Standard Bigraded Hilbert Functions

ABSTRACT

We show Macaulay-type bounds and persistence results for bigraded Hilbert function along rays. Global behavior of the bigraded Hilbert function and very supportive sets for bigraded Hilbert polynomials are also discussed.

We indicate by several results that direct generalizations of the standard graded theory are not true. In particular, we give a counterexample to an immediate generalization of Gotzmann's Regularity Theorem and Persistence Theorem. It is also made clear that maximal growth of a bigraded Hilbert function from bidegree (u, v) to bidegree (u + 1, v) usually is obtained at the expense of the growth from bidegree (u, v) to bidegree (u, v + 1).

ACKNOWLEDGMENTS

Work on this paper started while the author visited UC Berkeley. I thank Gregory Smith for several very useful discussions. In addition, Smith provided interesting examples from computations in Macaulay 2. I also benefited from general conversations about multigradings with Mark Haiman, Diane Maclagan and Bernd Sturmfels. The author was partially supported by Stiftelsen för Internationalisering av Höogre ut-bildning och Forskning (STINT) and Wenner-Gren stiftelserna.

Notes

Communicated by W. Bruns.