  The group PGL(3) of linear transformations of the projective plane acts
naturally on the projective space parametrizing curves of a given degree. In
this note we begin the study of the orbits of smooth curves under this action:
we construct a resolution of the closure of the orbit of a given curve, and we
use it to compute its degree. This turns out to depend only on the degree of
the curve, the order of its automorphism group, and on the number and type of
its flexes. This paper will appear on the Journal of Algebraic Geometry.
