Given a projective curve $C$ defined by the equation $X^5 + Y^5 +Z^5 = 0$, I would like to produce a global holomorphic 1-form $\omega$ such that for some fixed $P\in C$, I have $\operatorname{ord}_P(\omega) = 1$.
In Algebraic Curves and Riemann Surfaces, Miranda defines a gluing condition of locally defined $1$-forms $f(z)\,dz$ on $V_1$, $g(w)\,dw$ on $V_2$ by requiring that $$ g(w) = f(T(w))T'(w) $$ where $T: V_2 \to V_1, w\mapsto z$, which I believe would be relevant for my approach to this question, which I now outline:
Assuming $P = (a:b:1)\in C\cap U_2$, we have local coordinate $z$ defined by $$ (X:Y:Z)\mapsto \frac{X-aZ}{Z} $$ Then $f(z)\,dz = z\,dz = \left(\frac{X-aZ}{Z}\right)\,d\left(\frac{X-aZ}{Z}\right)$ is a local holomorphic 1-form on $C\cap U_2$, which has order $1$ at $P$. I was then hoping to produce a suitable local coordinate $w$ on $C\setminus (C\cap U_2) \subseteq C\cap U_0$ and define a holomorphic transformation $T$ from this new coordinate. However, the "obvious" (at least to me) candidate for $w$ is $\frac{Z}{X}$, which seems not to transform in a holomorphic way to $\frac{X-aZ}{Z}$ (since $C\cap U_0$ contains points of the form $(1:\zeta:0)$ for some $\zeta^5 = -1$).
Can someone please tell me whether my approach is totally wrong, or whether I should somehow be using the equation of the curve to find a clever change of coordinate?
This is somewhat redundant in the wake of Ted Shifrin's comment which provides quite a general approach for projective curves, but to take this off the unanswered list, I will post my attempt at a solution for this specific problem:
First, we assume that the point $P = (a:b:c)$ lies in the affine open $U_2$ (i.e. $c = 1$), so $Z\neq 0$ and take affine coordinates $x = \frac{X}{Z}, y = \frac{Y}{Z}$. The equation defining the curve becomes $$ x^5 + y^5 + 1 = 0 $$ Taking differentials gives $x^4dx + y^4dy = 0$, which allows us to define a holomorphic 1-form on $C\cap U_2$ by $$ \frac{dx}{y^4} = -\frac{dy}{x^4} $$ Furthermore, assuming we are working in the case $a\neq 0$ we can multiply this by $(x-a)$ to get a holomorphic 1-form on $C\cap U_2$ which has order 1 at $P$: \begin{equation}\label{holo-def} (x-a)\frac{dx}{y^4} = -(x-a)\frac{dy}{x^4} \end{equation} (If $a= 0$, we can just multiply by $(y-b)$ instead.) We still have to check that this 1-form is holomorphic outside $C\cap U_2$, i.e. on points where $Z = 0$. To check this, we work in the affine open $U_0$ and define new affine coordinates $s = \frac{Y}{X}, t = \frac{Z}{X}$. First note that we can connect the different local coordinates on the overlap $(C\cap U_2 )\cap (C\cap U_0)$: $$ x = \frac{1}{t} \text{ and } y = \frac{s}{t} $$ We can rewrite the 1-form above as $$ (x-a)\frac{dx}{y^4} = \left(\frac{1}{t} - a\right)\frac{d(\frac{1}{t})}{(\frac{s}{t})^4} = -t(1-at)\frac{dt}{s^4} $$ which is holomorphic on $C\cap U_0$ (except at points where $s = 0$, but we can apply the same trick to those as we did at points $y = 0$ on $C\cap U_2$).