Uniform bounds on multigraded regularity
Authors:
Diane Maclagan and Gregory G. Smith
Journal:
J. Algebraic Geom. 14 (2005), 137-164
DOI:
https://doi.org/10.1090/S1056-3911-04-00385-6
Published electronically:
July 20, 2004
MathSciNet review:
2092129
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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.
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[ACN]ACdN 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.
[AK]altmanKleiman A. B. Altman and S. L. Kleiman, Compactifying the Picard scheme, Adv. in Math. 35 (1980) 50–112.
[Ape]apel J. Apel, On a conjecture of R. P. Stanley; part II - Quotients modulo monomial ideals, J. Algebraic Combin. 17 (2003) 57–74.
[Bat]Batyrev V. V. Batyrev, On the classification of toric Fano $4$-folds, J. Math. Sci. (New York) 94 (1999) 1021–1050.
[BH]BH W. Bruns and J. Herzog, Cohen-Macaulay rings, Cambridge Studies in Advanced Mathematics 39, Cambridge University Press, Cambridge, 1993.
[BS]BrodmannSharp 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.
[Co1]Cox D. A. Cox, The homogeneous coordinate ring of a toric variety, J. Algebraic Geom. 4 (1995) 17–50.
[Co2]cox2 ---, Recent developments in toric geometry, Algebraic geometry—Santa Cruz 1995, 389–436, Proc. Sympos. Pure Math. 62, Amer. Math. Soc., Providence, RI, 1997.
[EL]EL L. Ein and R. Lazarsfeld, Syzygies and Koszul cohomology of smooth projective varieties of arbitrary dimension, Invent. Math. 111 (1993) 51–67.
[Ful]fulton W. Fulton, Introduction to toric varieties, Annals of Mathematics Studies 131, Princeton University Press, Princeton, NJ, 1993.
[GLP]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.
[Got]gotzmann G. Gotzmann, Eine Bedingung für die Flachheit und das Hilbertpolynom eines graduierten Ringes, Math. Z. 158 (1978) 61–70.
[Gr1]GreenGot 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.
[Gr2]Green ---, Generic initial ideals, Six lectures on commutative algebra (Bellaterra, 1996), 119–186, Progr. Math. 166, Birkhäuser, Basel, 1998.
[Gro]G A. Grothendieck, Fondements de la géométrie algébrique. [Extraits du Séminaire Bourbaki, 1957–1962.], Secrétariat mathématique, Paris, 1962.
[M2]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]HS M. Haiman and B. Sturmfels, Multigraded Hilbert schemes, J. Algebraic Geom. 13 (2004), 725–769.
[HTr]HoangTrung N. D. Hoang and N. V. Trung, Hilbert polynomials of non-standard bigraded algebras, Math. Z. 245 (2003) 309–334.
[HTh]HoTh1 S. Hoşten and R. R. Thomas, Standard pairs and group relaxations in integer programming, J. Pure Appl. Algebra 139 (1999) 133–157.
[Kle]Kleiman S. L. Kleiman, Toward a numerical theory of ampleness, Ann. of Math. (2) 84 (1966) 293–344.
[Lat]LattE 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]antichains D. Maclagan, Antichains of monomial ideals are finite, Proc. Amer. Math. Soc. 129 (2001) 1609–1615.
[MS]MaclaganSmith1 D. Maclagan and G. G. Smith, Multigraded Castelnuovo-Mumford regularity, J. Reine Angew. Math. 571 (2004), 179–212.
[Mum]Mumford David Mumford, Lectures on curves on an algebraic surface, Princeton University Press, Princeton, N.J., 1966.
[RS]ReevesStillman A. Reeves and M. E. Stillman, Smoothness of the lexicographic point, J. Algebraic Geom. 6 (1997) 235–246.
[Rob]Robertsbook P. C. Roberts, Multiplicities and Chern classes in local algebra, Cambridge Tracts in Mathematics 133, Cambridge University Press, Cambridge, 1998.
[Sch]schrijver A. Schrijver, Theory of linear and integer programming, John Wiley & Sons Ltd., Chichester, 1986.
[Sim]clean R. S. Simon, Combinatorial properties of “cleanness”, J. Algebra 167 (1994) 361–388.
[Sta]StanleySD R. P. Stanley, Linear Diophantine equations and local cohomology, Invent. Math. 68 (1982) 175–193.
[St1]SturmfelsVPF B. Sturmfels, On vector partition functions,J. Combin. Theory Ser. A 72 (1995) 302–309.
[St2]sturmfels ---, Gröbner bases and convex polytopes, American Mathematical Society, Providence, RI, 1996.
[STV]STV B. Sturmfels, N. V. Trung, and W. Vogel, Bounds on degrees of projective schemes, Math. Ann. 302 (1995) 417–432.
[SW]SturmfelsWhite B. Sturmfels and N. White, Computing combinatorial decompositions of rings, Combinatorica 11 (1991) 275–293.
[Vie]Viehweg E. Viehweg, Quasi-projective moduli for polarized manifolds, Springer-Verlag, Berlin, 1995.
[Zie]Ziegler G. M. Ziegler, Lectures on polytopes, Graduate Texts in Mathematics 152, Springer-Verlag, New York, 1995.
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
MR Author ID:
607134
Email:
maclagan@math.stanford.edu
Gregory G. Smith
Affiliation:
Department of Mathematics and Statistics, Queen’s University, Kingston, Ontario, Canada K7L 3N6
MR Author ID:
622959
Email:
ggsmith@mast.queensu.ca
Received by editor(s):
May 14, 2003
Received by editor(s) in revised form:
December 31, 2003
Published electronically:
July 20, 2004
Additional Notes:
Both authors were partially supported by the Mathematical Sciences Research Institute in Berkeley, California