Asymptotic linearity of regularity and a*-invariant of powers of ideals

TL;DR

This paper studies the long-term linear behavior of algebraic invariants like regularity and a*-invariant of powers of ideals in projective schemes, linking these to fiber invariants of a specific geometric projection.

Contribution

It establishes a connection between the asymptotic linearity of ideal invariants and the geometric properties of fibers in a blowup projection.

Findings

Asymptotic linearity of regularity and a*-invariant is characterized.

Fiber invariants influence the linear growth of ideal invariants.

Provides geometric insights into algebraic invariant behavior for powers of ideals.

Abstract

Let X = Proj R be a projective scheme over a field k, and let I be an ideal in R generated by forms of the same degree d. Let Y --> X be the blowing up of X along the subscheme defined by I, and let f: Y --> Z be the projection of Y given by the divisor dH - E, where E is the exceptional divisor of the blowup and H is the pullback of a general hyperplane in X. We investigate how the asymptotic linearity of the regularity and a*-invariant of I^q (for q large) is related to invariants of fibers of f.