Midspring Math Meeting

Fernando Cukierman (Buenos Aires, AR): Varieties of Complexes and Foliations
Let F(r, d) denote the algebraic variety parametrizing integrable differential 1-forms of degree d in the complex projective space of dimension r. We consider the problem of describing the irreducible components of F(r, d). Let C(n₀, ..., n_N) denote the variety parametrizing differential complexes on fixed vector spaces of dimensions n₀, ..., n_N. We shall review the definition and basic properties of C(n₀, ..., n_N), including the description of its irreducible components. We will represent F(r, d) as a linear section of certain C(n₀, ..., n_N). Finally, we discuss the relation with the irreducible components of F(r, d).

Gianpietro Pirola (Pavia): The Paracanonical System
The paracanonical system is the Hilbert scheme that contains canonical divisors. The problem of its irreducibility will be addressed, by establishing in which circumstances the canonical system is "exorbitant" and when it is a component of the paracanonical system. Based on a collaboration with Margarida Mendes Lopes and Rita Pardini.

Francesca Tovena (Roma Tor Vergata): Holomorphic Homogeneous Vector Fields and Meromorphic Connections
We study the dynamics of homogeneous vector fields in Cⁿ, via the geodesic flow of a suitable meromorphic connection. As an application, in dimension two we obtain a description of the dynamics in a full neighbourhood of the origin for a class of holomorphic maps tangent to the identity. This is a joint work with M. Abate.

Angelo Lopez (Roma 3): Effective non-vanishing Conjectures
Shokurov's effective non-vanishing theorem lies at the heart of Mori theory. It states that given a klt pair (X, D) and a nef divisor L such that L - (K_X + D) is big and nef, then there exists an m such that h⁰(mL) > 0. In 2000 Kawamata, reworking a conjecture of Ionescu, speculated that one can take m = 1. We will show how the existence of "tigres" in ample linear systems can shed light on the conjecture, and, perhaps, on Fujita's conjecture too. We will then prove a special, for the time being, case.

Gloria della Noce (Pavia): On the Picard Number of Singular Fano Varieties
Let X be a Fano variety of arbitrary dimension and let D be a prime divisor of X. In a recent paper, C. Casagrande proved that, if X is smooth, then ρ(X) - ρ(D) < 9, where ρ denotes the Picard number. Moreover, if ρ(X) - ρ(D) > 3, then X is isomorphic to a product S × Y, where S is a Del Pezzo surface. In this talk I will face the same problem in the singular case. I will show that, under suitable assumptions on the singularities of X, the inequality ρ(X) - ρ(D) < 9 still holds. Moreover, if ρ(X) - ρ(D) > 3, then X has a finite morphism to a product S × Y and ρ(X) = ρ(S) + ρ(Y). The main consequence of this result concerns the three dimensional case, where, under certain assumptions of the singularities of X, it allows us to find effective bounds for ρ(X).

Caterina Cumino (concerning accommodation or other)
Participants should make their own reservation (but we can help) Hotel suggestions

Letterio Gatto, Caterina Cumino, Simon G. Chiossi: (DISMA, Politecnico di Torino)