Gröbner theory and tropical geometry on spherical varieties

Abstract

Let $G$ be a connected reductive algebraic group. We develop a Gröbner theory for multiplicity-free $G$-algebras, as well as a tropical geometry for subschemes in a spherical homogeneous space $G/H$. We define the notion of a spherical tropical variety and prove a fundamental theorem of tropical geometry in this context. We also propose a definition for a spherical amoeba in $G/H$. Our work partly builds on the previous work of Vogiannou on spherical tropicalization and in some ways is complementary.

Some sections throughly revised. In particular, added a subsection about fan structure on spherical tropical varieties. In the spherical tropical varieties sections, previously we used a G-line bundle on G/H, we now use a very ample G-line bundle on a projective spherical embedding of G/H. Typos corrected and presentation improved in several places. 43 pages, 1 figure