Hilbert functions of fat point subschemes of the plane: the two-fold way

Details Hilbert functions of fat point subschemes of the plane: the two-fold way Hilbert functions of fat point subschemes of the plane: the two-fold way

2011-01-26

arxiv.org/abs/1101.5140v1

Mathematics

Algebraic Geometry

21 pages

Scientific paper

Two approaches for determining Hilbert functions of fat point subschemes of $\mathbb P^2$ are demonstrated. A complete determination of the Hilbert functions which occur for 9 double points is given using the first approach, extending results obtained in a previous paper using the second approach. In addition the second approach is used to obtain a complete determination of the Hilbert functions for $n\geq 9$ $m$-multiple points for every $m$ if the points are smooth points of an irreducible plane cubic curve. Additional results are obtained using the first approach for $n\geq 9$ double points when the points lie on an irreducible cubic (but now are not assumed to be smooth points of the cubic).

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