LIMIT LINEAR SERIES, THE IRRATIONALITY OF Mg, AND OTHER APPLICATIONS

We describe degenerations and smoothings of linear series on some reducible algebraic curves. Applications include a proof that the moduli space of curves of genus g has general type for all g > 24, a proof that the monodromy action is transitive on the set of linear series of dimension r and degree d on a general curve of genus g when p := g — (r + l)(g — d+r) = 0, a proof that there exist Weierstrass points with every semigroup of a certain class—in particular, on curves of genus g, all those semigroups with weight w < g/2 occur and a proof that the monodromy group acts as the full symmetric group on the g — g Weierstrass points of the general curve. Curves will here be reduced, connected, and complex algebraic. The study of general curves (Brill-Noether theory, etc.) and of moduli of curves depends on the degeneration of smooth curves to singular ones. Originally, the singular curves used were irreducible curves with nodes ([GH] is a recent avatar) or, more recently, cusps [E-Hl], but from the work of Mumford and others on the moduli space of stable curves it is apparent that reducible curves should be considered as well. Unfortunately the degeneration of a linear series on a curve which degenerates to a reducible curve has not been well understood except in the particularly simple case of pencils; there the "limit" of the linear series, after removing base points, corresponds to an admissible covering, in the sense of Beauville, Knudsen and Harris-Mumford [B, K, H-M], of a curve of genus 0. The potential of a general theory is indicated, for example, by work of Gieseker [G]. In this announcement we describe the limits of linear series on some reducible curves and give some applications. We call a curve tree-like if its irreducible components meet only two at a time, in ordinary nodes, in such a way that its dual graph (a vertex for each component, an edge for each intersection between distinct components) has no loops. We say that a curve is of compact type if its (generalized) Jacobian is compact, or, equivalently, if it is tree-like and its irreducible components are all nonsingular. Received by the editors November 8, 1983. 1980 Mathematics Subject Classification. Primary 14H10. The authors are grateful to the NSF, and the second author is grateful to the Alfred P. Sloan Foundation, for partial support of this work. © 1984 American Mathematical Society 0273-0979/84 $1.00 + $.25 per page 277 278 DAVID EISENBUD AND JOE HARRIS DEFINITION. A limit gd on a tree-like curve y is a collection of g r ds, one on each irreducible component Z of Y, Lz a line bundle of degree d on Z, Vz C H°(Z,Lz) an r + 1-dimensional subspace such that whenever two components of Y meet in a point, say p = Z\ n Z2> there is for each a G Vzx a r G Vz2 such that ordp o + ordp r = d. The following result is implicit in [E-H3]: THEOREM 1. Let 0 be a discrete valuation ring, and letX —• Spec 0 be a family of curves with irreducible geometric general fiber X^ and reduced, special fiber of compact type. Given a line bundle L and a gd k(rj) + s V c H°(X^, L) on X^j, there is a family ir': X' -* SpecO' obtained from X by base change, blow-ups of points in the central fiber, and normalizations, with reduced, special fiber Y of compact type such that : (1) For each irreducible component Z C Y there is an extension Lz of L to X with deg(LZ|Z) = d, deg(£Z|z') = 0 for irreducible components Z' # Z.