Linear series with ρ < 0 \rho<0 via thrifty Lego building

The moduli space parameterizing algebraic curves with a linear series of degree d and rank r has expected relative dimension . Classical Brill–Noether theory concerns the case ; we consider the non-surjective case . We prove the existence of components of this moduli space with the expected relative dimension when , or , where is a constant depending on the rank of the linear series such that as . These results are proved via a two-marked-point generalization suitable for inductive arguments, and the regeneration theorem for limit linear series.