0
votes
0answers
51 views

How can we compute the order of 1-form on Riemann surfaces

Let X be a hyperellictic curve defined by $y^2=h(x)$. Let $\pi:X\rightarrow\mathbb{P}^1$ be the double covering map seding $(x,y)$ to $x$. Let $\omega=\pi^*(dx/h(x))$. How can we compute the orders of ...
3
votes
2answers
55 views

Poles of abelian differentials

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 ...
4
votes
2answers
147 views

zeroes of forms on Riemann surfaces

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 ...
5
votes
1answer
106 views

How to prove that this kind of differential form exists on an algebraic curve?

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 ...