Brill-Noether theory studies the existence and deformations of curves in projective spaces; its basic object of study is W(g,r,d), the moduli space of smooth projective genus g curves with a choice of degree d line bundle having at least (r+1) independent global sections. The geometry of W(g,r,d) depends on the number ρ = g-(r+1)(g-d+r). The Brill-Noether theorem, proved by Griffiths and Harris, states that when ρ is nonnegative, the map from W(g,r,d) to M_g is surjective, and a general fiber has dimension ρ. One may naturally conjecture that for ρ < 0, W(g,r,d) is finite over a locus of codimension -ρ in M_g. This conjecture fails, but seemingly only when -ρ is large compared to g. I discuss a proof that this conjecture holds for at least one component of W(g,r,d) in cases where 0 < -ρ < r/(r+2) g - 3r.
Tuesday, January 28, at 4:00pm in HBH 227
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