Kahler differentials on a smooth projective plane curve Let $C = \{f=0\} \subset \mathbb{P}_k^2$ be a smooth plane curve of degree $d$.  
I'm trying to find an explicit basis for $H^0(C,\Omega^1_{C/k})$. I know it should be $\frac{(d-1)(d-2)}{2}$ - dimensional. What I'm trying to find is an explicit collection of rational 1-forms on $C$ (basically elements in $k(C) \large{\frac{dx}{\partial_y f}}$) which when restricted to $C$ give a basis for all 1-forms. Sadly most of what I tried didn't get me anywhere and so I have no interesting attempts to share. Help would be really appreciated. 
 A: To find the basis explicitly you can use the Poincare residue map
$$
\text{res}: H^0(\mathbb{P}^2, \Omega^2_{\mathbb{P}^2}(C)) \to H^0(C, \Omega^1_C),
$$
which in this case is an isomorphism.
Let $g(x_1, x_2) = f(1,x_1,x_2)$, any 2-form $\omega$ on $\mathbb{P}^2$ with a single pole along $C$ can be written locally in coordinates $x_1$, $x_2$ as
$$
\omega = t(x_1, t_2) \frac{dx_1 \wedge dx_2}{g(x_1,x_2)},
$$
then the residue of $\omega$ is 
$$
\text{res}(\omega) = t(x_1, x_2) \frac{dx_2}{g_{x_1}(x_1,x_2)}.
$$
Section $dx_1 \wedge dx_2$ has a pole of order 3 on the hyperplane $H$ at infinity ($K_{\mathbb{P}^2} = -3H$), and $f$ has a pole of order $d$ along $H$, so $\omega$ is holomorphic when $t$ is a rational function with a possible pole of order $\leq d-3$ along $H$, such functions are polynomials in $x_1$ and $x_2$ of order $\leq d-3$. There are $l+1$ monomials of degree $l$ in two variables, so there are $\sum_{l=0}^{d-3} (l+1)=\frac{(d-1)(d-2)}{2}$ monomials of degree $\leq d-3$. The explicit basis of $H^0(C, \Omega^1_C)$ is given by residues of $\omega$'s with all monomials of such degrees.
