Moduli of elliptic $K3$ surfaces: monodromy and Shimada root lattice strata

TL;DR

This paper classifies positive-dimensional ambi-typical strata in the moduli space of elliptic K3 surfaces, linking Shimada root lattice classifications with monodromy stratifications, and explores their relation to lattice-polarised K3 moduli.

Contribution

It provides a complete classification of all positive-dimensional ambi-typical strata, connecting two stratifications and their relation to lattice-polarised K3 surfaces.

Findings

Classification of all positive-dimensional ambi-typical strata.

Connection established between Shimada root strata and monodromy strata.

Computational results on 1-dimensional ambi-typical strata included.

Abstract

In this paper we investigate two stratifications of the moduli space of elliptically fibred K3 surfaces. The first comes from Shimada's classification of connected components of elliptically fibred K3 surfaces and is closely related to the root lattice of the fibration. The second is the monodromy stratification defined by Bogomolov, Petrov and Tschinkel. The main result of the paper is a classification of all positive-dimensional ambi-typical strata, that is strata which are both Shimada root strata and monodromy strata. We further discuss the connection with moduli spaces of lattice-polarised K3 surfaces. The paper contains an appendix by M. Kirschmer providing computational results on the 1-dimensional ambi-typical strata.