Moduli of elliptic $K3$ surfaces: Monodromy and Shimada root lattice strata \n (with an appendix by Markus Kirschmer)

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 the moduli of elliptically fibred K3 surfaces and is closely related to the root lattices 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 also discuss the relationship with moduli spaces of lattice-polarised K3 surfaces. The appendix by M. Kirschmer contains computational results about the 1-dimensional ambi-typical strata.