Birational geometry of Algebraic Varieties

Birational geometry of Algebraic Varieties Birational Geometry of Varieties

Pisa, May 5-7 2006

BGV

Abstracts

Finite subgroups of the Cremona group
Arnaud Beauville (Nice)

The Cremona group is the group of birational automorphisms of P² Attempts to describe its finite subgroups go back to the 19th century, and have been improved by many authors. The modern approach is based on Mori theory; I will explain what this gives, and what remains to be done.

Surfaces with K²≤ 3χ-1 and finite fundamental groups
Margarida Mendes Lopes (IST, Lisbon)

If a surface of general type has a finite (algebraic) fundamental group then it is regular. Xiao Gang proved in 1985 that for minimal surfaces of general type S satisfying K²≤ 3χ-1 either the fundamental group is finite is finite or then S has a base point free pencil of hyperelliptic genus 3 curves with at least 4 double fibres. In this talk I will report on work in colaboration with R. Pardini concerning the order of the finite fundamental groups that can occur in this range of invariants. It turns out the the biggest possible order is 9, and fundamental groups of order 9 occur only if S is a Campedelli surface. I will also present the complete classification and some properties of Campedelli surfaces with fundamental groups of order 9.

Rational cubic hypersurfaces
Massimiliano Mella (Ferrara)

All known explicit rationality constructions for smooth cubic hypersurfaces are based on varieties with one apparent double point. This force the dimension of the hypercubic to be even. A sligth change in this approach allows to construct rational smooth hypercubic in any dimension greater or equal to 7. This leads to different perceptions of the rationality problem for cubic hypersurfaces.

On Fano manifolds
Marco Andreatta (Trento)

I will present some recent reults on Fano manifolds connected to a conjecture of S. Mukai and to classification problems.

Deformations of real and complex structures
on principal torus bundles over tori.
Paola Frediani (Pavia)

We will present some results obtained in collaboration with Fabrizio Catanese on the deformations of principal holomorphic torus bundles over tori. We consider the Kodaira Spencer map of the complete Appell-Humbert family parametrising such bundles and we show that we obtain a connected component of the space of complex structures each time that the base dimension is two, the fibre dimension is one, and a suitable topological condition is verified. We also describe the family of real structures s on these bundles X, and prove its connectedness when the complex dimension is at most three. From these results it follows that the differentiable type determines the deformation type of the pair (X,s), provided we have complex dimension at most three, fibre dimension one, and a certain 'reality' condition on the fundamental group is satisfied.

On the topological index of irregular surfaces
Miguel A. Barja (U. Politècnica de Catalunya, Barcelona)

Let X be a complex smooth projective variety of dimension n. We say that is "generalized Lagrangian" if there exist linearly independent global 1-forms ω₁,ω₂,...,ω_2n on X such that ω₁∧ ω₂+ ω₃∧ ω₄+ ... + ω_2n-1∧ ω_2n=0 (as a global 2-form on X) and V=<ω₁,...,ω_2n> generically generates the cotangent bundle of X. Let δ(X)=(c₁(x)²-2c₂(X))/2 be the degree 2 part of the Chern character of X. If X is generalized Lagrangian and V generates the cotangent budle of X in codimension 1, I will show that δ(X) is pseudo-effective. When X is a surface (and then δ(X) is, up to a positive constant, the "topological index" τ of X), we can obtain the same result also in other cases, depending on the geometry of the base locus F_V of the linear subsystem |∧²V| of the canonical system of X.
Finally I will show that minimal smooth surfaces of general type with p_g=5, q=4 verify τ (X) ≥0.
This is a joint work with J.C. Naranjo and G.P. Pirola

On varieties which are uniruled by lines
Andreas Knutsen

I will talk about joint work with Carla Novelli and Alessandra Sarti. It is well-known that an irreducible nondegenerate complex variety X in Pⁿ of degree d satisfies d ≥ n - dim X+1. Varieties for which d is ``small'' compared to n have been the objects of intensive study throughout the years and one of the common features is that such varieties are covered by rational curves, i.e. uniruled. More generally one can study the degree of the covering curves with respect to any big and nef line bundle H on the variety and one says that X is uniruled of H-degree m if all the covering curves D satisfy D.H ≤m. Our result is the following:
Let (X, H) be a pair consisting of a reduced and irreducible variety X of dimension k ≥ 3 and a globally generated big line bundle H on X with d:= H^k and n:= h⁰(X, H)-1 such that d<2(n-k)-4. Then X is uniruled of H-degree one, except if (k,d,n)=(3,27,19) and a #-minimal model of (X, H) is (P³,O_P³(3)). The result is optimal for k=3 and the proof is in fact done by induction on k, using the #-minimal model program on threefolds.

Degenerations to unions of planes and interpolation theorems
Rick Miranda (Colorado State University)

Several recent results on interpolation will be described, all coming from interesting degeneration techniques. First, the linear system of plane curves of degree d having a square number of general points of multiplicity m all have the expected dimension: this was recently proved by Evain, and we give a new proof based on work with Ciliberto. Second, a degeneration of the Veronese to unions of planes provides a very quick proof of the multiplicity two theorem, which is similar combinatorially to the proofs using the techniques of Lorentz and Lorentz, but much more geometric in flavor.

Graded rings and algebraic varieties
Miles Reid (Warwick)

Many of the traditional and modern constructions of algebraic varieties are closely related to graded ring methods. The lecture will discuss some old and new results in this direction.

Triangular groups and some of their applications
Fabrizio Catanese (Bayreuth)

I would like to talk on joint works with Ingrid Bauer, respectively with Soenke Rollenske. A triangular group is a finite group which is a quotient of the triangle group T(m,n,r), a subgroup of index two in the group generated by reflections along the sides of a hyperbolic triangle with angles π/m, π/n, π/r. These groups determine algebraic curves defined over number fields, and can be used to construct many interesting surfaces, and Kodaira fibrations with high slope. I will explain the slope question for Kodaira fibrations, and the packing problem for curve maps. If time permits I will also report on Beauville surfaces, their classification, and the action of the absolute Galois group on their moduli space.

GV sheaves, Fourier-Mukai, and generic vanishing
Mihnea Popa (Chicago)

The classical Kodaira and Kawamata-Viehweg vanishing theorems have some partial analogues, the Generic Vanishing Theorems discovered by Green and Lazarsfeld, when the positivity hypotheses on line bundles are weakened. I will explain how abstract Fourier-Mukai transforms allow one to relate in a formal sense generic vanishing theorems to classical vanishing theorems. In particular I will generalize (and give algebraic proofs of) the previously known generic vanishing results. I will also show how the same techniques produce higher rank generic vanishing on moduli spaces of sheaves considered by Mukai and Bridgeland (among others), and apply to other more unexpected situations too. This is joint work with G. Pareschi.

Almost Fano threefolds with pseudoindex > 1
Cinzia Casagrande (Pisa)

A complex projective variety X is almost Fano is its anticanonical divisor is Cartier, nef and big. We define its pseudo-index as the minimal positive anticanonical degree of rational curves in X. As in the Fano case, one expects that almost Fano varieties with large pseudo-index are simpler. We describe some properties of almost Fano and Fano threefolds with Gorenstein canonical singularities and pseudo-index > 1; in particular we compute their maximal Picard number. This is a joint work with P. Jahnke and I. Radloff (University of Bayreuth).

Cyclic coverings and Seshadri constants on smooth surfaces
Luis Fuentes Garcia (Universidade da Coruña)

We study the Seshadri constants of cyclic coverings of smooth surfaces. The existence of an automorphism on these surfaces can be used to produce Seshadri exceptional curves. We apply this method to cyclic coverings of the projective plane of degree n. When 2 ≤n≤9, explicit values are obtained. We relate this problem with the Nagata conjecture.

Weak Fano 3-folds
Alessio Corti (Cambridge)

This talk describes work of my student Anne-Sophie Kaloghiros. I will discuss some general tools to work with weak Fano 3-folds and sketch some applications.

The Hodge structure of intersection cohomology
Luca Migliorini (Bologna)

we will discuss the Hodge structure defined on the intersection cohomology of a singular variety and how it can be realized as a sub Hodge structure of the cohomology of a resolution. This depends on the choice of a polarization of the resolution and we will discuss this dependance.

On the Chow ring of certain algebraic hyper-Kaehler manifolds
Claire Voisin (Paris, Jussieu)

We study and generalize a conjecture by Beauville concerning polynomial relations involving Chern classes and classes in the Neron-Severi group of X, where X is algebraic hyper-kaehler. (Beauville considered polynomial relations between classes in NS(X).) The generalized conjecture states that these relations already hold in CH(X). We prove it for X =Hilbⁿ(S), S a K3 surface, and n<11, and also for the Fano variety of lines of a cubic fourfold.