The geometry of Frobenius on toric varieties
Javier Carvajal-Rojas, Emre Alp \"Ozavc{\i}

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 -factorial projective toric variety. To do so, we define its Frobenius support as well as the notions of -effectiveness for divisors and -cycles. As it turns out, the interaction of the corresponding cone of -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 .
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Taxonomy
TopicsCommutative Algebra and Its Applications · Algebraic Geometry and Number Theory · Polynomial and algebraic computation
