Examples violating Golyshev's canonical strip hypotheses

TL;DR

This paper presents the first known examples of smooth Fano and Calabi-Yau varieties that violate Golyshev's canonical strip hypotheses, using moduli spaces of rank 2 bundles on high-genus curves.

Contribution

It provides the first explicit counterexamples to the canonical strip hypothesis, expanding understanding of Hilbert polynomial root locations in algebraic geometry.

Findings

Counterexamples to the canonical strip hypothesis are constructed.

Fano examples have Picard rank 1, index 2, and are rational.

Violations also occur in related varieties.

Abstract

We give the first examples of smooth Fano and Calabi-Yau varieties violating the (narrow) canonical strip hypothesis, which concerns the location of the roots of Hilbert polynomials of polarised varieties. They are given by moduli spaces of rank 2 bundles with fixed odd-degree determinant on curves of sufficiently high genus, hence our Fano examples have Picard rank 1, index 2, are rational, and have moduli. The hypotheses also fail for several other closely related varieties.