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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Resolutions of ideals of quasiuniform fat point subschemes of $\mathbf P^2$


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
Published electronically: October 4, 2002
MathSciNet review: 1932715
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Abstract: The notion of a quasiuniform fat point subscheme $Z\subset\mathbf P^2$is introduced and conjectures for the Hilbert function and minimal free resolution of the ideal $I$ defining $Z$ 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 $m$th symbolic power $I(m;n)$ of an ideal defining $n$ general points of $\mathbf P^2$ when both $m$ and $n$ are large (in particular, for infinitely many $m$ for each of infinitely many $n$, and for infinitely many $n$ for every $m>2$). 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 $k$. As an incidental result, a bound for the regularity of $I(m;n)$ 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: http://dx.doi.org/10.1090/S0002-9947-02-03124-0
PII: S 0002-9947(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