Continuous Linear Series
Eduardo Esteves, Antonio Nigro, Pedro Rizzo
TL;DR
This work develops a framework for degenerations of linear series on a nodal curve $X$ consisting of two components $Y$ and $Z$ meeting at $P$. It introduces continuous linear series as ${\mathbf G}_{\mathbf m}$-invariant, maximal families of generalized linear series carried by two-punctured semistable chains of rational curves, and proves representability by a projective moduli space $G^r_d(X)$. The construction relies on the twister family $\mathcal F$ and twisted linear series along chain maps, embedding the moduli into a Hilbert scheme via a $\mathbf G_m$-equivariant framework; it unifies Osserman’s exact limit linear series with refinements of Eisenbud–Harris and extends to level-$\delta$ limit linear series. The main results establish (i) the representability of the continuous linear series functor and its open-closed Hilbert subscheme realization, and (ii) a second main theorem linking level-$\delta$ data to continuous linear series through explicit morphisms $\Psi_\delta^N$, producing a stratified, compact moduli that behaves well in families. Collectively, the approach provides a natural, projective compactification of existing moduli spaces and a robust framework for degeneration theory in this setting.
Abstract
We parameterize by a fine moduli space all degenerations of linear series to a singular curve which is the union of two smooth components meeting transversally at a single point. For this we introduce a novel object in the study of degenerations of linear series, which is the continuous linear series. Our moduli space can be regarded as a Hilbert quotient, in the terminology introduced by Kapranov, and is a new compactification of Osserman moduli space of exact limit linear series, and consequently, of Eisenbud and Harris moduli space of refined limit linear series on the curve.