ON -DIVISIBLE GROUPS WITH SATURATED NEWTON POLYGONS

This paper concerns the classification of isogeny classes of$p$-divisible groups with saturated Newton polygons. Let$S$be a normal Noetherian scheme in positive characteristic$p$with a prime Weil divisor$D$. Let${\mathcal{X}}$be a$p$-divisible group over$S$whose geometric fibers over$S\setminus D$(resp. over$D$) have the same Newton polygon. Assume that the Newton polygon of${\mathcal{X}}_{D}$is saturated in that of${\mathcal{X}}_{S\setminus D}$. Our main result (Corollary 1.1) says that${\mathcal{X}}$is isogenous to a$p$-divisible group over$S$whose geometric fibers are all minimal. As an application, we give a geometric proof of the unpolarized analogue of Oort’s conjecture (Oort, J. Amer. Math. Soc.17(2) (2004), 267–296; 6.9).

Cartier and Weil divisors on varieties with quotient singularities

It is well-known that the notions of Weil and Cartier Q-divisors coincide for V-manifolds. The main goal of this paper is to give a direct constructive proof of this result providing a procedure to express explicitly a Weil divisor as a rational Cartier divisor. The theory is illustrated on weighted projective spaces and weighted blow-ups.

Classification of non-Gorenstein Q-Fano d-folds of Fano index greater than d − 2

A d-dimensional normal projective variety X is called a Q-Fano d-fold if it has only terminal singularities and if the anti-canonical Weil divisor – Kx is ample. The singularity index I = I(X) of X is defined to be the smallest positive integer such that – IKX is Cartier. Then there is a positive integer r and a Cartier divisor H such that – IKX ~ rH. Taking the largest number of such r, we call r/I the Fano index of X.