Resolutions of ideals of fat points with support in a hyperplane
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- by Giuliana Fatabbi, Brian Harbourne and Anna Lorenzini PDF
- Proc. Amer. Math. Soc. 134 (2006), 3475-3483 Request permission
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 $\operatorname {char}(K)=0$.References
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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
- MR Author ID: 217048
- 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
- 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
- © Copyright 2006 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 134 (2006), 3475-3483
- MSC (2000): Primary 13D02, 13D40; Secondary 14M05, 14M20
- DOI: https://doi.org/10.1090/S0002-9939-06-08514-5
- MathSciNet review: 2240658