Brill-Noether theory studies the existence and deformations of curves in
projective spaces; its basic object of study is Wd,grβ, the
moduli space of smooth genus g curves with a choice of degree d line bundle
having at least (r+1) independent global sections. The Brill-Noether theorem
asserts that the map Wd,grββMgβ is
surjective with general fiber dimension given by the number Ο=gβ(r+1)(gβd+r), under the hypothesis that 0β€Οβ€g. One may
naturally conjecture that for Ο<0, this map is generically finite onto a
subvariety of codimension βΟ in Mgβ. This conjecture fails in
general, but seemingly only when βΟ is large compared to g. This paper
proves that this conjecture does hold for at least one irreducible component of
Wd,grβ, under the hypothesis that 0<βΟβ€r+2rβgβ3r+3. We conjecture that this result should hold for all 0<βΟβ€g+C for some constant C, and we give a purely combinatorial conjecture that
would imply this stronger result.Comment: 16 page