Newton–Okounkov body, Rees algebra, and analytic spread of graded families of monomial ideals

Abstract

Let I = { I k } k ∈ N \mathcal {I}=\{I_k\}_{k \in \mathbb {N}} be a graded family of monomial ideals. We use the Newton–Okounkov body of I \mathcal {I} to: (a) give a characterization for the Noetherian property of the Rees algebra of the family I \mathcal {I} ; and (b) present a combinatorial interpretation for the analytic spread of I \mathcal {I} . We also apply these results to investigate and give bounds for the generation type and the Veronese degree of the symbolic Rees algebra of a monomial ideal.