Arithmetic on Singular Del Pezzo Surfaces

The study of singular cubic surfaces is quite an old subject, since their classification (over C) goes back to Schlafli [39] and Cay ley [8]. However, a recent account by Bruce and Wall [6] has shown that modern singularity theory can give much insight into this classification. One of the main themes of the present paper is that this approach is also useful over an arbitrary perfect field k for studying the fc-birational properties of singular cubic surfaces, and of certain other singular surfaces which are defined below. Recall that an absolutely irreducible algebraic variety V, defined over k, is said to be k-rational (respectively, k-unirational) if its function field k(V) is (respectively, is contained in) a purely transcendental extension of k. Throughout this paper k denotes an algebraic closure of k, and V = Vxkk. We say that V is rational if V is ^-rational, and we write P" for P£. Unless stated otherwise, the notation V <= P£ implies that V is a projective sub variety of P£, defined over k. In Part I we prove: