A Laurent phenomenon for $\mathrm{OGr}(5,10)$ and explicit mirror symmetry for the Fano $3$-fold $V_{12}$

Abstract: The $5$-periodic birational map $(x, y) -> (y, (1+y)/x)$ can be interpreted as a mutation between five open torus charts in a del Pezzo surface of degree $5$, coming from a cluster algebra structure on the Grassmannian $\mathrm{Gr}(2,5)$. This can used to construct a rational elliptic fibration which is the Landau-Ginzburg mirror to $\mathrm{dP}_5$. I will briefly recap this, and then explain the following $3$-dimensional generalisation: the $8$-periodic birational map $(x, y, z) -> (y, z, (1+y+z)/x)$ can be used to exhibit a Laurent phenomenon for the orthogonal Grassmannian $\mathrm{OGr}(5,10)$ and construct a completely explicit $K3$ fibration which is mirror to the Fano $3$-fold $V_{12}$, as well as some other Fano $3$-folds.

algebraic geometrycombinatorics

Audience: researchers in the topic

( slides | video )

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Online Nottingham algebraic geometry seminar

Series comments: Online geometry seminar, typically held on Thursday. This seminar takes place online via Microsoft Teams on the Nottingham University "Algebraic Geometry" team.

For recordings of past talks, copies of the speaker's slides, or to be added to the Team, please visit the seminar homepage at: kasprzyk.work/seminars/ag.html

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