[Paper Review] Kawamata-Viehweg vanishing fails for log del Pezzo surfaces in char. 3

Fabio Bernasconi|arXiv (Cornell University)|Sep 26, 2017

Algebraic Geometry and Number Theory5 citations

TL;DR

This paper constructs a log del Pezzo surface in characteristic 3 that violates the Kawamata-Viehweg vanishing theorem, demonstrating that the theorem does not hold in positive characteristic for such surfaces. As a consequence, it establishes the existence of a klt threefold singularity in characteristic 3 that is not Cohen-Macaulay, challenging the expected behavior of vanishing theorems in positive characteristic algebraic geometry.

ABSTRACT

We construct a log del Pezzo surface in characteristic 3 violating the Kawamata-Viehweg vanishing theorem. As a consequence we show that there exists a Kawamata log terminal threefold singularity which is not Cohen-Macaulay in characteristic 3.

Motivation & Objective

• To investigate the validity of the Kawamata-Viehweg vanishing theorem for log del Pezzo surfaces in positive characteristic.
• To determine whether the failure of vanishing theorems in positive characteristic affects the Cohen-Macaulay property of klt singularities.
• To construct explicit counterexamples in characteristic 3 where standard vanishing fails.
• To extend the understanding of singularities in positive characteristic algebraic geometry, particularly in the context of klt and Cohen-Macaulay conditions.

Proposed method

• Construction of a specific log del Pezzo surface defined over a field of characteristic 3.
• Verification of the log del Pezzo condition by checking the anticanonical divisor's ampleness and log terminal singularities.
• Computation of cohomology groups to demonstrate the failure of Kawamata-Viehweg vanishing, specifically showing H^1(X, K_X + L) ≠ 0 for an ample line bundle L.
• Use of the failure of vanishing to deduce non-Cohen-Macaulayness of a related threefold singularity via geometric and cohomological arguments.
• Application of known results linking vanishing theorems to Cohen-Macaulay properties in the context of klt singularities.
• Leveraging characteristic 3-specific phenomena, such as Frobenius splitting behavior and non-reduced structures, to construct the counterexample.

Experimental results

Research questions

• RQ1 Does the Kawamata-Viehweg vanishing theorem hold for log del Pezzo surfaces in characteristic 3?
• RQ2 Can a failure of Kawamata-Viehweg vanishing be used to construct a klt threefold singularity that is not Cohen-Macaulay in positive characteristic?
• RQ3 What specific geometric or cohomological properties of surfaces in characteristic 3 lead to the breakdown of standard vanishing theorems?
• RQ4 Are there intrinsic features of log del Pezzo surfaces in characteristic 3 that make them more prone to violating vanishing theorems?
• RQ5 How does the failure of vanishing in dimension two relate to the Cohen-Macaulay property of associated threefold singularities?

Key findings

• A log del Pezzo surface in characteristic 3 is constructed for which Kawamata-Viehweg vanishing fails, specifically showing non-vanishing of H^1(X, K_X + L) for an ample line bundle L.
• The constructed surface exhibits a failure of the standard cohomological vanishing expected in characteristic zero, highlighting a key difference in positive characteristic geometry.
• This failure implies that the associated threefold singularity, constructed via a cone construction, is not Cohen-Macaulay.
• The example demonstrates that the klt condition does not imply Cohen-Macaulayness in characteristic 3, contrary to what holds in characteristic zero.
• The result shows that the behavior of singularities and vanishing theorems in positive characteristic is fundamentally different from the complex case.