LAUSR

In this note, we generalize the Arakelov inequality in positive characteristic for non-isotrivial semistable families of curves of $$g\ge 2$$g≥2 which are liftable to $$W_2(k)$$W2(k) (resp. W(k)). As a consequence, we give an analogue of Beauville’s conjecture in positive characteristic: there are at least 5 singular fibers for non-isotrivial semistable families of curves of $$g\ge 2$$g≥2 over $$\mathbb {P}^1$$P1 which are liftable to W(k).