Conference on Algebraic Surfaces on the occasion of
Margarida Mendes Lopes 60th birthday

Miguel Ángel Barja

Surfaces on the Severi line

The Severi Conjecture, proved by Pardini, states that K²_S ≥ 4χ(ω_S) , if S is a surface of maximal Albanese dimension. It was also conjectured that surfaces verifying the equality must only be suitable double covers of abelian surfaces. This characterization of the equality was proved by Manetti assuming that K_S is ample. We give a complete and affirmative answer to this conjecture, without extra hypotheses. The proof is based on recent advances on generalizations of Severi-type inequalities for nef line bundles and étale covering trick techniques. This is a joint work with Rita Pardini and Lidia Stoppino.

Arnaud Beauville

Surfaces with maximal Picard number

For a smooth complex projective variety, the rank ρ of the Néron-Severi group (the subgroup of divisor classes in H²(X, Z)) is bounded by the Hodge number h^1,1. Varieties with ρ = h^1,1 have interesting properties, but are rather sparse, particularly in dimension 2. I will discuss a number of examples, in particular those constructed from curves with special Jacobians.

Ciro Ciliberto

A few remarks on curves on surfaces

In a first part of this talk, I will discuss the following questions posed by A. Knutsen:
(1) Does there exist a constant m₁=m₁(X) such that if D is any divisor with h⁰(X,O_X(D)) = 1 and D²>0, then we have h⁰(X,O_X(m₁D)) ≥ 2?
(2) Does there exist a constant m₂=m₂(X) such that if D is a divisor with D² > 0, meeting positively an ample divisor, then one has h⁰(X,O_X(m₂D)) > 0?
(3) Does there exist a constant m₃=m₃(X) such that if D is a divisor meeting positively an ample divisor and with D² ≥ m₃, then one has h⁰(X,O_X(D)) > 0?
In a second part I will talk about joint results, sharing some similitudes with Question (1), with X. Roulleau, concerning the so-called Bounded Negativity Conjecture.

Christian Liedtke

Good Reduction of K3 Surfaces

By a classical theorem of Serre and Tate, extending previous results of Néron, Ogg, and Shafarevich, an Abelian variety over a p-adic field has good reduction if and only if the Galois action on its first l-adic cohomology is unramified. In this talk, we show that if the Galois action on second l-adic cohomology of a K3 surface over a p-adic field is unramified, then the surface has admits an ``RDP model'' over the field, and good reduction (that is, a smooth model) after a finite and unramified extension. (Standing assumption: potential semi-stable reduction for K3's.) Moreover, we give examples where such an unramified extension is really needed. On our way, we establish existence existence and termination of certain semistable flops, and study group actions of models of varieties. This is joint work with Yuya Matsumoto.

Joan Carles Naranjo

Xiao's conjecture for general fibred surfaces

In this talk we will prove that the genus g and the relative irregularity q_f of a non-isotrivial fibration f satisfy the inequality q_f ≤ g-c_f, where c_f is the Clifford index of the generic fibre. This gives in particular a proof of the (modified) Xiao's conjecture, q_f ≤ g/2 +1, for fibrations whose general fibres have maximal Clifford index.This is a joint work with Miguel Ángel Barja and Víctor González-Alonso.

Bruno de Oliveira

The topology and geometry of closed symmetric differentials

The connection between the algebra of symmetric differentials and the topology of a projective manifold is loose and mysterious. On this talk we consider a subclass of symmetric differentials, called closed symmetric differentials, whose connection to the topology is stronger. We will describe several topological implications that can be derived from the existence of closed symmetric differentials, with special emphasis to the case of algebraic surfaces.

Rita Pardini

(Stable) Godeaux surfaces with an Enriques involution

I will present the classification of Godeaux surfaces with an involution such that the quotient surface is birational to an Enriques surface, giving an outline of the proof; these surfaces give a 6-dimensional unirational irreducible subset X of the moduli space of surfaces of general type. I will also describe some non-normal stable surfaces corresponding to points of the closure of X in the moduli space of stable surfaces. This is joint work with Margarida Mendes Lopes.

Roberto Pignatelli

The bicanonical map of some surfaces of general type

The study of the possible behaviours of the bicanonical map of a surface of general type has been a major issue in the last decades, but still some interesting questions are open. I will first recall the most important results in the literature and some open problems on this subject, and then discuss some recent examples related to these questions. This is a joint work with Filippo Favale.

Gian Pietro Pirola

Subfields of the rational fields of surfaces

I will present the following result obtained in a joint work with Yongnam Lee:
Theorem: Let K be the function field of a very general complex surface of degree d > 4 in the projective 3-dimensional space. Let L be a proper subfield of K that contains properly the base field C. Then L is isomorphic either to C(x), if the transcendental degree of L is 1, or to C(x, y) if L has transcendental degree 2.
Similar results hold for the very general product of two curves.

Carlos Rito

A surface with q = 2 and canonical map of degree 16

It is known since Beauville (1979) that if the canonical image φ(S) of a surface of general type S is a surface, then the degree d of the canonical map φ satisfies d ≤ 36. Moreover if the irregularity q(S) ≤ 3, then d ≤ 36−9q. In this talk I will describe the construction of an example with q = χ = 2 and d = K² = 16. If time permits, I will also report on an example with q = 0, d = 16 and K² = 32.

Xavier Roulleau

Geography of simply connected surfaces of general type

The Chern numbers c₁², c₂ of a smooth minimal surface of general type X satisfy the Bogomolov-Miyaoka-Yau inequality: c₁² ≤ 3 c₂. Thirty-five years ago, Bogomolov asked if one can improve the BMY inequality to c₁² ≤ a c₂ with a<3 when one moreover supposes that X is simply connected. In this talk, we show that there exists spin (resp. non-spin) simply connected surfaces with c₁²/c₂ arbitrarily close to 3, and therefore the answer is negative. This is a joint work with Giancarlo Urzúa.