MULTIPLE POINT SCHEMES FOR CORANK 1 MAPS WASHINGTON LUIZ MARAR AND DAVID MOND Introduction In the theory of complex isolated complete intersection singularities (ICIS) a central role is played by the Milnorfibration. Briefly, if (A^O) £ (C n+ *,0) is the germ of an ^-dimensional ICIS, let / : (C n+ *, 0) -> (C*, 0) be an analytic map germ with X o =f' 1 (0). Then there is a class of representatives of/(the 'good representatives') f:U->V, such that if A £ Kis the set of critical values, /induces a C°°-locally trivial fibre bundle U-f~\A) -• V- A. The fibre X t =f~\t) is called the Milnorfibreof/; up to isomorphism it depends only on the germ (X o , 0), and one should think of it as the stable (smooth) object near to the unstable (singular) X o . It turns out that X t has homology only in dimensions 0 and n, and H n (X t , Z) is a free abelian group. Its rank fi is the Milnor number of X o , which can be calculated algebraically. For example, if k = 1 then n = dim c & n+1 /(df/d Xl ,...,df/dx n+1 ). Finitely determined unstable map-germs /: (C n , 0) -> (C p , 0) (p > n), with (n,p) nice dimensions in the sense of Mather [11], can similarly be approximated by stable map-germs and, as in the case of an ICIS, one can knit together such approximations into a locally trivial bundle. It is an interesting problem to determine the topology of such a stable approximation, in terms of data calculable from the original map- germ/ In this paper we show that information on the topology of a stable approximation is provided by the multiple point schemes of the map-germ/(defined below, in Section 1); a substantial part of the paper is devoted to proving a characterisation of stability and finite determinacy of corank 1 map-germs, Theorem 2.14: a corank 1 map-germ is finitely determined if and only if each multiple point scheme of dimension at least 1 is an ICIS, and is stable if moreover each non-empty multiple point scheme is smooth. From this it follows that the multiple point schemes of stable approximations to a finitely determined corank 1 germ / are Milnor fibres of the corresponding multiple point schemes of/ and thus that their topology is of an especially simple kind, as described above. In Section 3 we use this fact to obtain formulae for the Euler characteristic of the image X t of a stable approximation in the cases when (n,p) = (2,3) and (3,4), in terms of the Milnor numbers of the relevant multiple point schemes, and we discuss the relation between these invariants and others which have been described elsewhere. From our formula for x(X t ) in the case when (n,p) = (2,3) one deduces (empirically) that for all of the quasi-homogeneous map-germs/: (C 2 ,0) -> (C 3 ,0) listed in [12], one haS cod(sf e ,f)= X (X t )-l, (1) where the left-hand side is the dimension of the base of a miniversal unfolding of/ Received 22 July 1988. 1980 Mathematics Subject Classification (1985 Revision) 58C27. J. London Math. Soc. (2) 39 (1989) 553-567