The decomposition group of a line in the plane
  We show that the decomposition group of a line $L$ in the plane, i.e. the subgroup of plane birational transformations that send $L$ to itself birationally, is generated by its elements of degree 1 and one element of degree 2, and that it does not decompose as a non-trivial amalgamated product. 
Isac Hed\'en and Susanna Zimmermann
math.AG