The embedding theorem in Hurwitz–Brill–Noether theory

We generalize the Embedding Theorem of Eisenbud–Harris from classical Brill–Noether theory to the setting of Hurwitz–Brill–Noether theory. More precisely, in classical Brill–Noether theory, the embedding theorem states that a general linear series of degree d d and rank r r on a general curve of genus g g is an embedding if r ≥ 3 r ≥ 3 . If f : C → P 1 f C → P^1 is a general cover of degree k k , and L L is a line bundle on C C , recent work of the authors shows that the splitting type of f ∗ L f_* L provides the appropriate generalization of the pair ( r , d ) (r, d) in classical Brill–Noether theory (see K. Cook-Powell and D. Jensen [Michigan Math. J. 71 (2022), pp. 19–45]; K. Cook-Powell and D. Jensen [Adv. Math. 398 (2022)]; E. Larson, H. Larson, and I. Vogt [Geom. Topol. 29 (2025), pp. 193–257]; and H. K. Larson [Invent. Math. 224 (2021), pp. 767–790]). In the context of Hurwitz–Brill–Noether theory, the condition r ≥ 3 r ≥ 3 is no longer sufficient to guarantee that a general such linear series is an embedding. We show that the additional condition needed to guarantee that a general linear series | L | |L| is an embedding is that the splitting type of <mml:math xmlns:mml="http://www.