Curves having one place at infinity and linear systems on rational surfaces

Abstract

Denoting by ${\mathcal{L}}_d(m_0,m_1,...,m_r)$ the linear system of plane curves passing through $r+1$ generic points $p_0,p_1,...,p_r$ of the projective plane with multiplicity $m_i$ (or larger) at each $p_i$, we prove the Harbourne-Hirschowitz Conjecture for linear systems ${\mathcal{L}}_d(m_0,m_1,...,m_r)$ determined by a wide family of systems of multiplicities $\mathbf{m}=(m_i{)}_{i=0}^r$ and arbitrary degree $d$. Moreover, we provide an algorithm for computing a bound of the regularity of an arbitrary system $\mathbf{m}$ and we give its exact value when $\mathbf{m}$ is in the above family. To do that, we prove an $H^1$-vanishing theorem for line bundles on surfaces associated with some pencils ``at infinity''.

This is a revised version of a preprint of 2004

References in corpus (1)

• Reduction method for linear systems of plane curves with base fat points