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arXiv · 2105.07843

The 3-dimensional Lyness map and a self-mirror log Calabi-Yau 3-fold

Tom Ducat
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Abstract

The 2-dimensional Lyness map is a 5-periodic birational map of the plane which may famously be resolved to give an automorphism of a log Calabi-Yau surface, given by the complement of an anticanonical pentagon of $(-1)$-curves in a del Pezzo surface of degree 5. This surface has many remarkable properties and, in particular, it is mirror to itself. We construct the 3-dimensional big brother of this surface by considering the 3-dimensional Lyness map, which is an 8-periodic birational map. The variety we obtain is a special (non-$\mathbb Q$-factorial) affine Fano 3-fold of type $V_{12}$, and we show that it is a self-mirror log Calabi-Yau 3-fold.

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Tom Ducat. 2021-05-17. The 3-dimensional Lyness map and a self-mirror log Calabi-Yau 3-fold. https://arxiv.org/abs/2105.07843

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