$X$ is compact closed riemann surface and $\Omega(X)$ space of homolomorphic 1-forms over $X$, $H_a=\{\omega\in\Omega(X):\omega(a)=0\}$. Then $\dim(H_a)$ is either $g$ or $g-1$.
$\textbf{Q1:}$Why $\dim(H_a)$ is either $g$ or $g-1$? I tried riemann roch by considering the following. $\Omega(X)\cong O_{K_X}$ where $K_X$ is any canonical divisor. Then $H_a$ is the subspace s.t. vanishing at $a$ at least once. So $H_a=O(K_X-a)$. However $l(a)=\deg(a)+1-g+l(K_x-a)=2-g+l(K_x-a)$. Then I do not know what is $l(a)$ and $l(K_x-a)$.
Ref: Forster, Riemann Surface on Jacobi Inversion Problem.