Suppose $C$ is a smooth compact complex curve, and let $J$ be its Jacobian. Then there is a theta divisor $\Theta$ in $J$. Such divisor depend on a point $p\in C$.

Question. How to calculate the dimension of the set of divisors on $J$ linearly equivalent to $\Theta$? In other words, what is $dim( H^0(J,\cal O(\Theta)))$?

Suppose $C_g$ is a smooth compact complex curve (of genus $g$), and let $J$ be its Jacobian. Recall that the Jacobian $J$ of a curve $C_g$ is a complex torus that can by obtained by contractions of all rational curves on the $g$-th symmetric power of $C_g$, e.i., $Sym^g(C_g)$. Recall also that there is a theta divisor $\Theta$ in $J$, depending on a point $p\in C_g$. The divisor $\Theta$ is the image in $J$ of the set of points $(p,p_1,...,p_{g-1})$ with $p$ fixed.

Question. How to calculate the dimension of the set of divisors on $J$ linearly equivalent to $\Theta$? In other words, what is $dim( H^0(J,\cal O(\Theta)))$?