Kouchnirenko type formulas for local invariants of plane analytic curves

TL;DR

This paper derives formulas relating local invariants of plane analytic curves, such as double points, through toric modifications, providing a new way to compute these invariants using geometric transformations.

Contribution

It introduces Kouchnirenko type formulas that connect local double point invariants of plane curves with their images under toric modifications.

Findings

Formulas linking double points before and after toric modifications

Explicit computation of local invariants using intersection data

Enhanced understanding of plane curve singularities

Abstract

Let f(x,y)=0 be an equation of plane analytic curve defined in the neighborhood of the origin and let $\pi:M\to(\Cn^2,0)$ be a local toric modification. We give a formula which connects a number of double points \delta_0(f)$ with a sum $\sum_p \delta_p(\tilde f)$ which runs over all intersection points of the proper preimage of f=0 with the exceptional divisor.