(Brill--Noether number)
Let \(C\) be a smooth projective curve of genus \(g\) over an algebraically closed field, and let \(r \geq 0\) and \(d\) satisfy \(g - d + r \geq 0\). The locus \(W^r_d(C) \subseteq \Pic^d(C)\) of line bundles \(L\) with \(h^0(C, L) \geq r + 1\) is a closed subscheme, cut out locally as a degeneracy locus of a map of vector bundles. Its expected dimension is the Brill–Noether number \[\rho(g, r, d) = g - (r+1)(g - d + r).\] The curve \(C\) is Brill–Noether general if for every such \(r\) and \(d\), \(W^r_d(C)\) is empty when \(\rho(g,r,d) < 0\) and has dimension \(\rho(g,r,d)\) when \(\rho(g,r,d) \geq 0\).
(Brill--Noether)
Let \(g\), \(r\), \(d\) be as above.
If \(\rho(g,r,d) \geq 0\), then every smooth projective curve of genus \(g\) has a \(g^r_d\), and every component of \(W^r_d(C)\) has dimension at least \(\rho(g,r,d)\) (Kempf, Kleiman–Laksov).
A general curve of genus \(g\) is Brill–Noether general (Griffiths–Harris).
For a general curve and \(\rho(g,r,d) \geq 0\), \(W^r_d(C) \setminus W^{r+1}_d(C)\) is smooth of dimension \(\rho(g,r,d)\) (Gieseker–Petri).
A \(g^1_d\) has \(\rho(g,1,d) = 2d - g - 2\). So every curve of genus \(g\) has a map of degree \(\lceil (g+2)/2 \rceil\) to \(\PP^1\), and the general curve has no map of smaller degree: its gonality is \(\lfloor (g+3)/2 \rfloor\). For \(d = 2\), \(\rho = 2 - g\), which is negative for \(g \geq 3\); this is why every curve of genus \(2\) is hyperelliptic while the general curve of genus \(g \geq 3\) is not.
Dependencies
D-CRVGONLinear series $g^r_d$, and the gonality of a curveuses