4 papers
On the Nagata conjecture
Joaquim Roé
T. Szemberg proposed in 2001 a generalization to arbitrary varieties of M. Nagata's 1959 open conjecture, which claims that the Seshadri constant of r>9 very general points of the…
Varieties of clusters and Enriques diagrams
Joaquim Roe
We study the geometry of the varieties of clusters X_r introduced by Kleiman in the 70's, showing that for every Enriques diagram D of r vertices the subset Cl(D) of the clusters w…
Tacnodes and cusps
Joaquim Roe
Let Z be a zero-dimensional subscheme of the projective plane consisting of the union of r>5 double points, I its defining ideal sheaf. It is known that I has the expected cohomolo…
On the existence of plane curves with prescribed multiple points
Joaquim Roe
We address the problem of determining the degree a plane curve must have in order to pass with multiplicity m through r points in general position. A conjecture of Nagata states th…