On some invariants of hypersurface singularities

TL;DR

This paper introduces and studies a new invariant of hypersurface singularities, the log canonical threshold of the product of the maximal ideal and the Jacobian ideal, exploring its properties and relation to minimal exponents.

Contribution

It defines a new invariant for hypersurface singularities, analyzes its properties, and investigates its relationship with minimal exponents, providing examples and partial answers to related questions.

Findings

The invariant satisfies most formal properties of the log canonical threshold.

Examples illustrate the behavior of the invariant.

Partial positive results on bounds for minimal exponents.

Abstract

Given a hypersurface defined by $f$ in a smooth complex algebraic variety $X$, and a point $P$ on this hypersurface, we consider the invariant $\beta_P(f)$ given by the log canonical threshold at $P$ of ${\mathfrak m}_P\cdot J_f$, where ${\mathfrak m}_P$ is the ideal defining $P$ and $J_f$ is the Jacobian ideal of $f$. We show that this invariant satisfies most of the formal properties of the log canonical threshold of $f$ and give some examples. Dano Kim asked whether this invariant always gives an upper bound for the minimal exponent of $f$ at $P$. Motivated by this, we raise another question about minimal exponents, give a positive answer to a weaker version, and discuss some examples.