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 P¹ 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.
Cook-Powell et al. (Mon,) studied this question.