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Asymptotic behaviour of arithmetically Cohen-Macaulay blow-ups


Authors: Huy Tài Hà and Ngô Viêt Trung
Journal: Trans. Amer. Math. Soc. 357 (2005), 3655-3672
MSC (2000): Primary 14M05, 13H10, 13A30, 14E25
DOI: https://doi.org/10.1090/S0002-9947-05-03758-X
Published electronically: January 21, 2005
MathSciNet review: 2146643
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Abstract: This paper addresses problems on arithmetic Macaulayfications of projective schemes. We give a surprising complete answer to a question poised by Cutkosky and Herzog. Let $Y$ be the blow-up of a projective scheme $X = \operatorname{Proj} R$ along the ideal sheaf of $I \subset R$. It is known that there are embeddings $Y \cong \operatorname{Proj} k[(I^e)_c]$for $c \ge d(I)e + 1$, where $d(I)$ denotes the maximal generating degree of $I$, and that there exists a Cohen-Macaulay ring of the form $k[(I^e)_c]$(which gives an arithmetic Macaulayfication of $X$) if and only if $H^0(Y,\mathcal{O}_Y) = k$, $H^i(Y,\mathcal{O}_Y) = 0$ for $i = 1,..., \dim Y-1$, and $Y$ is equidimensional and Cohen-Macaulay. We show that under these conditions, there are well-determined invariants $\varepsilon$ and $e_0$ such that $k[(I^e)_c]$ is Cohen-Macaulay for all $c > d(I)e + \varepsilon$ and $e > e_0$, and that these bounds are the best possible. We also investigate the existence of a Cohen-Macaulay Rees algebra of the form $R[(I^e)_ct]$. If $R$ has negative $a^*$-invariant, we prove that such a Cohen-Macaulay Rees algebra exists if and only if $\pi_*\mathcal{O}_Y = \mathcal{O}_X$, $R^i\pi_*\mathcal{O}_Y = 0$ for $i > 0$, and $Y$ is equidimensional and Cohen-Macaulay. Moreover, these conditions imply the Cohen-Macaulayness of $R[(I^e)_ct]$ for all $c > d(I)e + \varepsilon$ and $e> e_0$.


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Additional Information

Huy Tài Hà
Affiliation: Department of Mathematics, University of Missouri, Columbia, Missouri 65201
Address at time of publication: Department of Mathematics, Tulane University, 6823 St. Charles Ave., New Orleans, Louisiana 70118
Email: tai@math.missouri.edu, tai@math.tulane.edu

Ngô Viêt Trung
Affiliation: Institute of Mathematics, 18 Hoang Quoc Viet, Hanoi, Vietnam
Email: nvtrung@math.ac.vn

DOI: https://doi.org/10.1090/S0002-9947-05-03758-X
Keywords: Blow-up, Rees algebra, Cohen-Macaulay, projective embedding
Received by editor(s): January 10, 2004
Published electronically: January 21, 2005
Additional Notes: The second author was partially supported by the National Basic Research Program of Vietnam
Article copyright: © Copyright 2005 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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