A connection between birational automorphisms of the plane and linear systems of curves

In this paper, we prove that there exists a one-to-one correspondence between birational automorphisms of the plane and pairs of pencils of curves intersecting in a unique point. As a consequence, we show how to construct birational automorphisms of the plane of a certain degree d (fixed in advance) from some curves generating two linear systems of curves of degrees d and d˜, where d˜=d−2 for d>2, and d˜=1 otherwise. In addition, we also get the inverse of the birational automorphism constructed, and we show that its degree is obtained from the degree of the linear system of curves. As a special case, we show how these results can be stated to polynomial birational automorphisms of the plane.