Generic tropical initial ideals of Cohen-Macaulay algebras

We study the generic tropical initial ideals of a positively graded Cohen-Macaulay algebra R over an algebraically closed field k. Building on work of Römer and Schmitz, we give a formula for each initial ideal, and we express the associated quasivaluations in terms of certain I-adic filtrations. As a corollary, we show that in the case that R is a domain, every initial ideal coming from the codimension 1 skeleton of the tropical variety is prime, so “generic presentations of Cohen-Macaulay domains are well-poised in codimension 1.”