Let $ V $ be the weighted projective variety defined by a weighted homogeneous ideal $ J $
and $ C $ a maximal cone in the Gröbner fan of $ J $ with $ m $ rays. We construct a flat …

Let Y be a (partial) minimal model of a scheme V with a cluster structure (of type A, X or of a
quotient of A or a fibre of X). Under natural assumptions, for every choice of seed we …

We study toric degenerations arising from Gröbner degenerations or the tropicalization of
partial flag varieties. We produce a new family of toric degenerations of partial flag varieties …

Matching fields were introduced by Sturmfels and Zelevinsky to study certain Newton
polytopes, and more recently have been shown to give rise to toric degenerations of various …

The concept of the Gröbner fan for a polynomial ideal, introduced by Mora and Robbiano in
1988, provides a robust polyhedral framework where maximal cones correspond to the …

The aim of this paper is to introduce a certain number of tools and results suitable for the
study of valuations of higher rank on function fields of algebraic varieties. This will be based …

We introduce the notion of a polyptych lattice, which encodes a collection of lattices related
by piecewise linear bijections. We initiate a study of the new theory of convex geometry and …

In this survey I summarize the constructions of toric degenerations obtained from valuations
and Gr\" obner theory and describe in which sense they are equivalent. I show how adapted …

Toward building tropical analogues of adic spaces, we study certain spaces of prime
congruences as a topological semiring replacement for the space of continuous valuations …

Abstract The Gelfand-Tsetlin and the Feigin–Fourier–Littelmann–Vinberg polytopes for the
Grassmannians are defined, from the perspective of representation theory, to parametrize …