Geometriae Dedicata 64: 157–191, 1997. 157 c 1997 Kluwer Academic Publishers. Printed in the Netherlands. Classification of Bertini’s Series of Varieties of Dimension Less than or Equal to Four ENRICO ROGORA Dipartimento di Matematica, Universit` a di Roma ‘La Sapienza’, Piazzale A. Moro 5, I-00185 Roma. e-mail: Rogora@mat.uniroma1.it (Received: 15 May 1995; revised version: 15 November 1995) Abstract. In this paper I give a classification of irreducible projective varieties of dimension less than or equal to four, according to a new classification scheme. No assumption is made about singularities. Mathematics Subject Classifications (1991): Primary, 14N05; Secondary, 51N35. Key words: Bertini’s series, dual variety. 1. Introduction In this paper, by a variety I shall mean a projective subvariety of some projective space over the complex numbers. No assumption is made about singularities. Let P be an irreducible variety and let P be its dual variety. The defect of , denoted by , is dim 1 For any hyperplane , the contact locus of with , denoted by , is the closure of the set of smooth points of for which the embedded tangent space is contained in . I shall denote by the point in P representing a hyperplane . Let be an 1 -dimensional linear space. For every integer there exists a natural isomorphism : G 1 P G P (1) which sends to the -dimensional subspace P , intersection of all hyperplanes of P which belong to . Let P be any variety. For the general tangent hyperplane , it is well known that the contact locus is a linear space of dimension equal to the defect of , and that , where denotes the embedded tangent space to at the point corresponding to (see, for example, [12, par. 4, p. 16] or [6, The author has been supported by the Italian C.N.R. and M.U.R.S.T. Ch. Busuttil/corr. J.N.B. (Kb. 1/2) **PREPROOFS** INTERPRINT: PIPS Nr.: 113615 MATHKAP geom1426.tex; 18/03/1997; 9:48; v.5; p.1