A linear series on a curve C is a line bundle L together with a subspace V of the global sections of L. Eisenbud and Harris develeoped a theory of limit linear series which explans how (L, V) degenerate when C does. I was under the impression that one of the nice things about limit linear series is it allows you to prove statements by induction on the genus of a curve. What are some examples of such statements that you can prove like this?
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Not sure off the top of my head whether any of the Eisenbud-Harris papers literally used induction. One example is my paper "Linked Grassmannians and crude limit linear series" which gives a simple inductive proof of the Brill-Noether theorem using limit linear series. One can give similar arguments for existence of certain maps with prescribed ramification in characteristic p (see Theorem 7.1 of version 1 on the arxiv of "Linear series and existence of branched covers).
But more broadly, one might say that the "inductive" structure of limit linear series is encapsulated by the fact that one can describe limit linear series component by component (in contrast to the situation for higher rank vector bundles, where one has to worry about gluing maps). Many arguments can either be phrased in terms of induction or by degenerating immediately to a maximally degenerate case (e.g., a comb curve, or a chain of elliptic curves), and are essentially the same either way. But the point either way is that the machinery reduces to studying linear series on the individual components of the degeneration, which have smaller genus.
