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arXiv · 2506.02994

The geometry of Frobenius on toric varieties

Javier Carvajal-Rojas·Emre Alp Özavcı
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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 $1$.

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Javier Carvajal-Rojas, Emre Alp Özavcı. 2025-06-03. The geometry of Frobenius on toric varieties. https://arxiv.org/abs/2506.02994

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