On some invariants of hypersurface singularities

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 $β_P(f)$ 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.

Theorems & Definitions (31)

• Definition 2.1
• Remark 2.2
• Remark 2.3
• Example 2.4
• Remark 2.5
• Example 2.6
• Example 2.7
• Example 2.8
• Example 2.9
• Proposition 2.10