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Uniform bounds on multigraded regularity

Author(s): Diane Maclagan; Gregory G. Smith
Journal: J. Algebraic Geom. 14 (2005), 137-164.
Posted: July 20, 2004
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Abstract | References | Additional information

Abstract: We give an effective uniform bound on the multigraded regularity of a subscheme of a smooth projective toric variety $X$ with a given multigraded Hilbert polynomial. To establish this bound, we introduce a new combinatorial tool, called a Stanley filtration, for studying monomial ideals in the homogeneous coordinate ring of $X$. As a special case, we obtain a new proof of Gotzmann's regularity theorem. We also discuss applications of this bound to the construction of multigraded Hilbert schemes.

References:

[ACN]
A. Aramova, K. Crona, and E. De Negri, Bigeneric initial ideals, diagonal subalgebras and bigraded Hilbert functions, J. Pure Appl. Algebra 150 (2000) 215-235. MR 1769356 (2001f:13046)

[AK]
A. B. Altman and S. L. Kleiman, Compactifying the Picard scheme, Adv. in Math. 35 (1980) 50-112. MR 0555258 (81f:14025a)

[Ape]
J. Apel, On a conjecture of R. P. Stanley; part II - Quotients modulo monomial ideals, J. Algebraic Combin. 17 (2003) 57-74. MR 1958009 (2004e:13027)

[Bat]
V. V. Batyrev, On the classification of toric Fano $4$-folds, J. Math. Sci. (New York) 94 (1999) 1021-1050. MR 1703904 (2000e:14088)

[BH]
W. Bruns and J. Herzog, Cohen-Macaulay rings, Cambridge Studies in Advanced Mathematics 39, Cambridge University Press, Cambridge, 1993. MR 1251956 (95h:13020)

[BS]
M. P. Brodmann and R. Y. Sharp, Local cohomology: an algebraic introduction with geometric applications, Cambridge Studies in Advanced Mathematics 60, Cambridge University Press, Cambridge, 1998. MR 1613627 (99h:13020)

[Co1]
D. A. Cox, The homogeneous coordinate ring of a toric variety, J. Algebraic Geom. 4 (1995) 17-50. MR 1299003 (95i:14046)

[Co2]
-, Recent developments in toric geometry, Algebraic geometry--Santa Cruz 1995, 389-436, Proc. Sympos. Pure Math. 62, Amer. Math. Soc., Providence, RI, 1997. MR 1492541 (99d:14054)

[EL]
L. Ein and R. Lazarsfeld, Syzygies and Koszul cohomology of smooth projective varieties of arbitrary dimension, Invent. Math. 111 (1993) 51-67. MR 1193597 (93m:13006)

[Ful]
W. Fulton, Introduction to toric varieties, Annals of Mathematics Studies 131, Princeton University Press, Princeton, NJ, 1993. MR 1234037 (94g:14028)

[GLP]
L. Gruson, R. Lazarsfeld, and C. Peskine, On a theorem of Castelnuovo, and the equations defining space curves, Invent. Math. 72 (1983) 491-506. MR 0704401 (85g:14033)

[Got]
G. Gotzmann, Eine Bedingung für die Flachheit und das Hilbertpolynom eines graduierten Ringes, Math. Z. 158 (1978) 61-70. MR 0480478 (58:641)

[Gr1]
M. Green, Restrictions of linear series to hyperplanes, and some results of Macaulay and Gotzmann, Algebraic curves and projective geometry (Trento, 1988), 76-86, Lecture Notes in Math. 1389, Springer, Berlin, 1989. MR 1023391 (90k:13021)

[Gr2]
-, Generic initial ideals, Six lectures on commutative algebra (Bellaterra, 1996), 119-186, Progr. Math. 166, Birkhäuser, Basel, 1998. MR 1648665 (99m:13040)

[Gro]
A. Grothendieck, Fondements de la géométrie algébrique. [Extraits du Séminaire Bourbaki, 1957-1962.], Secrétariat mathématique, Paris, 1962. MR 0146040 (26:3566)

[M2]
D. R. Grayson and M. E. Stillman, Macaulay 2, a software system for research in algebraic geometry, Available at http://www.math.uiuc.edu/Macaulay2.

[HS]
M. Haiman and B. Sturmfels, Multigraded Hilbert schemes, J. Algebraic Geom. 13 (2004), 725-769.

[HTr]
N. D. Hoang and N. V. Trung, Hilbert polynomials of non-standard bigraded algebras, Math. Z. 245 (2003) 309-334. MR 2013503

[HTh]
S. Hosten and R. R. Thomas, Standard pairs and group relaxations in integer programming, J. Pure Appl. Algebra 139 (1999) 133-157. MR 1700541 (2000f:13052)

[Kle]
S. L. Kleiman, Toward a numerical theory of ampleness, Ann. of Math. (2) 84 (1966) 293-344. MR 0206009 (34:5834)

[Lat]
J. A. De Loera, R. Hemmecke, J. Tauzer, and R. Yoshida, LattE, a computer software for Lattice point Enumeration, Available at http://www.math.ucdavis.edu/latte.

[Mac]
D. Maclagan, Antichains of monomial ideals are finite, Proc. Amer. Math. Soc. 129 (2001) 1609-1615. MR 1814087 (2002f:13045)

[MS]
D. Maclagan and G. G. Smith, Multigraded Castelnuovo-Mumford regularity, J. Reine Angew. Math. 571 (2004), 179-212.

[Mum]
David Mumford, Lectures on curves on an algebraic surface, Princeton University Press, Princeton, N.J., 1966. MR 0209285 (35:187)

[RS]
A. Reeves and M. E. Stillman, Smoothness of the lexicographic point, J. Algebraic Geom. 6 (1997) 235-246. MR 1489114 (98m:14003)

[Rob]
P. C. Roberts, Multiplicities and Chern classes in local algebra, Cambridge Tracts in Mathematics 133, Cambridge University Press, Cambridge, 1998. MR 1686450 (2001a:13029)

[Sch]
A. Schrijver, Theory of linear and integer programming, John Wiley & Sons Ltd., Chichester, 1986. MR 0874114 (88m:90090)

[Sim]
R. S. Simon, Combinatorial properties of ``cleanness'', J. Algebra 167 (1994) 361-388. MR 1283293 (96c:13018)

[Sta]
R. P. Stanley, Linear Diophantine equations and local cohomology, Invent. Math. 68 (1982) 175-193. MR 0666158 (83m:10017)

[St1]
B. Sturmfels, On vector partition functions,J. Combin. Theory Ser. A 72 (1995) 302-309. MR 1357776 (97b:52014)

[St2]
-, Gröbner bases and convex polytopes, American Mathematical Society, Providence, RI, 1996. MR 1363949 (97b:13034)

[STV]
B. Sturmfels, N. V. Trung, and W. Vogel, Bounds on degrees of projective schemes, Math. Ann. 302 (1995) 417-432. MR 1339920 (96i:13029)

[SW]
B. Sturmfels and N. White, Computing combinatorial decompositions of rings, Combinatorica 11 (1991) 275-293. MR 1122013 (92f:13031)

[Vie]
E. Viehweg, Quasi-projective moduli for polarized manifolds, Springer-Verlag, Berlin, 1995. MR 1368632 (97j:14001)

[Zie]
G. M. Ziegler, Lectures on polytopes, Graduate Texts in Mathematics 152, Springer-Verlag, New York, 1995. MR 1311028 (96a:52011)

Additional Information:

Diane Maclagan
Affiliation: Department of Mathematics, Stanford University, Stanford, California 94305
Address at time of publication: Department of Mathematics, Rutgers University, Hill Center-Busch Campus, Piscataway, New Jersey 08854
Email: maclagan@math.stanford.edu

Gregory G. Smith
Affiliation: Department of Mathematics and Statistics, Queen's University, Kingston, Ontario, Canada K7L 3N6
Email: ggsmith@mast.queensu.ca

PII: S 1056-3911(04)00385-6
Received by editor(s): May 14, 2003
Received by editor(s) in revised form: December 31, 2003
Posted: July 20, 2004
Additional Notes: Both authors were partially supported by the Mathematical Sciences Research Institute in Berkeley, California

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