Mixed Łojasiewicz exponents and log canonical thresholds of ideals

Abstract

Abstract. We study the Lojasiewicz exponent and the log canonical threshold of ideals of O n when restricted to generic subspaces of C n of different dimensions. We obtain effective formulas of the resulting numbers for ideals with monomial integral closure. An inequality relating these numbers is also proven.

Multiplicity and Łojasiewicz Exponent of Generic Linear Sections of Monomial Ideals

“…The following result is motivated by [12, Théorème 1.1] and, in turn, the case J = m is the motivation of the problem considered in this article. This result can be seen as a particular case of [6,Theorem 4.7] (see [6,Corollary 4.8]). We also refer to [12,Remarque 4.3(3)] for the deduction of inequality (2.1) using different techniques in a slightly different context.…”

“…The next result gives a description of the sequence L * 0 (I) in terms of Γ + (I) when I is a monomial ideal of finite colength of O n . Theorem 2.3 [6]. Let I be a monomial ideal of O n of finite colength.…”

“…We also refer to [12,Remarque 4.3(3)] for the deduction of inequality (2.1) using different techniques in a slightly different context. Proposition 2.4 [6]. Let (R, m) be a quasi-unmixed Noetherian local ring and let I and J be ideals of R of finite colength.…”

“…We remark that this inequality was generalised in [6,Theorem 4.7]. There now arises the problem of characterising when equality holds in (1.3) and understanding the structure of the ideals satisfying that equality.…”

Invariants for bi-Lipschitz equivalence of ideals

“…. , I n are ideals of O n (see [7,11,12] for details). In particular, if I is an ideal of finite colength, we can speak about the sequence…”