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

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

2006-07-26

arxiv.org/abs/math/0607677v1

Mathematics

Algebraic Geometry

This is a revised version of a preprint of 2004

Scientific paper

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 $\bold{m}=(m_i)_{i=0}^r$ and arbitrary degree $d$. Moreover, we provide an algorithm for computing a bound of the regularity of an arbitrary system $\bold{m}$ and we give its exact value when $\bold{m}$ 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''.

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