The Limiting Polytope of the Generic Initial System of a Complete Intersection

TL;DR

This paper investigates the asymptotic behavior of the generic initial ideals of powers of a complete intersection, revealing that their limiting polytope depends solely on the intersection's type.

Contribution

It characterizes the limiting polytope of the generic initial system for complete intersections, linking it explicitly to the type of the intersection.

Findings

The limiting polytope depends only on the type of the complete intersection.

The asymptotic behavior of the generic initial ideals is explicitly described.

The structure of the generic initial system is captured by a well-defined polytope.

Abstract

Consider a complete intersection I of type (d_1,..., d_r) in a polynomial ring over a field of characteristic 0. We study the graded system of ideals {gin(I^n)}_n obtained by taking the reverse lexicographic generic initial ideals of the powers of I and describe its asymptotic behavior. This behavior is nicely captured by the limiting polytope which is shown to depend only on the type of the complete intersection.