Linear series with ρ < 0 via thrifty Lego building.

The moduli space G g , d r → M g parameterizing algebraic curves with a linear series of degree 푑 and rank 푟 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 ≥ − C r ⁢ g + O ⁢ (g 5 / 6) , where C r is a constant depending on the rank of the linear series such that C r → 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. [ABSTRACT FROM AUTHOR]