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Moscow Mathematical Journal

Volume 14, Issue 3, July–September 2014  pp. 429–471.

The Automorphism Group of a Variety with Torus Action of Complexity One

Authors:  I. Arzhantsev (1), J. Hausen (2), E. Herppich (2), and A. Liendo (3)
Author institution:(1) Department of Higher Algebra, Faculty of Mechanics and Mathematics, Moscow State University, Leninskie Gory 1, Moscow 119991, Russia and National Research University Higher School of Economics, School of Applied Mathematics and Information Science, Pokrovsky blvd. 11, Moscow 109028, Russia
(2) Mathematisches Institut, Universität Tübingen, Auf der Morgenstelle 10, 72076 Tübingen, Germany
(3) Instituto de Matemática y Física, Universidad de Talca, Casilla 721, Talca, Chile


We consider a normal complete rational variety with a torus action of complexity one. In the main results, we determine the roots of the automorphism group and give an explicit description of the root system of its semisimple part. The results are applied to the study of almost homogeneous varieties. For example, we describe all almost homogeneous (possibly singular) del Pezzo 𝕂-surfaces of Picard number one and all almost homogeneous (possibly singular) Fano threefolds of Picard number one having a reductive automorphism group with twodimensional maximal torus.

2010 Mathematics Subject Classification. 14J50, 14M25, 14J45, 13A02, 13N15.

Keywords: Algebraic variety, torus action, automorphism, Cox ring, Mori Dream Space, locally nilpotent derivation, Demazure root.

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