Limit points of one-parameter subgroups for additive actions on hypersurfaces

TL;DR

This paper investigates the behavior of additive group actions on projective hypersurfaces, characterizing those hypersurfaces where certain limit conditions of subgroup actions are satisfied.

Contribution

It identifies all projective hypersurfaces admitting additive actions that meet the OP-condition, advancing understanding of group actions on algebraic varieties.

Findings

Characterization of hypersurfaces with additive actions satisfying the OP-condition

Complete classification of such hypersurfaces

Insights into limit points of one-parameter subgroups in additive actions

Abstract

By an additive action on an algebraic variety $X$ over $\mathbb{C}$, we mean an action of the group $\mathbb{G}_a^n = \mathbb{C}^n $ on $X$ with an open orbit. We study limit points of one-dimensional subgroups of $\mathbb{G}_a^n$ for additive actions on projective hypersurfaces. We say that an additive action on $X$ satisfies the OP-condition if for every point $x\in X $ that does not lie in the open orbit $O$ there is a point $y \in O$ and a vector $v \in \mathbb{G}_a^n$ such that $ \lim_{t\to\infty} tv\circ y = x$. We find all projective hypersurfaces on which there is an additive action satisfying the OP-condition.