A Characterization of Toric Varieties in Characteristicp

Abstract

If X is a smooth toric variety over an algebraically closed field of positive characteristic and L is an invertible sheaf on X, then F∗L, the pushforward of L along the Frobenius morphism of X, splits into a direct sum of invertible sheaves [Tho00], [Bog98]. We show that this property characterizes smooth projective toric varieties. INTRODUCTION If X is a smooth toric variety over an algebraically closed field of positive characteristic and L is an invertible sheaf on X, then F∗L, the push-forward of L along the Frobenius morphism of X, splits into a direct sum of invertible sheaves [Tho00], [Bog98]. In this article, we show that this property in fact characterizes smooth projective toric varieties. Theorem 1. Let X be a projective connected scheme over an algebraically closed field k of characteristic p > 0. ThenX is a smooth toric variety if and only if, for every invertible sheaf L on X, F∗L is a direct sum of invertible sheaves. Although the “only if” part of this result has been known already, we include a short proof for completeness (Theorem 2). The proof of the “if” part (Theorem 4) is a bit more involved: we first reduce to the case when the Picard group of X is free abelian and look at the Cox ring R of X. The assumption on X means that R is flat over R, so R is regular by Kunz’s criterion. It follows that R has to be a polynomial ring, so X is a toric variety. We may expect Theorem 1 to hold more generally for X proper. However, we use the characterization of toric varieties in terms of their Cox rings in the form stated in [KW11] which requires X to be projective. It is worth noting that similar characterizations have been obtained [BH07, Corollary 4.4] for X satisfying the A2 condition: every two points have a common affine neighborhood. These theorems, depending on various results on group actions, have been stated only in characteristic zero, therefore we have to be very careful applying them to actions of tori in positive characteristic. 1. PROOF OF THE “ONLY IF” PART We provide an a bit shorter and more explicit proof of Thomsen’s result [Tho00]. The proof gives a very short computation even for projective spaces (in which case one can use the Horrocks splitting criterion and the projection formula to prove that the direct image is a direct sum of invertible sheaves). The key point of our approach is to consider Frobenius push-forwards of all invertible sheaves at once. 1