K3 surfaces from configurations of six lines in $\mathbb{P}^2$ and mirror symmetry I

TL;DR

This paper explores the mirror symmetry of K3 surfaces derived from six-line configurations in projective plane, analyzing hypergeometric systems and their compactifications to deepen understanding of their geometric and arithmetic properties.

Contribution

It constructs a compactification of the parameter space for K3 surfaces related to the hypergeometric system E(3,6) and establishes local isomorphisms with GKZ systems, proposing a framework for general hypergeometric systems.

Findings

Constructed a resolution of the Baily-Borel-Satake compactification.

Identified local isomorphisms between E(3,6) and GKZ systems.

Proposed conjectures for hypergeometric systems on Grassmannians.

Abstract

From the viewpoint of mirror symmetry, we revisit the hypergeometric system $E(3,6)$ for a family of K3 surfaces. We construct a good resolution of the Baily-Borel-Satake compactification of its parameter space, which admits special boundary points (LCSLs) given by normal crossing divisors. We find local isomorphisms between the $E(3,6)$ systems and the associated GKZ systems defined locally on the parameter space and cover the entire parameter space. Parallel structures are conjectured in general for hypergeometric system $E(n,m)$ on Grassmannians. Local solutions and mirror symmetry will be described in a companion paper \cite{HLTYpartII}, where we introduce a K3 analogue of the elliptic lambda function in terms of genus two theta functions.