Let
be a curve
of genus
.
A fundamental problem in the theory of algebraic curves is to understand maps
of specified
degree
.
When
is general, the moduli space of such maps is well understood by the main theorems of
Brill–Noether theory. Despite much study over the past three decades, a similarly
complete picture has proved elusive for curves of fixed gonality. Here we
complete such a picture, by proving analogs of all of the main theorems of
Brill–Noether theory in this setting. As a corollary, we prove a conjecture of
Eisenbud and Schreyer regarding versal deformation spaces of vector bundles
on .
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