Brill–Noether theory

Remarks

The locus $W^r_d(X)$ is a closed subvariety of $\operatorname{Pic}^d(X)$, which is a copy of the Jacobian, of dimension $g$. The number $\rho$ is the expected dimension of $W^r_d(X)$: it is $g$ minus the number of conditions $(r + 1)(g - d + r)$ that the existence of $r + 1$ sections imposes, as predicted by Riemann–Roch. Existence theorem (Kempf, Kleiman–Laksov): if $\rho \ge 0$, then $W^r_d(X)$ is nonempty for every curve $X$. Griffiths–Harris theorem: for a general curve $X$ of genus $g$, $W^r_d(X)$ is empty if $\rho < 0$, and it has dimension $\rho$ if $0 \le \rho \le g$. Example: for $g = 3$, $r = 1$, $d = 2$, one gets $\rho = 3 - 2 \cdot 2 = -1$, so a general curve of genus $3$ has no pencil of degree $2$ and is not hyperelliptic. Example: for $r = 0$ and $0 \le d \le g$, one gets $\rho = d$, and $W^0_d(X)$ is the image of the $d$-fold symmetric product of $X$, of dimension $d$.