The nonexistence of negative weight derivations on positive dimensional isolated singularities: Generalized Wahl conjecture

Abstract

Let $R = \mathbb{C} [ x_1, x_2, \dotsc , x_n ] / (f)$ where $f$ is a weighted homogeneous polynomial defining an isolated singularity at the origin. Then $R$ and $\operatorname{Der} (R,R)$ are graded. It is well-known that $\operatorname{Der} (R,R)$ does not have a negatively graded component. Wahl conjectured that this is still true for $R = \mathbb{C} [ x_1, x_2, \dotsc, x_n] / (f_1, f_2, \dotsc, f_m)$ which defines an isolated, normal and complete intersection singularity and $f_1, f_2, \dotsc, f_m$ weighted homogeneous polynomials with the same weight type $(w_1, w_2, \dotsc, w_n)$. Here we give a positive answer to the Wahl Conjecture and its generalization (without the condition of complete intersection singularity) for $R$ when the degree of $f_i, 1 \leq i \leq m$ are bounded below by a constant $C$ depending only on the weights $w_1, w_2, \dotsc, w_n$. Moreover this bound C is improved when any two of $w_1, w_2, \dotsc, w_n$ are coprime. Since there are counter-examples for the Wahl Conjecture and its generalization when $f_i$ are low degree, our theorem is more or less optimal in the sense that only the lower bound constant can be improved.