The geometry of Frobenius on toric varieties

TL;DR

This paper provides a geometric framework for understanding the positivity of the Frobenius-trace kernel on ${Q}$-factorial projective toric varieties, linking it to the structure of cones of $F$-effective curves and Mori contractions.

Contribution

It introduces the concepts of Frobenius support and $F$-effectiveness, connecting Frobenius properties with the Mori cone and extremal contractions in toric varieties.

Findings

Frobenius-trace kernel is ample iff Picard rank is 1.

Interaction of $F$-effective cone with Mori cone reflects contraction types.

Provides geometric criteria for Frobenius-trace kernel positivity.

Abstract

We give a geometric description of the positivity of the Frobenius-trace kernel on a $\mathbb{Q}$-factorial projective toric variety. To do so, we define its Frobenius support as well as the notions of $F$-effectiveness for divisors and $1$-cycles. As it turns out, the interaction of the corresponding cone of $F$-effective curves with the Mori cone of curves reflects the type of extremal Mori contractions that the variety can undergo. As a corollary, we obtain that the Frobenius-trace kernel is ample if and only if the Picard rank is $1$.