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][B. Osserman, 2006]. So we extend his treatment to a more general setup. In particular, we construct a projective moduli spaceG d , δ r ( 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][E. Esteves, 2013] by associating with each exact level‐δ limit linear series g on X a closed subscheme P ( g ) ⊆ X ( d )of the d th 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 .