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Let g, r, d be positive integers such that the Brill-Noether number, p(g, r, d) := g-(r+1)(g-d+r) = 0. Weprovethatif r+1 $& g-d+r, then for suitable families of curves (C/B), the monodromy of the family Gr(C/B) -B is at least the alternating group. Our techniques are combinatorial, and similar to those used by Bercov and Proctor in their paper [BP]. INTRODUCTION Let g, r, d be positive integers such that the Brill-Noether number, p(g, r, d) := g (r + 1) (g d + r) = 0 . It is known (see for example [ACGH]), that a general curve of genus g has exactly N(g, r, d) g! J (g -d + r + i)! distinct gr'S. In their paper [EH], Eisenbud and Harris proved that there is a family of smooth curves F/B such that the family G r(F/B) --+ B has fibers consisting of N(g, r, d) distinct points, and that the monodromy of the family acts transitively on the fibers. If r = 1 or g d + r = 2 (the case of a pencil, and its residual), they showed that the monodromy was in fact SN. The condition on the family F/B in [EH] is that the family must contain as stable limits all curves of the form shown in Figure 1, for all relative positions of the points p1, ... Pg ' In this paper we prove the following theorem. Theorem 1. For a family F/B satisfying the above condition, the monodromy of G r(F/B) -B is at least the alternating group as long as r + 1 7 g d + r. If r + 1 = g d + r then a gd is residual to a gd . In this case, one expects the residual pairing of points in a fiber to be preserved by the monodromy, so the action would not be doubly transitive. Received by the editors May 25, 1990 and, in revised form, July 25, 1990. 1980 Mathematics Subject Classification (1985 Revision). Primary 14H10. ( 1991 American Mathematical Society 0002-9939/91 $1.00 + $.25 per page

The monodromy of certain families of linear series is at least the alternating group