On the isotriviality of families of elliptic surfaces

Abstract

We study non-isotrivial projective families of elliptic surfaces of Kodaira dimension one, over complex projective curves. If the base is an elliptic curve, we show that the family must have a singular fibre, and that over the projective line it must have at least three singular fibres. Similar results, for families of surfaces of general type, have been obtained by Migliorini and Kov\'acs, and they are well-known for projective families of surfaces of Kodaira dimension zero. Revised version: We corrected some minor errors and ambiguities, and we completed the list of references.