## Asymptotic behaviour of arithmetically Cohen-Macaulay blow-ups

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- by Huy Tài Hà and Ngô Viêt Trung PDF
- Trans. Amer. Math. Soc.
**357**(2005), 3655-3672 Request permission

## 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 $e> e_0$.## References

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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
- ORCID: 0000-0002-6002-3453
- Email: tai@math.missouri.edu, tai@math.tulane.edu
**Ngô Viêt Trung**- Affiliation: Institute of Mathematics, 18 Hoang Quoc Viet, Hanoi, Vietnam
- MR Author ID: 207806
- Email: nvtrung@math.ac.vn
- 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
- © Copyright 2005
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication. - 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
- MathSciNet review: 2146643