Generic Initial Ideals of Arithmetically Cohen–Macaulay Projective Subschemes

Abstract

For an arithmetically Cohen–Macaulay closed subscheme X in ℙ ⁿ with the saturated defining ideal I _X  ⊂ k[x ₀,…, x _n ], let gin(I _X ) be the generic initial ideal under the reverse lexicographic order. In this article, we find the minimal system of generators of gin (I _X ) in terms of characters f _i ′s, which will be defined in (3.1). Then we obtain the formula for deg(X) in terms of characters. For a curve C in ℙ ⁿ , we get the formula for the arithmetic genus in terms of characters of a general hyperplane section and the number of sporadic zeros. As an application, we give a new proof of the following upper bound on the regularity given in Ahn and Migliore (2007) and Nagel (1989): if X ⊂ ℙ ⁿ is an arithmetically Cohen–Macaulay closed subscheme, then

where s = {l | (I _X ) _l  ≠ 0}. Furthermore we give an equivalent condition for an arithmetically Cohen–Macaulay closed subscheme X ⊂ ℙ ⁿ to satisfy the equality from the view point of the generic initial ideal of I _X , this shows that the upper bound is sharp.

Keywords:

• Arithmetically Cohen–Macaulay scheme
• Arithmetic genus
• Borel-fixed monomial ideal
• Degree
• Generic initial idea
• Hilbert function
• Minimal system of generators
• Regularity

2000 Mathematics Subject Classification:

• 13D02
• 13D40
• 13P10
• 14M05

ACKNOWLEDGMENTS

We thank the referee for the careful reading of the manuscript and many helpful remarks. The first and second authors are partially supported by BK21, and the third author is supported by National Institute for Mathematical Sciences.

Notes

Communicated by W. Bruns.