Q:

Differential Form on a Riemann Surface

The following problem is basically from Miranda's "Algebraic Curves and Riemann Surfaces", which I am reading on my own; if there are any rules against posting textbook problems, my apologies!
Let $X$ be a smooth projective curve defined by the homogeneous polynomial $F(x,y,z)=0$, with $\deg F = d \geq 3$. Let $f(x,y) = F(x,y,1)$. Show that if $p(u,v)$ is a polynomial of degree at most $d-3$, then $p(u,v) \frac{du}{\partial f/ \partial v}$ defines a holomorphic 1-form on the compact Riemann surface X. If $X$ is not smooth, but has nodes, then this form is a holomorphic 1-form on the resolution. 
I see that this is a holomorphic $1$-form on the affine curve defined by $f$, since the charts are just projection to the $x$ or $y$ coordinate; in the former case the form is evidently holomorphic and in the latter case the form transforms to $p(u,v) \frac{dv}{\partial f/ \partial u}$. However, I'm a bit confused as to the computations involved in checking this on the other affine curves, and what extra argument is needed for the nodes case. 

A:

You have two questions: how to change variables, and how to handle nodes.  The case of changing variables, for hyperelliptic curves without nodes, is in chapter III.5.5 of Shafarevich's Basic algebraic geometry.  The general case is on page 105 of Phillip Griffiths' Introduction to algebraic curves (China notes).  The discussion of the nodes there needs supplementing with the argument from page 98, i.e. a proof that the derivative vanishes simply on both the separate branches lying over the node, rather than at the point in the plane, as suggested there in a footnote.  Basically a fraction is holomorphic if the numerator vanishes as much as the denominator.  Here the equation for the curve vanishes twice at the node so the derivative in the denominator vanishes once and is canceled by the vanishing of the adjoint polynomial.  But you still have to finesse the point raised in Griffiths' footnote as mentioned above about the order of vanishing of the pullback to the normalization.  Warm up on some specific examples.

