A limit linear series moduli scheme

Abstract

We develop a new, more functorial construction for the basic theory of limit linear series, which provides a compactification of the Eisenbud-Harris theory, and shows promise for generalization to higher-dimensional varieties and higher-rank vector bundles. We also give a result on lifting linear series from characteristic p to characteristic 0. 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.

Linked Grassmannians and crude limit linear series

“…The goal of the present paper is to obtain sufficiently sharp upper bounds for the dimensions of spaces of crude limit series that we can apply the theoretical machinery of [7] to the loci of crude limit series in addition to refined limit series. Our estimates will allow us to prove the following theorem.…”

“…The theory of [7] has a boundary which maps naturally to the Eisenbud-Harris crude limit series, but frequently with positive-dimensional fibers. Because of this distinction, we will refer to the boundary elements of the latter construction as crude limit series, and the boundary described by Eisenbud and Harris as EH-crude limit series.…”

“…Dimension estimates are central to both theories of limit linear series, and while EH-crude limit series are easily amenable to making such estimates inductively, the crude limit series of [7] are more combinatorially complicated, and were not closely analyzed in [7]. The goal of the present paper is to obtain sufficiently sharp upper bounds for the dimensions of spaces of crude limit series that we can apply the theoretical machinery of [7] to the loci of crude limit series in addition to refined limit series.…”

“…In [7], a new construction of spaces of limit linear series is introduced, which is functorial and provides a compactification of the Eisenbud-Harris space, agreeing with it on the open locus of "refined" limit series. Coincidentally, concrete motivation for such a construction has been provided by Khosla [6], who produces, given a suitable proper stack of limit linear series, an infinite family of effective virtual divisors in M g , and shows that whenever they are divisors, they give counter-examples to the Harris-Morrison slope conjecture.…”

“…In [7], a new construction of limit linear series is presented which functorializes and compactifies the original construction of Eisenbud and Harris, using a new space called the linked Grassmannian. The boundary of the compactification consists of crude limit series, and maps with positivedimensional fibers to crude limit series of Eisenbud and Harris.…”