Limit linear series: Basic theory

In this paper we introduce techniques for handling the degeneration of linear series on smooth curves as the curves degenerate to a certain type of reducible curves, curves of compact type. The technically much simpler special case of 1-dimensional series was developed by Beauville 2], Knudsen 21–23], Harris and Mumford 17], in the guise of “admissible covers”. It has proved very useful for studying the Moduli space of curves (the above papers and Harris 16]) and the simplest sorts of Weierstrass points (Diaz 4]). With our extended tools we are able to prove, for example, that:

1. The Moduli spaceM _g of curves of genusg has general type forg≧24, and has Kodaira dimension ≧1 forg=23, extending and simplifying the work of Harris and Mumford 17] and Harris 16].
2. Given a Weierstrass semigroup Γ of genusg and weightw≦g/2 (and in a somewhat more general case) there exists at least one component of the subvariety ofM _g of curves possessing a Weierstrass point of semigroup Γ which has the “expected” dimension 3g-2?w (and in particular, this set is not empty).
3. The fundamental group of the space of smooth genusg curves having distinct “ordinary” Weierstrass points acts on the Weierstrass points by monodromy as the full symmetric group.
4. Ifr andd are chosen so that $$\rho : = g - (r + 1)(g - d + r) = 0,$$ then the general curve of genusg has a certain finite number ofg _d ^r’ s 15, 20]. We show that the family of all these, allowing the curve to vary among general curves, is irreducible, so that the monodromy of this family acts transitively. If4=1, we show further that the monodromy acts as the full symmetric group.
5. Ifr andd are chosen so that $$\rho = - 1,$$ then the subvariety ofM _g consisting of curves posessing ag _d ^r has exactly one irreducible component of codimension 1.
6. For anyr, g, d such that ρ≦0, the subvariety ofM _g consisting of curves possessing ag _d ^r has at least one irreducible component of codimension—ρ so long as $$\rho \geqq \left\{ \begin{gathered} - g + r + 3 (r odd) \hfill \\ - \frac{r}{{r + 2}}g + r + 3 (r even). \hfill \\ \end{gathered} \right.$$

In this paper we present the basic theory of “limit linear series” necessary for proving these results. The results themselves will be taken up in our forthcoming papers 8-12]. Simpler applications, not requiring the tools developed in this paper but perhaps clarified by them, have already been given in our papers 5-7].