The Ideal Generation Problem for Fat Points

Details The Ideal Generation Problem for Fat Points The Ideal Generation Problem for Fat Points

1997-03-27

arxiv.org/abs/alg-geom/9703035v1

Mathematics

Algebraic Geometry

PlainTeX, 17 pages

Scientific paper

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 $Z=m_1p_1+ ... +m_rp_r$ for general sets of points $p_i$ of $P^2$. 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., $m_1= ... =m_r$) case, we determine all failures of this maximal rank property for $r\le 9$, and we develop evidence for the conjecture that no other failures occur for $r > 9$.

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