Joaquim Roé

On the existence of plane curves with imposed multiple points

Abstract We prove that a plane curve of degree d with r points of multiplicity m must have d≥m(r−1) ∏ i=2 r−1 1− i i 2 +r−1 , d> r−1 − π 8 m.

Discrete behavior of Seshadri constants on surfaces

Working over C, we show that, apart possibly from a unique limit point, the possible values of multi-point Seshadri constants for general points on smooth projective surfaces form a discrete set. In addition to its theoretical interest, this result is of practical value, which we demonstrate by giving significantly improved explicit lower bounds for Seshadri constants on P2 and new results about ample divisors on blow ups of P2 at general points.

Varieties of clusters and Enriques diagrams

We study the geometry of the varieties of clusters X_r introduced by Kleiman in the 70s, showing that for every Enriques diagram D of r vertices the subset Cl(D) of the clusters with Enriques diagram D is locally closed. We study also the relative positions of the subvarieties Cl(D), showing that they do not form a stratification and giving criteria for adjacencies between them.

Recent developments and open problems in linear series

In the week 3--9, October 2010, the Mathematisches Forschungsinstitut at Oberwolfach hosted a mini workshop Linear Series on Algebraic Varieties. These notes contain a variety of interesting problems which motivated the participants prior to the event, and examples, results and further problems which grew out of discussions during and shortly after the workshop. A lot of arguments presented here are scattered in the literature or constitute folklore. It was one of our aims to have a usable and easily accessible collection of examples and results.

Conditions imposed by tacnodes and cusps

The study of linear systems of algebraic plane curves with fixed imposed singularities is a classical subject which has recently experienced important progress. The Horace method introduced by A. Hirschowitz has been successfully exploited to prove many H1-vanishing theorems, even in higher dimension. Other specialization techniques, which include degenerations of the plane, are due to Z. Ran and C. Ciliberto and R. Miranda. G. M. Greuel, C. Lossen and E. Shustin use a local specialization procedure together with the Horace method to give the first asymptotically proper general existence criterion for singular curves of low degree. In this paper we develop a specialization method which allows us to compute the dimension of several linear systems as well as to substantially improve the bounds given by Greuel, Lossen and Shustin for curves with tacnodes and cusps.

The geometry of the real planar polynomial differential systems having their orbits embedded in conics

We classify and provide the global phase portraits in the Poincaré disc of all real planar polynomial differential systems having their orbits embedded in conics. This is achieved via the real affine classification of the pencils of conics, and each type corresponds to an equisingularity type of pencil. All such polynomial vector fields have degree less than or equal to 3. Also, when the degree is 3, infinity is filled with singular points.

Emptiness of homogeneous linear systems with ten general base points

In this paper, we discuss a proof of existence of log minimal models or Mori fibre spaces for klt pairs

Maximal rank for planar singularities of multiplicity 2

(X/Z,B)

An analytic invariant of ordinary curve singularities

with

Linear systems with multiple base points in ℙ2

B