Advanced Studies in Pure Mathematics

Unipotent group actions on projective varieties

Rajendra V. Gurjar, Kayo Masuda, and Masayoshi Miyanishi

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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

Algebraic Varieties and Automorphism Groups, K. Masuda, T. Kishimoto, H. Kojima, M. Miyanishi and M. Zaidenberg, eds. (Tokyo: Mathematical Society of Japan, 2017), 119-162

Received: 25 August 2014
Revised: 10 November 2015
First available in Project Euclid: 21 September 2018

Permanent link to this document euclid.aspm/1537498707

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 14R20: Group actions on affine varieties [See also 13A50, 14L30]
Secondary: 14J45: Fano varieties

regular vector field unipotent group stratified $G_a$-action Fano variety


Gurjar, Rajendra V.; Masuda, Kayo; Miyanishi, Masayoshi. Unipotent group actions on projective varieties. Algebraic Varieties and Automorphism Groups, 119--162, Mathematical Society of Japan, Tokyo, Japan, 2017. doi:10.2969/aspm/07510119.

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