Linear systems with multiple base points in P2

2001/01/12 by Brian Harbourne, Harbourne, Brian, Joaquím Roé +2
Computer Science · Mathematics · #13P10 #14C99 #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #Commutative Algebra (math.AC) #Commutative Algebra and Its Applications #FOS: Mathematics #Polynomial and algebraic computation #math.AC #math.AG #msc:13P10 #msc:14C99

paper · pdf · doi:10.48550/arxiv.math/0101109

Final version, to appear in Advances in Geometry. Largely rewritten, now includes results determining Hilbert functions (resolutions, resp.) for infinitely many multiplicities for every square (even square, resp.) formerly included in math.AG/0104254, 18 pages PlainTeX, includes four figures

openalex publication_date 2001/01/12 · arxiv created 2003/03/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Given positive integers m₁, m₂, ..., m_n, and n general points p_i of \bf CP², bounds are given for the least degree t among plane curves passing through each point p_i with multiplicity at least m_i, and for the least t such that the n multiple points impose independent conditions on curves of degree t, often improving substantially what was previously known. As an application, the Hilbert function (resp., minimal free resolution) is determined for symbolic powers I^(m) for the ideal I defining n general points of \bf CP² for infinitely many m for each square n (resp., for infinitely many m for each even square n). Four graphs are included showing other values of m and n for which results are given.

Related

• New effective bounds on the dimension of a linear system in \mathbb P²
• Linear Systems of Plane Curves with Base Points of Equal Multiplicity
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• Quasi-homogeneous linear systems on P2 with base points of multiplicity 7, 8, 9, 10
• Multiple points in P² and degenerations to elliptic curves