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Betti numbers for fat point ideals in the plane: A geometric approach


Authors: Alessandro Gimigliano, Brian Harbourne and Monica Idà
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
Published electronically: September 9, 2008
MathSciNet review: 2452836
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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.


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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
Email: bharbour@math.unl.edu

Monica Idà
Affiliation: Dipartimento di Matematica, Università di Bologna, 40126 Bologna, Italy
Email: ida@dm.unibo.it

DOI: https://doi.org/10.1090/S0002-9947-08-04599-6
Keywords: Graded Betti numbers, fat points, splitting types.
Received by editor(s): December 29, 2006
Received by editor(s) in revised form: June 15, 2007
Published electronically: September 9, 2008
Article copyright: © Copyright 2008 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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