Asymptotic invariants of generic initial ideals

Generic initial ideals (gins in short) were systematically introduced by Galligo in 1974 under the name of Grauert invariants since they appeared apparently first in works of Grauert and Hironaka. Ever since they are of interest in commutative algebra and indirectly in algebraic geometry. Recently Mayes in a series of articles associated to gins geometric objects called limiting shapes. The construction resembles that of Okunkov bodies but there are some differences as well. This work is motivated by Mayes articles and explores the connections between gins, limiting shapes and some asymptotic invariants of homogeneous ideals, e.g. asymptotic regularity, Waldschmidt constant and some new invariants, which seem relevant from geometric point of view. In this note we generalize Mayes ideas to graded families of ideals. We work out, sometimes surprising, properties of defined objects and invariants. For example we establish the existence of limiting shapes in dimension 2 which are polygons with an arbitrary high number of edges and vertices with irrational coordinates.