Derivations of negative degree on quasihomogeneous isolated complete intersection singularities

TL;DR

This paper investigates derivations of negative degree on quasihomogeneous isolated complete intersection singularities, confirming Wahl's conjecture in certain cases and providing counterexamples in others.

Contribution

It proves Wahl's conjecture for specific classes of singularities and constructs counterexamples for higher embedding dimensions.

Findings

Wahl's conjecture holds for singularities of order at least 3 or embedding dimension at most 5.

Counterexamples exist for embedding dimension greater than 5.

The paper advances understanding of derivations on singularities based on their dimension and order.

Abstract

J. Wahl conjectured that every quasihomogeneous isolated normal singularity admits a positive grading for which there are no derivations of negative weighted degree. We confirm his conjecture for quasihomogeneous isolated complete intersection singularities of either order at least 3 or embedding dimension at most 5. For each embedding dimension larger than 5 (and each dimension larger than 3), we give a counter-example to Wahl's conjecture.