Plane curves with small linear orbits I

The "linear orbit" of a plane curve of degree d is its orbit in Pd(d+3)/2 u nder the natural action of PGL(3). In this paper we compute the degree of t he closure of the linear orbits of most curves with positive dimensional st abilizers. Our tool is a nonsingular variety dominating the orbit closure, which we construct by a blow-up sequence mirroring the sequence yielding an embedded resolution of the curve. The results given here will serve as an ingredient in the computation of the analogous information for arbitrary pl ane curves. Linear orbits of smooth plane curves were studied by the author s in J. of Alg. Geom., 2 (1993), 155-184.