2 00 5 Maximal rank for planar singularities of multiplicity 2 Joaquim Roé

On the existence of plane curves with imposed multiple points

We prove that a plane curve of degree d with r points of multiplicity m must have d ≥ m (r − 1) r−1

Limit linear systems and applications

A system of plane curves defined by prescribing n points of multiplicity e in general position is regular if n ≥ 4e. The proof uses computation of limits of linear systems acquiring fixed divisors, an interesting problem in itself.

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ar X iv : m at h / 03 12 41 6 v 2 [ m at h . A G ] 1 6 O ct 2 00 4 LINKS AND ANALYTIC INVARIANTS OF SUPERISOLATED SINGULARITIES

Using superisolated singularities we present examples and counterexamples to some of the most important conjectures regarding invariants of normal surface singularities. More precisely, we show that the " Seiberg-Witten invariant conjecture " (of Nicolaescu and the third author), the " Universal abelian cover conjecture " (of Neumann and Wahl) and the " Geometric genus conjecture " fail (at lea...

Discrete Behavior of Seshadri Constants on Surfaces Brian

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 P 2 and new results abo...