2016/10/02 by Kaveh, Kiumars, Manon, Christopher · 4 citations
#13P10 #14T05 #Algebraic Geometry (math.AG) #Commutative Algebra (math.AC) #FOS: Mathematics
paper · doi:10.48550/arxiv.1610.00298
Given a finitely generated algebra A, it is a fundamental question whether A has a full rank discrete (Krull) valuation \mathfrakv with finitely generated value semigroup. We give a necessary and sufficient condition for this, in terms of tropical geometry of A. In the course of this we introduce the notion of a Khovanskii basis for (A, \mathfrakv) which provides a framework for far extending Gröbner theory on polynomial algebras to general finitely generated algebras. In particular, this makes a direct connection between the theory of Newton-Okounkov bodies and tropical geometry, and toric degenerations arising in both contexts. We also construct an associated compactification of Spec(A). Our approach includes many familiar examples such as the Gel'fand-Zetlin degenerations of coordinate rings of flag varieties as well as wonderful compactifications of reductive groups. We expect that many examples coming from cluster algebras naturally fit into our framework.