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

Abstract

Abstract The moduli space G g , d r → M g \mathcal{G}^{r}_{g,\smash{d}}\to\mathcal{M}_{g} parameterizing algebraic curves with a linear series of degree 𝑑 and rank 𝑟 has expected relative dimension ρ = g − ( r + 1 ) ⁢ ( g − d + r ) \rho=g-(r+1)(g-d+r) . Classical Brill–Noether theory concerns the case ρ ≥ 0 \rho\geq 0 ; we consider the non-surjective case ρ < 0 \rho<0 . We prove the existence of components of this moduli space with the expected relative dimension when 0 > ρ ≥ − g + 3 0>\rho\geq-g+3 , or 0 > ρ ≥ − C r ⁢ g + O ⁢ ( g 5 / 6 ) 0>\rho\geq-C_{r}g+\mathcal{O}(g^{5/6}) , where C r C_{r} is a constant depending on the rank of the linear series such that C r → 3 C_{r}\to 3 as r → ∞ 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.