Problems and progress: survey on fat points in P2

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

paper · pdf · doi:10.48550/arxiv.math/0101112

46 pages PlainTeX (includes 23 pages of Macaulay 2 scripts which can be copied and pasted directly into Macaulay)

arxiv created 2001/01/12 · arxiv updated 2009/11/30

Abstract

This paper surveys certain problems involving numerical characters for ideals I(Z) defining fat points subschemes Z=m₁p₁+...+m_np_n for general points p_i∈ \bf P². It also presents some new results, and includes a suite of MACAULAY 2 scripts for computing actual or conjectured values of (or bounds on) these characters. One such script, for example, is findres, which implements an algorithm due to Fitchett, Harbourne and Holay for computing the syzygy modules in a minimal free resolution of the ideal I(Z) for any such Z with n<=8; since findres does not rely on a Gröbner basis calculation, it is very fast and can handle very large multiplicities.

Related

• Resolutions of fat point ideals involving 8 general points of P2
• Generators for Symbolic Powers of Ideals Defining General Points of P²
• The Ideal Generation Problem for Fat Points
• An Algorithm for Fat Points on P2
• Resolutions of Ideals of Uniform Fat Point Subschemes of P²