Resolutions of ideals of quasiuniform fat point subschemes of

Authors:
Brian Harbourne, Sandeep Holay and Stephanie Fitchett

Journal:
Trans. Amer. Math. Soc. **355** (2003), 593-608

MSC (2000):
Primary 13P10, 14C99; Secondary 13D02, 13H15

DOI:
https://doi.org/10.1090/S0002-9947-02-03124-0

Published electronically:
October 4, 2002

MathSciNet review:
1932715

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Abstract: The notion of a *quasiuniform* fat point subscheme is introduced and conjectures for the Hilbert function and minimal free resolution of the ideal defining are put forward. In a large range of cases, it is shown that the Hilbert function conjecture implies the resolution conjecture. In addition, the main result gives the first determination of the resolution of the th symbolic power of an ideal defining general points of when both and are large (in particular, for infinitely many for each of infinitely many , and for infinitely many for every ). Resolutions in other cases, such as ``fat points with tails'', are also given. Except where an explicit exception is made, all results hold for an arbitrary algebraically closed field . As an incidental result, a bound for the regularity of is given which is often a significant improvement on previously known bounds.

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Additional Information

**Brian Harbourne**

Affiliation:
Department of Mathematics and Statistics, University of Nebraska-Lincoln, Lincoln, Nebraska 68588-0323

Email:
bharbour@math.unl.edu

**Sandeep Holay**

Affiliation:
Department of Mathematics, Southeast Community College, Lincoln, Nebraska 68508

Email:
sholay@southeast.edu

**Stephanie Fitchett**

Affiliation:
Florida Atlantic University, Honors College, Jupiter, Florida 33458

Email:
sfitchet@fau.edu

DOI:
https://doi.org/10.1090/S0002-9947-02-03124-0

Keywords:
Ideal generation conjecture,
symbolic powers,
resolution,
fat points,
maximal rank.

Received by editor(s):
December 31, 2000

Received by editor(s) in revised form:
May 2, 2002

Published electronically:
October 4, 2002

Additional Notes:
The first author benefitted from a National Science Foundation grant.

Article copyright:
© Copyright 2002
American Mathematical Society