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 Römer and Schmitz, we give formula for each ideal, express associated quasivaluations in terms certain I-adic filtrations. As corollary, show that case is domain, every ideal coming from codimension 1 skeleton variety prime, so “generic presentations domains are well-poised 1.”