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

2006/07/26 by Francisco Monserrat, F. Monserrat, Monserrat, F.
Computer Science · Mathematics · #14C20 #Advanced Differential Equations and Dynamical Systems #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #FOS: Mathematics #Polynomial and algebraic computation #math.AG #msc:14C20

paper · pdf · doi:10.48550/arxiv.math/0607677

This is a revised version of a preprint of 2004

arxiv created 2006/07/26 · openalex publication_date 2006/07/26 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Denoting by \mathcal L_d(m₀,m₁,...,m_r) the linear system of plane curves passing through r+1 generic points p₀,p₁,...,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(m₀,m₁,...,m_r) determined by a wide family of systems of multiplicities \boldm=(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 \boldm and we give its exact value when \boldm is in the above family. To do that, we prove an H¹-vanishing theorem for line bundles on surfaces associated with some pencils ``at infinity''.

Citations

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

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