Nonexistence of asymptotic GIT compactification

TL;DR

This paper presents examples of families of smooth canonically polarized varieties where Chow semistable limits vary under different pluricanonical embeddings, challenging assumptions about stability in GIT compactifications.

Contribution

It provides explicit examples demonstrating the nonexistence of an asymptotic GIT compactification for certain families of varieties.

Findings

Chow semistable limits do not stabilize under different embeddings.

Examples include smooth weighted pointed curves and hypersurfaces in P^3.

Challenges the existence of a universal GIT compactification.

Abstract

We provide examples of families of (log) smooth canonically polarized varieties, including smooth weighted pointed curves and smooth hypersurfaces in $P^3$ with large degree such that the Chow semistable limits under distinct pluricanonical embeddings do not stabilize.