Frobenius splitting for some abelian fiber spaces
Let \(X\) be a smooth projective variety of characteristic \(p>0\). \(X\) is called Frobenius split if the map \(O_X\to F_*O_X\) splits as \(O_X\)-module, where \(F\) denotes the absolute Frobenius map. In this paper, the author study Frobenius split property of fibrations over curves. The main result, Theorem 1.3, is a classification of Frobenius split abelian fiber spaces over curves which has no wild fibers in terms of the degree of the top higher direct image \(R^{n-1}\pi_*O_X\), the multiplicities of multiple fibers, non-ordinary points, and the characteristic of the base field. The idea of the proof is a characterization of Frobenius split property of \(X\to C\) via the relative Frobenius morphism \(F_{X/C}\) (Proposition 1.5).
scientific article; zbMATH DE number 6676882