Examples Violating Golyshev's Canonical Strip Hypotheses

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Pieter Belmans, Sergey Galkin, Swarnava Mukhopadhyay

Published: 01 Jan 2022, Last Modified: 18 Sept 2025Exp. Math. 2022EveryoneRevisionsBibTeXCC BY-SA 4.0
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 polarized 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.