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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Resolutions of ideals of fat points with support in a hyperplane


Authors: Giuliana Fatabbi, Brian Harbourne and Anna Lorenzini
Journal: Proc. Amer. Math. Soc. 134 (2006), 3475-3483
MSC (2000): Primary 13D02, 13D40; Secondary 14M05, 14M20
Published electronically: June 12, 2006
MathSciNet review: 2240658
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Abstract: Let $ Z'$ be a fat point subscheme of $ \mathbb{P}^{d}$, and let $ x_0$ be a linear form such that some power of $ x_0$ vanishes on $ Z'$ (i.e., the support of $ Z'$ lies in the hyperplane $ H$ defined by $ x_0=0$, regarded as $ \mathbb{P}^{d-1}$). Let $ Z(i)=H\cap Z'(i)$, where $ Z'(i)$ is the subscheme of $ \mathbb{P}^{d}$ residual to $ x_0^i$; note that $ Z(i)$ is a fat points subscheme of $ \mathbb{P}^{d-1}=H$. In this paper we give a graded free resolution of the ideal $ I(Z')$ over $ R'=K[{\mathbb{P}}^{d}]$, in terms of the graded minimal free resolutions of the ideals $ I(Z(i))\subset R=K[{\mathbb{P}}^{d-1}]$. We also give a criterion for when the resolution is minimal, and we show that this criterion always holds if $ \hbox{char}(K)=0$.


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Additional Information

Giuliana Fatabbi
Affiliation: Dipartimento di Matematica e Informatica, Università di Perugia, via Vanvitelli 1, 06123 Perugia, Italy
Email: fatabbi@dipmat.unipg.it

Brian Harbourne
Affiliation: Department of Mathematics, University of Nebraska, Lincoln, Nebraska 68588-0130
Email: bharbour@math.unl.edu

Anna Lorenzini
Affiliation: Dipartimento di Matematica e Informatica, Università di Perugia, via Vanvitelli 1, 06123 Perugia, Italy
Email: annalor@dipmat.unipg.it

DOI: http://dx.doi.org/10.1090/S0002-9939-06-08514-5
PII: S 0002-9939(06)08514-5
Received by editor(s): January 21, 2005
Received by editor(s) in revised form: July 7, 2005
Published electronically: June 12, 2006
Additional Notes: The authors thank MURST, whose national project Algebra Commutativa e Computazionale, and the University of Perugia, whose project Metodi algebrici e analitici nello studio delle varietà supported visits to Perugia by the second author, who also thanks the NSA and NSF for supporting his research. The authors also thank the referee for helpful suggestions.
Communicated by: Michael Stillman
Article copyright: © Copyright 2006 American Mathematical Society