Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



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
Published electronically: January 21, 2005
MathSciNet review: 2146643
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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 [Enhancements On Off] (What's this?)

  • 1. I. M. Aberbach, C. Huneke and N.V. Trung. Reduction numbers, Briancon-Skoda theorems, and depth of Rees rings. Compositio Math. 97 (1995), 403-434. MR 1353282 (96g:13002)
  • 2. Y. Aoyama. On the depth and the projective dimension of the canonical module. Japanese J. Math. 6 (1980), 61-66. MR 0615014 (82h:13007)
  • 3. A. Bertram, L. Ein and R. Lazarsfeld. Vanishing theorems, a theorem of Severi, and the equations defining projective varieties. J. Amer. Math. Soc. 4 (1991), no. 3, 587-602.MR 1092845 (92g:14014)
  • 4. M. Brodmann and R. Sharp. Local cohomology. Cambridge University Press, 1998. MR 1613627 (99h:13020)
  • 5. W. Bruns and J. Herzog. Cohen-Macaulay rings. Cambridge University Press, 1993. MR 1251956 (95h:13020)
  • 6. K. A. Chandler. Regularity of the powers of an ideal. Commun. Algebra. 25 (1997), 3773-3776. MR 1481564 (98i:13040)
  • 7. A. Conca, J. Herzog, N.V. Trung and G. Valla. Diagonal subalgebras of bi-graded algebras and embeddings of blow-ups of projective spaces. American Journal of Math. 119 (1997), 859-901. MR 1465072 (99d:13001)
  • 8. S.D. Cutkosky and H. Tài Hà. Arithmetic Macaulayfication of projective schemes. J. Pure Appl. Algebra. To appear.
  • 9. S.D. Cutkosky and J. Herzog. Cohen-Macaulay coordinate rings of blowup schemes. Comment. Math. Helv. 72 (1997), 605-617. MR 1600158 (99d:13028)
  • 10. S.D. Cutkosky, J. Herzog and N.V. Trung. Asymptotic behaviour of the Castelnuovo-Mumford regularity. Compositio Math. 118 (1999), 243-261. MR 1711319 (2000f:13037)
  • 11. A.V. Geramita and A. Gimigliano. Generators for the defining ideal of certain rational surfaces. Duke Mathematical Journal. 62 (1991), no. 1, 61-83. MR 1104323 (92f:14031)
  • 12. A.V. Geramita, A. Gimigliano and B. Harbourne. Projectively normal but superabundant embeddings of rational surfaces in projective space. J. Algebra. 169 (1994), no. 3, 791-804.MR 1302116 (96f:14044)
  • 13. A.V. Geramita, A. Gimigliano and Y. Pitteloud. Graded Betti numbers of some embedded rational $n$-folds. Math. Ann. 301 (1995), 363-380.MR 1314592 (96f:13022)
  • 14. S. Goto and Y. Shimoda, On the Rees algebras of Cohen-Macaulay rings. Lect. Notes in Pure and Appl. Math. 68, Marcel-Dekker, 1979, 201-231.MR 0655805 (84a:13021)
  • 15. H. Tài Hà. On the Rees algebra of certain codimension two perfect ideals. Manu. Math. 107 (2002), 479-501. MR 1906772 (2003d:13002)
  • 16. H. Tài Hà. Projective embeddings of projective schemes blown up at subschemes. Math. Z. 246 (2004), 111-124.MR 2031448
  • 17. R. Hartshorne. Algebraic Geometry. Graduate Text 52. Springer-Verlag, 1977. MR 0463157 (57:3116)
  • 18. S. Huckaba and C. Huneke. Rees algebras of ideals having small analytic deviation. Trans. Amer. Math. Soc. 339 (1993), no. 1, 373-402.MR 1123455 (93k:13008)
  • 19. E. Hyry. The diagonal subring and the Cohen-Macaulay property of a multigraded ring. Trans. Amer. Math. Soc. 351 (1999), no. 6, 2213-2232. MR 1467469 (99i:13005)
  • 20. E. Hyry and K. Smith. On a Non-Vanishing Conjecture of Kawamata and the Core of an Ideal. Amer. J. Math. 125 (2003), 1349-1410. MR 2018664
  • 21. B. Johnston and D. Katz. Castelnuovo regularity and graded rings associated to an ideal. Proc. Amer. Math. Soc. 123 (1995), 727-734. MR 1231300 (95d:13005)
  • 22. T. Kawasaki. On arithmetic Macaulayfication of local rings. Trans. Amer. Math. Soc. 354, 123-149. MR 1859029 (2002i:13001)
  • 23. V. Kodiyalam. Asymptotic behaviour of Castelnuovo-Mumford regularity. Proc. Amer. Math. Soc. 128 (2000), 407-411.MR 1621961 (2000c:13027)
  • 24. D. Mumford. Varieties defined by quadratic equations. C.I.M.E. III. (1969), 29-100.MR 0282975 (44:209)
  • 25. O. Lavila-Vidal. On the Cohen-Macaulay property of diagonal subalgebras of the Rees algebra. Manu. Math. 95 (1998), 47-58. MR 1492368 (99i:13006)
  • 26. O. Lavila-Vidal. On the existence of Cohen-Macaulay coordinate rings of blow-up schemes. Preprint.
  • 27. J. Lipman. Cohen-Macaulayness in graded algebras. Math. Res. Letters 1 (1994), 149-157. MR 1266753 (95d:13006)
  • 28. C. Polini and B. Ulrich. Neccessary and sufficient conditions for the Cohen-Macaulayness of blow-up algebras. Compositio Math. 119 (1999), no. 2, 185-207. MR 1723128 (2001f:13007)
  • 29. R. Sharp. Bass numbers in the graded case, $a$-invariant formula, and an analogue of Falting's annihilator theorem. J. Algebra. 222 (1999), no. 1, 246-270. MR 1728160 (2000j:13027)
  • 30. A. Simis, B. Ulrich, and W. Vasconcelos. Cohen-Macaulay Rees algebras and degrees of plolynomial equations. Math. Ann. 301 (1995), 421-444. MR 1324518 (96a:13005)
  • 31. I. Swanson. Powers of ideals. Primary decompositions, Artin-Rees lemma and regularity. Math. Ann. 307 (1997), 299-313. MR 1428875 (97j:13005)
  • 32. N.V. Trung. The largest non-vanishing degree of graded local cohomology modules. J. Algebra. 215 (1999), no. 2, 481-499. MR 1686202 (2000f:13038)
  • 33. N.V. Trung and S. Ikeda. When is the Rees algebra Cohen-Macaulay? Comm. Algebra. 17 (1989), no. 12, 2893-2922. MR 1030601 (91a:13009)
  • 34. N.V. Trung and H-J. Wang. On the asymptotic linearity of Castelnuovo-Mumford regularity. Preprint. arXiv:math.AC/0212161.

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2000): 14M05, 13H10, 13A30, 14E25

Retrieve articles in all journals with MSC (2000): 14M05, 13H10, 13A30, 14E25

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

Ngô Viêt Trung
Affiliation: Institute of Mathematics, 18 Hoang Quoc Viet, Hanoi, Vietnam

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.

American Mathematical Society