Betti numbers for fat point ideals in the plane: A geometric approach
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- by Alessandro Gimigliano, Brian Harbourne and Monica Idà PDF
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Abstract:
We consider the open problem of determining the graded Betti numbers for fat point subschemes $Z$ supported at general points of $\mathbf {P}^2$. We relate this problem to the open geometric problem of determining the splitting type of the pullback of $\Omega _{\mathbf {P}^2}$ to the normalization of certain rational plane curves. We give a conjecture for the graded Betti numbers which would determine them in all degrees but one for every fat point subscheme supported at general points of $\mathbf {P}^2$. We also prove our Betti number conjecture in a broad range of cases. An appendix discusses many more cases in which our conjecture has been verified computationally and provides a new and more efficient computational approach for computing graded Betti numbers in certain degrees. It also demonstrates how to derive explicit conjectural values for the Betti numbers and how to compute splitting types.References
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Additional Information
- Alessandro Gimigliano
- Affiliation: Dipartimento di Matematica e CIRAM, Università di Bologna, 40126 Bologna, Italy
- Email: gimiglia@dm.unibo.it
- Brian Harbourne
- Affiliation: Department of Mathematics, University of Nebraska, Lincoln, Nebraska 68588-0130
- MR Author ID: 217048
- Email: bharbour@math.unl.edu
- Monica Idà
- Affiliation: Dipartimento di Matematica, Università di Bologna, 40126 Bologna, Italy
- Email: ida@dm.unibo.it
- Received by editor(s): December 29, 2006
- Received by editor(s) in revised form: June 15, 2007
- Published electronically: September 9, 2008
- © Copyright 2008
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Trans. Amer. Math. Soc. 361 (2009), 1103-1127
- MSC (2000): Primary 14C20, 13P10; Secondary 14J26, 14J60
- DOI: https://doi.org/10.1090/S0002-9947-08-04599-6
- MathSciNet review: 2452836