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 G r 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 G r 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.

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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 . So we extend his treatment to a more general setup. In particular, we construct a projective moduli space Gd,δrfalse(Xfalse) 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 by associating with each exact level‐δ limit linear series frakturg on X a closed subscheme Pfalse(frakturgfalse)⊆X(d) of the dth symmetric product of X, and showing that, if frakturg 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.

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In this paper we develop combinatorial techniques for the case of string algebras with the aim to give a characterization of string complexes with infinite minimal projective resolution. These complexes will be called periodic string complexes. As a consequence of this characterization, we give two important applications. The first one, is a sufficient condition for a string algebra to have infinite global dimension. In the second one, we exhibit a class of indecomposable objects in the derived category for a special case of string algebras. Every construction, concept and consequence in this paper is followed by some illustrative examples.