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In this paper we investigate two stratifications of the moduli space ofelliptically fibred K3 surfaces. The first comes from Shimada's classificationof connected components of elliptically fibred K3 surfaces and is closelyrelated to the root lattice of the fibration. The second is the monodromystratification defined by Bogomolov, Petrov and Tschinkel. The main result ofthe paper is a classification of all positive-dimensional ambi-typical strata,that is strata which are both Shimada root strata and monodromy strata. Wefurther discuss the connection with moduli spaces of lattice-polarised K3surfaces. The paper contains an appendix by M. Kirschmer providingcomputational results on the 1-dimensional ambi-typical strata.

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