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On Bounding the Number of Generators for Fat Point Ideals on the Projective Plane

✍ Scribed by Stephanie Fitchett

Publisher
Elsevier Science
Year
2001
Tongue
English
Weight
145 KB
Volume
236
Category
Article
ISSN
0021-8693

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✦ Synopsis

Let X be the surface obtained by blowing up general points p 1 p n of the projective plane over an algebraically closed ground field k, and let L be the pullback to X of a line on the plane. If C is a rational curve on X with C β€’ L = d, then for every t there is a natural map C C t βŠ— X X L β†’ C C t + d given by multiplication on simple tensors. The ranks of such maps are determined as a function of t, d, and m, where m is the largest multiplicity of C at any of the points p i . If I is the ideal defining the fat point subscheme Z = m 1 p 1 + β€’ β€’ β€’ + m n p n βŠ‚ P 2 , and Ξ± is the least degree in which I has generators, then the ranks of the maps C C t βŠ— X X L β†’ C C t + d can be used for bounding the number of generators of I in degrees t > Ξ± + 1. Β© 2001 Academic Press

, where P i is the saturated homogeneous ideal defining the point p i . The ideal I is called a fat point ideal, and consists of the homogeneous polynomials in three variables which, for each i, vanish to order at least m i at p i . 502

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