[Paper Review] A characterization of toric varieties in characteristic p

Piotr Achinger|arXiv (Cornell University)|Mar 24, 2013

Algebraic Geometry and Number Theory10 references3 citations

TL;DR

This paper establishes that a smooth projective variety over an algebraically closed field of positive characteristic is a toric variety if and only if the pushforward of every invertible sheaf under the Frobenius morphism decomposes as a direct sum of invertible sheaves. The proof uses the Cox ring and Kunz's criterion to show that flatness over the Frobenius twist forces the Cox ring to be a polynomial ring, hence the variety is toric, thereby characterizing toric varieties via Frobenius splitting behavior in positive characteristic.

ABSTRACT

If $X$ is a smooth toric variety over an algebraically closed field of positive characteristic and $L$ is an invertible sheaf on $X$, it is known that $F_* L$, the push-forward of $L$ along the Frobenius morphism of $X$, is a direct sum of invertible sheaves. We show that this property characterizes smooth projective toric varieties.

Motivation & Objective

• To characterize smooth projective toric varieties in positive characteristic using the splitting behavior of Frobenius pushforwards of invertible sheaves.
• To prove that the Frobenius pushforward of every invertible sheaf on a smooth projective variety splits into a direct sum of invertible sheaves if and only if the variety is toric.
• To extend known results from characteristic zero to positive characteristic by adapting the Cox ring approach and using Kunz’s criterion for regularity.
• To clarify the limitations of generalizing the characterization to non-projective varieties due to reliance on the Cox ring formalism.

Proposed method

• Reduction to the case where the Picard group is free abelian by considering a finite étale cover with free Picard group.
• Use of the Cox ring $ R $ of $ X $, which is graded by the Picard group, and the assumption that $ F_*L $ splits for all $ L $ implies $ R $ is flat over $ R^p $.
• Application of Kunz’s criterion: flatness of $ R $ over $ R^p $ implies $ R $ is regular, hence a polynomial ring.
• Use of Lemma 5 to conclude that a regular, graded domain with a field in degree zero and a maximal homogeneous ideal is isomorphic to a polynomial ring.
• Construction of a finite étale cover $ Y \to X $ such that the pullback Picard group is free, enabling the use of the Cox ring formalism.
• Application of [KW11, Theorem 1.5] in positive characteristic (justified via Luna’s étale slice theorem for tori) to conclude $ Y $ is toric, hence $ X $ is toric via étale isomorphism.

Experimental results

Research questions

• RQ1 Does the property that $ F_*L $ splits into a direct sum of invertible sheaves for all invertible sheaves $ L $ characterize smooth projective toric varieties in positive characteristic?
• RQ2 Can the Cox ring approach be used to characterize toric varieties in positive characteristic, given that it typically relies on characteristic zero assumptions?
• RQ3 What is the role of the Frobenius morphism in distinguishing toric varieties from other smooth projective varieties in positive characteristic?
• RQ4 To what extent can the characterization be extended to non-projective varieties, given the reliance on the Cox ring formalism?

Key findings

• A smooth projective variety $ X $ over an algebraically closed field of characteristic $ p > 0 $ is a toric variety if and only if $ F_*L $ is a direct sum of invertible sheaves for every invertible sheaf $ L $ on $ X $.
• The Frobenius pushforward $ F_*\mathcal{O}_X(D) $ decomposes as a direct sum of invertible sheaves with multiplicities determined by the number of $ T $-invariant divisors with coefficients in $ \{0,1,\ldots,\ell-1\} $ representing $ D - \ell E $.
• The flatness of the Cox ring $ R $ over $ R^p $, induced by the splitting assumption, implies $ R $ is regular by Kunz’s criterion, hence $ R $ is a polynomial ring.
• The regularity of the Cox ring forces $ X $ to be a toric variety via the classification of toric varieties by their Cox rings.
• The proof relies on the projectivity of $ X $, as the Cox ring characterization in [KW11] requires projectivity and fails in general for proper varieties.
• The result does not extend to non-projective varieties due to the dependence on the Cox ring formalism, which is not available in the same form for non-projective schemes.