Workshop on Seshadri Constants

Programme (Schedule)

All the conferences will take place at the Faculty of Mathematics in the town center (Plaça Universitat).            

	27th Thursday	28th Friday	29th Saturday
9:00-9:30	Registration*
9:30-10:30	T. Bauer (B1)	G. Pareschi (B1)	J. Ross (B1)
10:30-11:00	Coffee Break	Coffee Break	Pause
11:00-11:30	A. Broustet (B1)	H. Tutaj-Gasinska (B1)	E. Schlesinger (B1)
11:45-12:45	L. Fuentes (B1)	M. Dumnicki (B1)	T. Szemberg (B1)
Lunch
15:00-16:00	S. Di Rocco (B1)	T. Eckl (B5)
16:00-17:00	A. Knutsen (B1)	C. Bocci (B5)
		21h Social dinner

* Historical Building, Plaça Universitat, Faculty of Mathematics, Room B1

Thomas Bauer: Seshadri constants and the generation of jets  In this talk I will report about joint work with T. Szemberg, in  which we explore the connection between Seshadri constants and  the generation of jets. It is well-known that one way to view Seshadri constants is to consider them as measuring the rate of growth of the number of jets that multiples of a line bundle generate. Here we ask, conversely, what we can say about the number of jets once the Seshadri constant is known. As an application of our results, we prove a characterization of projective space among all Fano varieties in terms of Seshadri constants.

Cristiano Bocci: Comparing powers and symbolic powers of ideals We develop tools, based on Seshadri constants and postulational invariants, to study the problem of containment of symbolic powers $I^{(m)}$ in powers $I^r$ for a homogeneous ideal $I\subset k[{\bf P}^N]$ in a polynomial ring $k[{\bf P}^N]$ in $N+1$ variables over an algebraically closed field $k$. This work was prompted by a question of Huneke: does $I^2$ contain $I^{(3)}$ whenever $I=I(S)$ is the ideal of a finite set $S$ of points in ${\bf P}^2$? As an application of our methods, we show that $I^2$ contains $I^{(3)}$ whenever $S$ is a finite generic set of points in ${\bf P}^2$, and we show that the containment theorems of Ein-Lazarsfeld-Smith and Hochster-Huneke are optimal for every fixed dimension and codimension.

Amaël Broustet: Minoration for ample line bundles in dimension 3 when the anticanonical bundle is nef. I will prove the conjecture of Ein Küchle and Lazarsfeld in the case of varieties of dimension 3 that have nef anticanonical bundle.

Sandra Di Rocco: Seshadri constants and toric geometry. The action of an algebraic torus on a non singular variety simplifies considerably the understanding of Seshadri constants. An overview of combinatorial criteria for generation of jets for toric varieties will be presented. In particular the interaction between local properties of a projective toric variety, like Seshadri constants at a fixed point, and the shape of an associated convex lattice polytope will be explained.

Marcin Dumnicki: Multipoint Seshadri constants and symplecting packings of P^2 I will focus on multipoint Seshadri constant for (X,L) = (P2,O_{P2}(1)) and points in general position. The constant can be bounded (from below) by showing emptyness of some families of linear systems of curves with multiplicities. I will show how the above problem can be transformed to a combinatorial problem on some subsets of the lattice Z2. The pictures, that arises during considerations, are closely related to explicitly given symplectic packings of P2. In easy cases this allows to show the direct relation between Seshadri constants and packing numbers.

Thomas Eckl: An asymptotic version of Dumnicki's algorithm for linear systems in $\mathbb{CP}^2$ Using Dumnicki's approach to showing non-specialty of linear systems consisting of plane curves with prescribed multiplicities in sufficiently general points on $\mathbb{CP}2$ we develop an asymptotic method to determine lower bounds for Seshadri constants of general points on $\mathbb{CP}2$. With this method we prove the lower bound 4/13 for 10 general points on $\mathbb{CP}2$.

Luis Fuentes: Finite maps and Seshadri constants We will explain some facts about finite maps and Seshadri constants. We will pay special attention to cyclic coverings. Finally, as an application, we exhibit some examples of explicit computation of Seshadri constants.

Andreas Knutsen: Moving curves and Seshadri constants on surfaces I will talk about bounds on the self-intersection of members of families of curves covering a projective surface and all having a point of multiplicity at least m. I will then talk about applications to Seshadri constants at very general points on surfaces. This is all joint work with W. Syzdek and T. Szemberg.

Giuseppe Pareschi: Generic vanishing, Fourier-Mukai transforms, and the Castelnuovo-de Franchis inequality Castelnuovo and de Franchis proved, more than a century ago, that a complex surface X such that p_g(X)<2q(X)-3 has an irrational pencil of genus at least two. I will describe an extension of this result to compact Kahler manifolds of arbitrary dimension. Suprisingly, the key ingredient of the proof is  Evans-Griffith's syzygy theorem, a classical result of commutative algebra, applied in a context relating generic vanishing of the cohomology and Fourier-Mukai transforms. This is a joint work with Mihnea Popa.

Julius Ross: Slope Stability of Varieties and Seshadri Constants I will introduce a notion of slope stability for varieties that is given in terms its subschemes and so analagous to stability of vector bundles. This is largely motivated by a body of work connecting stability (in the sense of geometric invariant theory) of varieites to the existence of special metrics.  I will emphasise the important role that the Seshadri constant of the subscheme plays for slope stability, illustrated by some examples.

Enrico Schlesinger: Gonality of a general ACM curve in projective space  Let C be an ACM (projectively normal) nonsingular curve in P3 not contained in a plane, and suppose C is general in its Hilbert scheme - this is irreducible once the postulation is fixed. Answering a question by Peskine  we show the gonality  of C is d-m,  where d is the degree of the curve, and m is the maximum order of a  multisecant to C. Furthermore m=4 except for rare cases, when the postulation of C forces every surface of minumum degree containing C to contain a line as well. We compute the value of m in these exceptional cases as well. Joint work with R. Hartshorne.

Tomasz Szemberg: Seshadri constants on surfaces of general type This talk is based on the joint paper with Thomas Bauer with the same title published in Manuscripta. We study Seshadri constants of the canonical bundle on minimal surfaces of general type. First, we prove that if the Seshadri constant $\eps(K_X,x)$ is between 0 and 1, then it is of the form $(m-1)/m$ for some integer $m\ge 2$. Secondly, we study values of $\eps(K_X,x)$ for a very general point $x$ and show that small values of the Seshadri constant are accounted for by the geometry of $X$. Since the paper appeared, there emerged a new lower bound on self-intersections of moving singular curves (presented at this workshop by A. Knutsen). Using this new bound I will show a slightly improved version of results from the publishedpaper.

Halszka Tutaj-Gasinska: Seshadri constants and packing numbers  In the talk I present  another view on the proof of Lazarsfeld's  theorem connecting Seshadri constants of $(X,L)$ with the maximal radius of a ball which can be embedded symplectically and holomorphically in $X$.