The set of fixed points of a unipotent group

TL;DR

This paper proves that fixed point sets of unipotent group actions on affine varieties are uniruled, and extends this to certain fixed point hypersurfaces under infinite algebraic group actions, revealing geometric structure.

Contribution

It establishes that fixed points of unipotent groups are uniruled varieties and generalizes this to fixed point hypersurfaces under infinite algebraic groups.

Findings

Fixed points of unipotent groups are K-uniruled varieties.

Fixed point hypersurfaces under infinite algebraic groups are K-uniruled.

Provides a geometric description of fixed point sets in algebraic group actions.

Abstract

Let $K$ be an algebraically closed field. Let $G$ be a non-trivial connected unipotent group, which acts effectively on an affine variety $X.$ Then every non-empty component $R$ of the set of fixed points of $G$ is a $K$-uniruled variety, i.e, there exists an affine cylinder $W\times K$ and a dominant, generically-finite polynomial mapping $\phi:W\times K\rightarrow R.$ We show also that if an arbitrary infinite algebraic group $G$ acts effectively on $K^n$ and the set of fixed points contains a hypersurface $H$, then this hypersurface is $K$-uniruled.