GOTZMANN NUMBERS OF GRADED k-ALGEBRAS AND BETTI NUMBERS OF THE ASSOCIATED LEX-SEGMENT IDEAL

ABSTRACT

Our work starts from the persistence index G(R/I) of the k-algebra R/I introduced by Presser (2002). It is the degree where the maximal growth of the Hilbert function begins to persist in the sense of Gotzmann.

The main results of this paper consist of two parts. First, by introducing an invariant T(R/I), we show that G(R/I) is equal to the Gotzmann number of the Hilbert polynomial for a saturated ideal I. For a closed subscheme X in ℙ ^r and its defining ideal I _X , we set G(X) as G(R/I X). Then, we present the Hilbert polynomial P(X) in terms of G(X) and inductively defined new invariants G _i (X)(0 ≤ i ≤ dim X). From this, we find a formula for C(X) = G(X) − G(X ∪ ℋ), where ℋ is a general hyperplane. In defining G, _i (X), our theorem on maximal growth and the hyperplane restriction theorem are used (Theorem 3.3).

In the second result, given a homogeneous saturated ideal I, we find some conditions under which the Betti numbers of I ^lex do not vanish. We focus our attention on the inequality in Green's hyperplane restriction theorem or Gotzmann numbers G _i (X)'s. Theorem 5.7 is one of the cases where the Betti numbers of I ^lex do not vanish. This is a specialization of Theorem 4.5 in Presser (2002) to the situation of saturated ideals.

Key Words:

• Betti number
• Gotzmann number
• Hilbert function and polynomial
• Lex segment ideal
• Persistence index

Mathematics Subject Classification:

• 13D02
• 13D40
• 14M05

ACKNOWLEDGMENT

The authors were supported in part by the Research Institute of Mathematics and BK21. They also thank the referee for careful reading and for pointing our inaccuracies.

Notes

^#Communicated by W. Bruns.