(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, ...,
nN) denote the variety
parametrizing differential complexes on fixed vector spaces of
dimensions n0, ..., nN. We shall review the definition and basic
properties of C(n0, ..., nN), including the description of its
irreducible components. We will represent F(r, d) as a linear section of certain C(n0, ..., nN). 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
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.
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 - (KX + 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.
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,
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).