Brill–Noether with ramification at unassigned points

We discuss, how via limit linear series and standard facts about divisors on moduli spaces of pointed curves, one can establish a non-existence Brill–Noether results for linear series with prescribed ramification at unassigned points. © 2013 Elsevier B.V. All rights reserved. In the course of developing their theory of limit linear series, among many other applications, Eisenbud and Harris [2,3] also considered the Brill–Noether problem with prescribed ramification at assigned points. For a smooth curve C of genus g , a point p ∈ C and a linear series l = (L, V ) ∈ Gd(C), one denotes by α(p) : 0 ≤ α 0(p) ≤ · · · ≤ α l r (p) ≤ d − r the ramification sequence of l at p. Having fixed points p1, . . . , pn ∈ C , Schubert indices ᾱj : 0 ≤ α j 0 ≤ · · · ≤ α j r ≤ d − r of type (r, d) for j = 1, . . . , n, the locus of linear series on C having prescribed ramification at p1, . . . , pn ∈ C , that is, Gd  C, (pj, ᾱj)  := {l ∈ Gd(C) : α (pj) ≥ ᾱj for j = 1, . . . , n} is a generalized determinantal variety of expected dimension ρ(g, r, d, ᾱ1, . . . , ᾱn) := ρ(g, r, d) − n 