Linear series with $ρ< 0$ via thrifty lego-building

Abstract

The moduli space G r g,d → Mg parameterizing algebraic curves with a linear series of degree d and rank r has expected relative dimension ρ = g−(r+1)(g−d+r). Classical Brill-Noether theory concerns the case ρ ≥ 0; we consider the non-surjective case ρ < 0. We prove the existence of components of this moduli space with the expected relative dimension when 0 > ρ ≥ −g + 3, or 0 > ρ ≥ −Crg + O(g 5/6 ), where Cr is a constant depending on the rank of the linear series such that Cr → 3 as r → ∞. These results are proved via a two-marked-point generalization suitable for inductive arguments, and the regeneration theorem for limit linear series.