# Wall-crossing for Newton-Okounkov bodies

(joint work with Laura Escobar.)

Let $X$ be an irreducible complex projective variety. Toric degenerations of such an $X$ are of interest because the toric geometry of the central fiber (which is a toric variety) can often yield insights about the geometry of the general fiber (which is isomorphic to the original variety $X$). Tropical geometry and the theory of Newton-Okounkov bodies are two methods which can produce toric degenerations, and in 2016, Kaveh and Manon showed that the two are related. Specifically, they showed that a toric degeneration corresponding to a maximal prime cone in the tropicalization of $X$, with central fiber $X_0$, can also be realized as one associated to a certain full-rank valuation (with one-dimensional leaves), with Newton-Okounkov body equal to the polytope of $X_0$. In this talk, we present a construction of two explicit geometric maps between the Newton-Okounkov bodies corresponding to two maximal prime cones in the tropicalization of $X$ which are *adjacent*, i.e., share a codimension-1 facet. The two maps, the ``shift'' and the ``flip'', acts as the identity along the shared codimension-1 facet in a suitable sense. We dub this a ``geometric wall-crossing for Newton-Okounkov bodies''. (This geometric wall-crossing result was also observed by Ilten and Manon in 2017 using the theory of complexity-1 $T$-varieties. ) Under a technical condition, we also produce a natural ``algebraic wall-crossing'' map on the underlying value semigroups (of the corresponding valuations). There are examples showing that this algebraic wall-crossing need not arise as the restriction of either of the geometric wall-crossing maps. If time permits, I will discuss the case of the tropical Grassmannian of 2-planes in $\mathbb{C}^m$, where we can prove that the algebraic wall-crossing map is the restriction of the geometric ``flip'' map.