Limit linear series: Basic theory

Abstract

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 ^r’ _d 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 ^r _d 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 ^r _d 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].