## Advanced Studies in Pure Mathematics

### Unipotent group actions on projective varieties

#### Abstract

The correspondence between $G_a$-actions on affine varieties and locally nilpotent derivations of the coordinate algebras is generalized in the projective case to the correspondence between stratified $G_a$-actions on smooth projective varieties $V$ and regular vector fields on $V$ which are effectively locally nilpotent with stratification. These notions with stratifications are inspired by explicit computations of $G_a$-actions on the projective space $\mathbb{P}^n$ as well as the Hirzebruch surface $\mathbb{F}_n$ and the associated regular vector fields. Using partly these observations, we investigate the existence of $\mathbb{A}^1$-cylinders in Fano threefolds with rank one.

#### Article information

Dates
Revised: 10 November 2015
First available in Project Euclid: 21 September 2018

https://projecteuclid.org/ euclid.aspm/1537498707

Digital Object Identifier
doi:10.2969/aspm/07510119

Mathematical Reviews number (MathSciNet)
MR3793364

Zentralblatt MATH identifier
1396.14061

Subjects