For colinked chains, the associated linked projective space is a reduced local complete intersection with rational equidimensional components, diagonal Hilbert polynomial, and the scheme satisfies a Riemann inequality on two-component curves.
Limit canonical series
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abstract
We describe the limits of canonical series along families of curves degenerating to a nodal curve which is general for its topology, in the weak sense that the branches over nodes on each of its components are in general position. We define a fan structure on the space of edge lengths on the dual graph of the limit curve, and construct a projective variety parametrizing the limits, organized in strata associated to the cones of this fan. This extends to all topologies the works by Eisenbud-Harris (Invent. Math. 87: 496-515, 1987) on curves of compact type and Esteves-Medeiros (Invent. Math. 149: 267-338, 2002) on two-component curves.
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Degenerations of maps to projective spaces
For colinked chains, the associated linked projective space is a reduced local complete intersection with rational equidimensional components, diagonal Hilbert polynomial, and the scheme satisfies a Riemann inequality on two-component curves.