Level-δ limit linear series

We consider all one-parameter families of smooth curves degenerating to a singular curve X and describe limits of linear series along such families. We treat here only the simplest case where X is the union of two smooth components meeting transversely at a point P. We introduce the notion of level-δ limit linear series on X to describe these limits, where δ is the singularity degree of the total space of the degeneration at P. If the total space is regular, that is, δ=1, we recover the limit linear series introduced by Osserman in [11]. So we extend his treatment to a more general setup. In particular, we construct a projective moduli space Gd,δr(X) parameterizing level-δ limit linear series of rank r and degree d on X, and show that it is a new compactification, for each δ, of the moduli space of Osserman exact limit linear series. Finally, we generalize [7] by associating with each exact level-δ limit linear series g on X a closed subscheme P(g)⊆X(d) of the dth symmetric product of X, and showing that, if g is a limit of linear series on the smooth curves degenerating to X, then P(g) is the limit of the corresponding spaces of divisors. In short, we describe completely limits of divisors along degenerations to such a curve X.