A note on wild fibers of elliptic surfaces

Abstract

is equal to d (resp. less than d). In characteristic 0, there does not exist a wild fiber by the cohomological flatness. In positive characteristic, however, the existence of wild fibers makes the situation complicated. The notion of wild fiber was introduced in Bombieri and Mumford [l], and Raynaud [S] examined the structure of wild fiber in detail. In this note, we consider elliptic surfaces obtained as quotients of the product of a curve and a supersingular elliptic curve by rational vector fields in positive characteristic. We calculate numerical invariants of wild fibers of such elliptic surfaces (cf. Theorem 3.5). Moreover, we give a characterization of such elliptic surfaces over the projective line P’ (cf. Theorem 4.2). To calculate numerical invariants, Raynaud’s results on wild fibers play an important role (cf. [5]). For the case of the product of a curve and an ordinary elliptic curve, we already treated this in [3]. 1. Preliminaries In this section, we recall some basic facts on elliptic surfaces and Raynaud’s theory on wild fibers. For details, see Bombieri and Mumford [l] and Raynaud [S].