|Abstract:||In 1970, Demazure gave a combinatorial description of the automorphism
group Aut(X) of a complete smooth toric variety X as a linear algebraic
group. The central concept is a root system associated with a complete
fan. Later Cox interprated and generatlized these results in terms of
the homogeneous coordinate ring R(X).
We describe the automoprhism group of a complete rational variety X with torus action of complexity one. Our description is based on a presentation of the Cox ring R(X) in terms of trinomials and on an interpretation of Demazure's roots as homogeneous locally nilpotent derivations of R(X). Also we obtain an explicit description of the root system of the semisimple part of Aut(X). The results are applied to the study of almost homogeneous varieties. This is a joint work with Jurgen Hausen, Elaine Herppich and Alvaro Liendo.