Level-δ limit linear series (2018)
We introduce the notion of level-δ limit linear series, which describe limits of linear series along families of smooth curves degenerating to a singular curve X. We treat here only the simplest case where X is the union of two smooth components meeting transversely at a point P. The integer δ stands for the singularity degree of the total space of the degeneration at P. If the total space is regular, we get level-1 limit linear series, which are precisely those introduced by Osserman [10]. We construct a projective moduli space Gr d,δ(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, an open subscheme G r,∗ d,1 (X) of the space Gr d,1 (X) already constructed by Osserman. Finally, we generalize [6] by associating to 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 P(g) is the limit of the spaces of divisors associated to linear series on smooth curves degenerating to g on X, if such degenerations exist. In particular, we describe completely limits of divisors along degenerations to such a curve X.