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- W4361194114 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 is at least 3. If (f colon C to mathbb{P}^1) 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. In the context of Hurwitz-Brill-Noether theory, the condition that r is at least 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 (mathbb{P}^r)." @default.
- W4361194114 created "2023-03-31" @default.
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- W4361194114 date "2023-03-27" @default.
- W4361194114 modified "2023-10-12" @default.
- W4361194114 title "The embedding theorem in Hurwitz-Brill-Noether Theory" @default.
- W4361194114 doi "https://doi.org/10.48550/arxiv.2303.15189" @default.
- W4361194114 hasPublicationYear "2023" @default.
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