Limits of PGL(3)-translates of plane curves, I

We classify all possible limits of families of translates of a fixed, arbitrary complex plane curve. We do this by giving a set-theoretic description of the projective normal cone (PNC) of the base scheme of a natural rational map, determined by the curve, from the P8P8 of 3×3 matrices to the PNPN of plane curves of degree  dd. In a sequel to this paper we determine the multiplicities of the components of the PNC. The knowledge of the PNC as a cycle is essential in our computation of the degree of the PGL(3)PGL(3)-orbit closure of an arbitrary plane curve, performed in [P. Aluffi, C. Faber, Linear orbits of arbitrary plane curves, Michigan Math. J. 48 (2000) 1–37].