Let $X$ be a smooth projective curve of genus $g$ over an algebraically closed field $k$. As a corollary of the Riemann-Roch theorem we know that for every abelian differential $\omega$ on $X$ we have ...
Let $P$ be a point on a Riemann surface. Does there exist a non-trivial differential form $\omega$ on $X$ such that $\omega$ vanishes at $P$? Does there exist a non-constant rational function $f$ on ...
The following is a problem in Miranda's Algebraic Curves and Riemann Surfaces. Given any algebraic curve $X$ and a point $p \in X$, show that there is a meromorphic $1$-form $\omega$ on $X$ whose ...