Abstracts

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.

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.

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.

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.

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.

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.

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

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.

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.

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.

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.

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.

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.

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.