Gotzmann numbers of graded k-algebras and Betti numbers of the associated lex-segment ideal

摘要

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 Is, 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 less than or equal to i less than or equal to dim X). From this, we find a formula for C(X) = G(X) - G(X boolean OR 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 1, 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.