### A limit linear series moduli scheme

We develop a new, more functorial construction for the basic theory of limit linear series, which provides a compactification of the Eisenbud-Harris theory. In an appendix, in order to obtain the necessary dimensional lower bounds on our limit linear series scheme we develop a theory of “linked Grassmannians”; these are schemes parametrizing sub-bundles of a sequence of vector bundles, which map into one another under fixed maps of the ambient bundles.