Multiple fibers on elliptic surfaces in positive characteristic

Abstract

In this note, we investigate elliptic surfaces in char. $p>0$ with multiple fibers of a supersingular elliptic curve. In particular, we show that wild fibers on elliptic surfaces over $\mathbf{P}^{1}$ with $c_{2}=0$ and the general fiber being a supersingular elliptic curve can be reduced to tame fibers by taking a purely inseparable covering of degree $p$ successively and as an application of it, we show that if such elliptic surface has only one multiple fiber, then its multiplicity is equal to $p$.