The Bruce-Roberts Numbers of a Function on an ICIS

2022/03/21 by Bárbara K. Lima-Pereira, Lima-Pereira, Bárbara K., J. J. Nuño‐Ballesteros +5 · 1 citation
Computer Science · Mathematics · #32S25 #32S50 #58K40 #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #Commutative Algebra and Its Applications #FOS: Mathematics #Polynomial and algebraic computation

paper · pdf · doi:10.48550/arxiv.2203.11186

openalex publication_date 2022/03/21 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We give formulas for the Bruce-Roberts number μ_BR(f,X) and its relative version μ_BR^-(f,X) of a function f with respect to an ICIS (X,0). We show that μ_BR^-(f,X)=μ(f^-1(0)∩ X,0)+μ(X,0)-τ(X,0), where μ and τ are the Milnor and Tjurina numbers, respectively, of the ICIS. The formula for μ_BR(f,X) is more complicated and also involves μ(f) and some lengths in terms of the ideals I_X and Jf. We also consider the logarithmic characteristic variety, LC(X), and its relative version, LC(X)^-. We show that LC(X)^- is Cohen-Macaulay and that LC(X) is Cohen-Macaulay at any point not in X×\0\. We generalize previous results presented by the authors when (X,0) has codimension one and by Bruce and Roberts when it is weighted homogeneous of any codimension.

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• Exploring a local variant of the Buchsbaum--Eisenbud--Horrocks conjecture

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