— Let g, r, and d be positive integers such that g=(r-{-1) (g—d-\-r), so that the general curve of genus g has only finitely many g^s. We will show in this paper that for suitable families of curves ^ -> B, the family of all cffs on all fibers of ^ -> B is irreducible. We do this by analyzing the monodromy action on the set of 97 on a fibre, using a degeneration to reducible curves and our technique of limit series [198?^]. In the case r = 1 we prove the sharper statement that the monodromy is the full symmetric group, a result motivated by a problem posed by Verdier, and applied by him in the study of harmonic maps from 2 to S (Verdier [198?]). If we take ^ to be the universal curve over a suitable open set B of the moduli space My then the family of c^'s is a finite cover of B, and the branch locus of this cover (in the case r=l), analyzed through the tools developed in this paper, plays a fundamental role in the even-genus case in our proof [198?^] that Jig has general type for all g 24.

/pdf/irreducibility-and-monodromy-of-some-families-of-linear-30o7vkicyj.pdf

Irreducibility and monodromy of some families of linear series

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 $100 + $25 per page

/pdf/the-monodromy-of-certain-families-of-linear-series-is-at-4v2ge9l0zb.pdf

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

/pdf/annales-scientifiques-de-l-e-n-s-1gfcvb9n7z.pdf

Annales scientifiques de l'é.n.s.

© Gauthier-Villars (Éditions scientifiques et médicales Elsevier), 1987, tous droits réservés. L’accès aux archives de la revue « Annales scientifiques de l’É.N.S. » (http://www. elsevier.com/locate/ansens) implique l’accord avec les conditions générales d’utilisation (http://www.numdam.org/legal.php). Toute utilisation commerciale ou impression systématique est constitutive d’une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright.

/pdf/solution-of-a-combinatorially-formulated-monodromy-problem-4g0zw0gikj.pdf

Solution of a combinatorially formulated monodromy problem of Eisenbud and Harris

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Solution of a combinatorially formulated monodromy problem of Eisenbud and Harris