GIT for $\hat U$-actions on algebraic $\mathbb C$-schemes

TL;DR

This paper generalizes the GIT $$-theorem from projective varieties to projective schemes over $$ and introduces an algebraic reformulation and a more efficient blowing-up process for the semistability condition.

Contribution

It extends the GIT $$-theorem to algebraic schemes over $$, reformulates the semistability condition algebraically, and simplifies the blowing-up process to achieve the condition in one step.

Findings

Extended GIT $$-theorem to projective schemes over $$

Reformulated semistability condition algebraically

Developed a single-step blowing-up method

Abstract

We extend the $\hat U$-theorem for projective varieties proved in arXiv:1601.00340 and arXiv:1607.04181 to projective schemes over $\mathbb C$. The condition "ss=s" required in arXiv:1601.00340 and arXiv:1607.04181 is reformulated algebraically for our setting. We also describe a blowing up for the condition "ss=s". Different from the sequence of blowing ups in arXiv:1607.04181 , we can get "ss=s" in one blowing up.