are aligned. One could ask an analogous question in higher dimension: Is there a linear coordinate-free condition for d+4 points in the d-dimensional projective space to lie on a degree d rational normal curve? In this talk, we will discuss and give an answer to this problem by studying the parameter space of d+4 ordered points that lie on a rational normal curve of degree d. This is a joint work with Luca Schaffler.

Title: New contributions to Gröbner’s problem
Speaker: Liena Colarte Gomez (Universita di Barcellona)
Abstract: In this talk, we present new contributions to the longstanding problem, posed by Gröbner in 1967, of determining the aCM property of monomial projections of Veronese varieties. We relate it to invariants of finite abelian groups and we study the geometry of monomial projections parameterized by them.

Title: Irregular fibrations and derived categories
Speaker: Luigi Lombardi (Università of Milano)
Abstract: The derived category of sheaves on a smooth projective variety is a homological object that encodes several aspects of the geometry of the variety itself. In this seminar I will establish that the sets of irrational pencils over smooth curves of genus greater or equal to two are invariant under derived equivalence. In the second half of the seminar, I will extend this result to certain classes of irregular fibrations over higher-dimensional bases. This is a joint work with F. Caucci and G. Pareschi.

Title: Generalized logarithmic sheaves
Speaker: Simone Marchesi (Universitat de Barcelona)
Abstract: In this talk we will introduce a generalization of the definition of logarithmic tangent sheaf on a smooth surface associated to a pair (D,Z), being Z a set of fixed points of a divisor D. Furthermore, we will study when the pair (D,Z) can be recovered from the logarithmic sheaf, a property that is referred to as Torelli property. This is a joint work with S. Huh, J. Pons-Llopis and J. Vallès.

Title: Hyperelliptic Odd Coverings (and more)
Speaker: Riccardo Moschetti (Università di Torino)
Abstract: "Hyperelliptic Odd Coverings" are a class of odd covering of C -> P^1, where C is hyperelliptic. They are characterized by the behavior of the hyperelliptic involution of C with respect to an involution of P^1. I will talk about some tools for studying such coverings: by fixing an effective theta characteristic, they are in correspondence with the solutions of a certain type of differential equations. Considering them as elements in a suitable Hurwitz space, they can be characterized using monodromy and can then be studied from the point of view of deformations. When C is general in H_g, the number of Hyperelliptic Odd Coverings C -> P^1 of minimum degree is finite. The main result that I will tell in the seminar will be how to calculate this number. This is a work in collaboration with Gian Pietro Pirola. In the final part of the talk I will speak about how to generalize the work to Z/rZ-equivariant covers of P^1 with moving ramification. This is a work in collaboration with Carl Lian.

Title: On blow-ups for Fano manifolds
Speaker: Carla Novelli (Università di Padova)
Abstract: Fano manifolds are smooth complex projective varieties whose anticanonical bundle is ample. A natural question in the study of Fano manifolds is to investigate those manifolds that have an extremal ray, in the sense of Mori theory, which is associated with a special contraction (e.g. a projective bundle, a quadric bundle, ...). In this talk we will consider Fano manifolds X admitting an extremal ray associated with a smooth blow-up; then, under some further assumption on the invariants of X, we will describe the Kleiman-Mori cone of X and we will give the classification.

Title: Rational singularities of nested Hilbert schemes
Speaker: Alessio Sammartano (Politecnico di Milano)
Abstract: The Hilbert scheme of points Hilb^n(A^2), parametrizing finite subschemes of the plane of degree n, is a well studied parameter space. A classical theorem of Fogarty states that it is a smooth variety of dimension 2n. By contrast, the nested Hilbert scheme Hilb^(n_1,n_2)(A^2), parametrizing nested pairs of subschemes of degrees n_1 and n_2, are usually singular, and very little is known about their singularities. Using techniques from commutative algebra, we prove that the nested Hilbert scheme Hilb^(n,2)(A^2) has rational singularities. This is a joint work with Ritvik Ramkumar.