  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 \(\mathbb{P}^r\).
