[Paper Review] Seshadri Constants and Fujita's Conjecture via Positive Characteristic Methods

Takumi Murayama|arXiv (Cornell University)|May 9, 2019

Algebraic Geometry and Number Theory160 references6 citations

TL;DR

This paper establishes a novel connection between Seshadri constants and Fujita's conjecture using positive characteristic methods, particularly the Frobenius morphism and test ideals. By leveraging the gamma construction and F-singularities, it proves that Fujita's conjecture holds in positive characteristic for smooth projective varieties under certain ampleness conditions, offering a new geometric characterization of projective space via F-purity and multiplier ideals.

ABSTRACT

In 1988, Fujita conjectured that there is an effective and uniform way to turn an ample line bundle on a smooth projective variety into a globally generated or very ample line bundle. We study Fujita's conjecture using Seshadri constants, which were first introduced by Demailly in 1992 with the hope that they could be used to prove cases of Fujita's conjecture. While examples of Miranda seemed to indicate that Seshadri constants could not be used to prove Fujita's conjecture, we present a new approach to Fujita's conjecture using Seshadri constants and positive characteristic methods. Our technique recovers some known results toward Fujita's conjecture over the complex numbers, without the use of vanishing theorems, and proves new results for complex varieties with singularities. Instead of vanishing theorems, we use positive characteristic techniques related to the Frobenius-Seshadri constants introduced by Mustaţă-Schwede and the author. As an application of our results, we give a characterization of projective space using Seshadri constants in positive characteristic, which was proved in characteristic zero by Bauer and Szemberg.

Motivation & Objective

• . To establish a new approach to Fujita's conjecture using positive characteristic algebraic geometry.
• . To characterize projective space via F-purity and multiplier ideals in positive characteristic.
• . To prove that Seshadri constants can be controlled via Frobenius and test ideal techniques.
• . To extend the ampleness criterion of de Fernex–K"uronya–Lazarsfeld to positive characteristic settings.
• . To demonstrate the openness of F-pure loci in schemes essentially of finite type over G-rings.

Proposed method

• . Applies the gamma construction to lift schemes over complete local rings to F-finite settings.
• . Uses the Frobenius morphism and its trace to define F-singularities such as F-purity and F-regularity.
• . Employs test ideals and multiplier ideals to measure singularities and control vanishing theorems.
• . Leverages Grothendieck duality and dualizing complexes in positive characteristic to analyze cohomological invariants.
• . Uses the pigeonhole principle in F-singularities to control the behavior of Frobenius powers.
• . Applies the theory of reductive group actions and equivariant resolution to reduce to local cases.

Experimental results

Research questions

• RQ1 . Can Fujita's conjecture be proven in positive characteristic using Frobenius methods?
• RQ2 . How do Seshadri constants behave under Frobenius and F-singularities?
• RQ3 . Is the F-pure locus open in schemes essentially of finite type over G-rings?
• RQ4 . Do strong F-regularity and split F-regularity coincide for rings over complete local rings?
• RQ5 . Can the ampleness criterion of de Fernex–K"uronya–Lazarsfeld be extended to positive characteristic?

Key findings

• . Fujita's conjecture holds in positive characteristic for smooth projective varieties when the line bundle is sufficiently ample, as shown via F-singularities and the gamma construction.
• . The F-pure locus is open in schemes essentially of finite type over a G-ring in positive characteristic, a key technical result.
• . For rings essentially of finite type over a complete local ring of positive characteristic, strong F-regularity implies split F-regularity.
• . The gamma construction allows lifting schemes to F-finite settings where F-singularities behave well and are preserved under base change.
• . Seshadri constants can be bounded below using F-singularities and test ideals, providing a new geometric invariant in positive characteristic.
• . The paper proves that a smooth projective variety with ample line bundle satisfying certain F-singularities conditions is isomorphic to projective space.