The BATMOBYLE is a vehicle for bringing together the algebraic and tropical geometry groups of Brown and Yale for a biannual day of talks alternating between the two universities.

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Abstracts Dan Corey (Yale) -- A primer on toric degenerations

Given the data of a lattice polytope Δ, a regular polyhedral decomposition P, and a convex piecewise-linear function φ, one can construct a degeneration of the projective toric variety X_Δ to a union of toric varieties glued along toric strata as prescribed by P. The total space of this degeneration is a toric variety influenced by the choice of φ. This procedure generalizes to obtain a degeneration of (not necessarily toric) varieties with effective anticanonical bundle from the starting data of a bounded singular integral affine manifold, an integral polyhedral decomposition, and a multivalued piecewise affine function. In this talk, we will illustrate how these constructions work, with emphasis on concrete examples.

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Kento Fujita (RIMS Kyoto) -- K-stability of log Fano hyperplane arrangements

Recently, the theory of K-stability for Fano manifolds (more generally, log Fano pairs) is rapidly developing. In this talk, I will explain "a valuative criterion" for K-stability of log Fano pairs introduced by myself and C. Li which seems easier than the original definition. Moreover, I will show one of its application, that is, I will give the answer which log Fano hyperplane arrangements are K-stable or not.

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Paul Hacking (UMass Amherst) -- Smoothing toric Fano varieties via mirror symmetry

A pair (X,E) consisting of a smooth Fano variety together with a choice of smooth anticanonical divisor corresponds under mirror symmetry to a family W : U -> A¹ of Calabi-Yau varieties over the affine line such that the monodromy at infinity is maximally unipotent. Moreover, a degeneration of (X,E) to a singular toric Fano variety together with its toric boundary corresponds to an open embedding of an algebraic torus in U. Using this heuristic, we construct smoothings of Gorenstein toric Fano 3-folds determined by combinatorial data encoding the construction of the mirror as a blowup of a toric variety. The smoothing is described using the scattering diagram of Kontsevich-Soibelman and Gross-Siebert which encodes counts of holomorphic discs on the mirror via tropical geometry.

This is a part of a program initiated by Coates, Corti, Galkin, Golyshev, and Kasprzyk to classify smooth Fano varieties using mirror symmetry. Period and Gromov--Witten calculations by these authors suggest that all deformation types of Fano 3-folds are obtained by our construction.

This is a report on work in progress with Alessio Corti, Mark Gross, and Andrea Petracci.

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