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

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.