THE EMBEDDING THEOREM IN HURWITZ–BRILL–NOETHER THEORY

Authors

• Kaelin Cook-Powell, Upland Specialty Insurance CompanyFollow
• David Jensen, University of KentuckyFollow
• Eric Larson, Brown UniversityFollow
• Hannah Larson, University of California BerkeleyFollow
• Isabel Vogt, Brown UniversityFollow

Abstract

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 and rank r on a general curve of genus g is an embedding if r ≥ 3. If f ∶ C → ℙ¹ is a general cover of degree k, and L is a line bundle on C, recent work of the authors shows that the splitting type of f_∗ L provides the appropriate generalization of the pair (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 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∣ is an embedding is that the splitting type of f_∗ L has at least three nonnegative parts. This new extra condition reflects the unique geometry of k-gonal curves, which lie on scrolls in ℙ^r.

Document Type

Article

Publication Date

1-1-2026

Notes/Citation Information

Publisher Copyright: © 2026 by the author(s).

Digital Object Identifier (DOI)

10.1090/btran/232

Archival?

Archival

Repository Citation

Cook-Powell, Kaelin; Jensen, David; Larson, Eric; Larson, Hannah; and Vogt, Isabel, "THE EMBEDDING THEOREM IN HURWITZ–BRILL–NOETHER THEORY" (2026). Mathematics Faculty Publications. 58.
https://uknowledge.uky.edu/math_facpub/58