The Ideal Generation Problem for Fat Points

1997/03/27 by Brian Harbourne, Harbourne, Brian
Computer Science · Mathematics · #Advanced Differential Equations and Dynamical Systems #Algebraic Geometry (math.AG) #Commutative Algebra and Its Applications #FOS: Mathematics #Polynomial and algebraic computation #alg-geom #math.AG

paper · pdf · doi:10.48550/arxiv.alg-geom/9703035

PlainTeX, 17 pages

arxiv created 1997/03/27 · openalex publication_date 1997/03/27 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

This paper is concerned with determining the number of generators in each degree for minimal sets of homogeneous generators for saturated ideals defining fat point subschemes Z=m₁p₁+ ... +m_rp_r for general sets of points p_i of P². For thin points (i.e., m_i=1 for all i), a solution is known, in terms of a maximal rank property. Although this property in general fails for fat points, we show it holds in an appropriate asymptotic sense. In the uniform (i.e., m₁= ... =m_r) case, we determine all failures of this maximal rank property for r≤ 9, and we develop evidence for the conjecture that no other failures occur for r > 9.

Related

• Generators for Symbolic Powers of Ideals Defining General Points of P²
• Resolutions of fat point ideals involving 8 general points of P2
• On the minimal free resolution for fat point schemes of multiplicity at most 3 in P²
• Problems and progress: survey on fat points in P2
• On the containment problem for fat points ideals