Differential operators on homogeneous plane curve singularities

Abstract

Let k be a field with characteristic zero. Let $f\in k[x,y]$ be a reduced homogeneous polynomial with degree $d\ge 1$. We set $I:=〈f〉$ and $B:=k[x,y]/I$. Let $n\ge 2$ be a positive integer. In this article, we compute the module of logarithmic differential operators along I, with order n, from the datum of the polynomial f. For $n=2$, we show that our method produces a closed-form presentation of this module, which yields, as a by-product, an effective proof of the Nakai conjecture for the reduced (not necessarily irreducible) homogeneous plane curve singularities. As a theoretical cornerstone of this computation, we begin by proving that the order filtration on the (left) B-module of the differential operators, with order n, denoted by ${\mathrm{Diff}}_k^n\left(B\right)$, actually defines a grading on ${\mathrm{Diff}}_k^n\left(B\right)$.