A multiplicity bound for graded rings and a criterion for the Cohen-Macaulay property

Abstract

Abstract. Let R be a polynomial ring over a field. We prove an upper bound for the multiplicity of R/I when I is a homogeneous ideal of the form I = J + (F ), where J is a Cohen-Macaulay ideal and F / ∈ J. The bound is given in terms of two invariants of R/J and the degree of F . We show that ideals achieving this upper bound have high depth, and provide a purely numerical criterion for the Cohen-Macaulay property. Applications to quasi-Gorenstein rings and almost complete intersections are given.

A Cayley–Bacharach theorem for points in Pn

“…In Section 3, we exhibit upper bounds for the multiplicity of almost complete intersections of any height combining a repeated use of Francisco's theorem with several other techniques (see Theorems 3.3 and 3.6). While our estimates are not in general as sharp as the ones predicted by [13,Conjecture CB12], they significantly improve the best-known upper bounds, due to Engheta [14] and later extended by Huneke, Mantero, McCullough and Seceleanu [23], in all those circumstances in which the latter are not already sharp (see Remark 3.5).…”

“…which is the bound given in [14,23]. When D σ, the results in [14,23] just give that e(S/a) ( We now further improve the bound of Theorem 3.3 by using that, if a = f + (G) is an almost complete intersection, then the ideal g = f : a defines a Gorenstein ring, hence it has symmetric Hilbert function. .…”

A tight bound on the projective dimension of four quadrics

“…There are examples in [19] achieving this bound for all possible integers h and n. This paper now answers this question affirmatively if n ≤ 4, while our previous paper [15] gave an affirmative answer if h ≤ 2. Note that if the question has a positive answer, it would give a bound on the projective dimension of ideals generated by n quadrics that is quadratic in n -much smaller than the known bounds of Ananyan-Hochster.…”

“…(2) When I is minimally generated by N quadrics and ht(I) = 2, we previously showed [15] that pd(S/I) ≤ 2N − 2. (See Theorem 2.12.)…”

The projective dimension of three cubics is at most 5

“…The first part of the following corollary was first proved by Engheta [9] and later generalized in [12]. The second part of the statement was proved in [12]. It is thus natural to divide the proof of the Main Theorem by the multiplicity of R/I.…”