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The moduli spaceGg,dr→Mg\mathcal{G}^{r}_{g,\smash{d}}\to\mathcal{M}_{g}parameterizing algebraic curves with a linear series of degree  and rank  has expected relative dimensionρ=g−(r+1)⁢(g−d+r)\rho=g-(r+1)(g-d+r). Classical Brill–Noether theory concerns the caseρ≥0\rho\geq 0; we consider the non-surjective caseρ&lt;0\rho&lt;0. We prove the existence of components of this moduli space with the expected relative dimension when0&gt;ρ≥−g+30&gt;\rho\geq-g+3, or0&gt;ρ≥−Cr⁢g+O⁢(g5/6)0&gt;\rho\geq-C_{r}g+\mathcal{O}(g^{5/6}), whereCrC_{r}is a constant depending on the rank of the linear series such thatCr→3C_{r}\to 3asr→∞r\to\infty. These results are proved via a two-marked-point generalization suitable for inductive arguments, and the regeneration theorem for limit linear series.

journal für die reine und angewandte mathematik (crelles journal)

Linear series with                                                                                                    ρ                              &lt;                              0                                                                                                 \rho&lt;0                                       via thrifty Lego building