Lattice duality for families of $K3$ surfaces associated to transpose duality

TL;DR

This paper demonstrates that certain transpose-dual pairs of singularities associated with K3 surfaces are actually lattice-dual, establishing a deeper geometric duality through lattice theory.

Contribution

It proves that specific transpose-dual pairs of singularities are also lattice-dual, clarifying their geometric relationship beyond previous polytope duality results.

Findings

Transpose-dual pairs are lattice-dual.

Lattice duality confirms polytope duality.

Enhances understanding of K3 surface dualities.

Abstract

The aim of this article is to show that the transpose-dual pairs in the sense of Ebeling-Ploog of singularities $(Z_{1,0},\, Z_{1,0})$, $(U_{1,0},\, U_{1,0})$, $(Q_{17},\, Z_{2,0})$, $(W_{1,0},\, W_{1,0})$ that are concluded to be polytope-dual by the author are actually lattice-dual.