A note on exceptional unimodal singularities and K3 surfaces

TL;DR

This paper explores the connection between certain singularities and K3 surfaces, revealing a basis for the Grothendieck group that links algebraic and geometric structures in complex geometry.

Contribution

It establishes a relationship between the derived categories of exceptional unimodal singularities and K3 surfaces, providing a new basis for the numerical Grothendieck group.

Findings

Identifies a basis of the numerical Grothendieck group for these singularities.

Links graded stable derived categories to K3 surface categories.

Provides insights into the algebraic structure of singularities and their geometric counterparts.

Abstract

This is a short note on the relation between the graded stable derived categories of 14 exceptional unimodal singularities and the derived category of K3 surfaces obtained as compactifications of the Milnor fibers. As a corollary, we obtain a basis of the numerical Grothendieck group similar to the one given by Ebeling and Ploog (arXiv:0809.2738).