Birational geometry of surfaces:
Abstracts

Miguel Angel Barja

I will present several recent results on continuous linear systems on
irregular surfaces and its extensions to higher dimensional
varieties. I will explain the eventual behavior of continuous linear
systems through multiplication maps and how to extend the continuous
rank to R-divisors in some directions. With these techniques I will
show how to extend Castelnuovo inequalities on the rank of
multiplication maps and how to derive old and new Clifford-Severi
inequalities relating the volume and the continuous rank of linear
systems (and so giving results on the geography of irregular surfaces
of maximal Albanese dimension). Finally I will show how to
characterize surfaces (and higher dimensional irregular varieties) on
the border of some of these inequalities, and present some open
problems. This ia joint work with Rita Pardini and Lidia Stoppino.

Francesco Bastianelli

We will discuss some birational invariants extending the notion of
gonality to algebraic varieties of arbitrary dimension, and somehow
measuring the failure of a given variety to be rational, rationally
connected or uniruled.

In particular, we will report on results in the case of surfaces, we
will present some of the techniques used to achieve them, and we will
list some open problems on these topics.

Andrea Bruno

I will talk about joint work with E. Arbarello and E. Sernesi and with
E. Arbarello, G. Farkas and G. Saccà.  Which canonically embedded
curves are hyperplane sections of K3 surfaces? In the first work,
following conjectures of Wahl, Mukai and Voisin, we give a complete
characterization of curves which are hyperplane sections of K3
surfaces (or of limits of such). This involves proving two conjectures
stated by Wahl in 1997.  In the second work we we find an explicit
family of curves, of any given genus, satisfying the
Brill-Noether-Petri condition.

Alberto Calabri

Cremona transformations are birational maps of P^n to itself and they
form the Cremona group Bir(P^n).  The subset Bir(P^n)_d of Bir(P^n)
formed by Cremona transformations of fixed degree d has a natural
structure of quasi-projective variety.  When n=2, we will review the
known properties of Bir(P^2)_d and we will then focus on open
problems.

Adrien Dubouloz

I will discuss a collection of elementary looking questions in affine
algebraic geometry in which projective cubic surfaces, smooth or not,
and their hyperplane sections play a central role: a flexibility
question concerning affine cylinders over rigid affine varieties in
relation to a variant of Zariski Cancellation Problem, the existence
of log-uniruled but not log-ruled smooth affine threefolds, the
existence of "exotic" compactifications of the affine 3-space into
Mori fiber spaces, and the existence of "plane like" affine surfaces
with infinite dimensional monoids of non proper étale endomorphisms.

Concettina Galati

During the talk we will explore open problems about explicit
compactifications of moduli spaces of surfaces and related
subjects. We will provide a brief summary of known results. In
particular, we will start by describing the partial compactification
$\overline{\mathcal K_p}$ of the moduli space $\mathcal K_p$ of
polarized $K3$ surfaces of genus $p$ due to Friedmann.

Andreas Leopold Knutsen

I will give an overview of known results concerning syzygies and
linear systems on smooth curves on surfaces, including some open
conjectures.

Margherita Lelli-Chiesa

I will report on some results obtained jointly with A. L. Knutsen and
G. Mongardi. They concern Severi varieties and Brill-Noether theory of
curves on abelian surfaces. Both analogies and differences with the
case of K3 sections will be highlighted. Some open problems will then
be discussed.

Juan Carlos Naranjo

Xiao's conjecture deals with the relation between the natural
invariants present on a fibred surface $f:S\to B$: the irregularity
$q$ of $S$, the genus  $b$ of the base curve $B$ and of the genus $g$
of the general fibre $F$ of $f$. It is well-known by a seminal work of
Beauville that $0\le q-b\le g$ and that $q-b=g$ if and only if $S$ is
birational to $B\times F$ being $f$ the first projection. Xiao
conjectured that in the non-product case the upper bound of $q-b$
should be around one half of $g$.
In this talk we will review some evidences of the conjecture given by
Xiao itself, Serrano, Cai, Pirola and others and we will give a new
result found in collaboration with Miguel Angel Barja and Victor
Gonzalez-Alonso: assume that $f$ is not isotrivial, then the
inequality $q-b \le g-c$ holds, where $c$ is the Clifford index of the
general fibre. This gives in particular a proof of the Xiao's
conjecture for fibrations whose general fibres have maximal Clifford
index.

Roberto Pignatelli

A pluricanonical system is the complete linear system of a multiple mK
of a canonical divisor K with m>1.  A twisted pluricanonical system is
the complete linear system of a divisor numerically equivalent to a
multiple mK of a canonical divisor K with m>1.

The pluricanonical systems of minimal surfaces of general type have
been deeply studied, mainly for their applications to the fine
classification. They may have base points only for m=2,3 and in very
few exceptional cases, although a complete classification of these
exceptional cases is still missing.

On the contrary, the slightly bigger class of the twisted
pluricanonical systems has not been studied much. We notice that
Reider's Theorem on base points of pluricanonical systems extends to
the case of twisted pluricanonical systems. We will discuss what is
known on base points of twisted pluricanonical systems, and we will
discuss some results obtained with F. Favale: an example of a surface
with a twisted bicanonical system with base points and canonical
degree three, and some steps towards a construction of a surface whose
bicanonical system has base points and canonical degree three.

Gianpietro Pirola

Let C be a smooth curve which is complete intersection of a quadric
and a degree k>2 surface in the 3 dimensional projective space.  Let
C(2) be its second symmetric power. of C.  We study the finite
generation of the extended canonical ring
R(?,K):=?(a,b)H^0(C(2),a?+bK), where ? is the image of the diagonal
and K is the canonical divisor.  We show that R(?,K) is finitely
generated if and only if the difference of the two linear series
defined on C by the rulings of the quadric is a torsion non-trivial
line bundle.  Then we show that this holds on an analytically dense
locus of the moduli space of such curves.  The results have been
obtained in a joint work with Antonio La Face and Michela Artebani.

Francesco Polizzi

We explain some new constructions of algebraic surfaces via finite
coverings. In particular, we present examples of surfaces with p_g=q=2
arising as double, triple and quadruple covers of abelian surfaces and
examples of triple planes with p_g=q=0.

These results were obtained in collaboration with M. Penegini,
R. Pignatelli, D. Faenzi and J. Valles.

Sönke Rollenske

Stable surfaces are the 2-dimensional analog of stable curves and
there is a moduli space of stable surfaces that compactifies the
Giesecker moduli space of canonical models of surfaces of general
type. In the talk I will survey joint work with Wenfei Liu, and Marco
Franciosi and Rita Pardini around the theme "Which results/techniques
on surfaces of general type can be extended to stable surfaces?" and
point out some problems for further research.

Lidia Stoppino

Let us consider a surface fibred over a curve. In this talk I will
give an overview of old and new results on the Hodge bundle E
associated to it (the pushforward of the relative canonical sehaf on
the base curve).
Firstly I will recall the positivity results of Fujita and Kollar,
then the first and second Fujita decompositions of E. I will give an
idea of the counterexamples to Fujita's conjecture on the
semiampleness of E given recently by Catanese and
Dettweiler. Eventually, I will describe some applications and open
problems.