On the boundedness of anti-canonical volumes of singular Fano $3$-folds in characteristic $p>5$

TL;DR

This paper proves a boundedness result for the anti-canonical volumes of certain singular Fano 3-folds over fields with characteristic greater than 5, extending the Weak BAB conjecture to positive characteristic.

Contribution

It establishes the boundedness of anti-canonical volumes for klt Fano 3-folds in characteristic p>5, confirming a version of the Weak BAB conjecture in this setting.

Findings

Anti-canonical volumes are bounded for the specified class of Fano 3-folds.

The result extends the Weak BAB conjecture to characteristic p>5.

Provides a new understanding of the geometry of Fano varieties in positive characteristic.

Abstract

In this article we prove the following version of the Weak-BAB conjecture for $3$-folds in char $p>5$: Fix a DCC set $I\subset [0, 1)$ and an algebraically closed field $k$ of characteristic $p>5$. Let $\mathfrak{D}$ be a collection of klt pairs $(X, \Delta)$ satisfying the following properties: (1) $X$ is a projective $3$-fold, (2) $\Delta$ is an $\mathbb{R}$-divisor with coefficients in $I$, (3) $K_X+\Delta\equiv 0$, and (4) $-K_X$ is ample. Then the set $\{\mbox{vol}_X(-K_X) \ | \ (X, \Delta)\in\mathfrak{D}\mbox{ for some }\Delta\}$ is bounded from above.