Holomorphic differential form

On a Riemann surface, a holomorphic differential form is a holomorphic section of the canonical bundle. In a local complex coordinate $z$ it has the form $f(z)dz$ with $f$ holomorphic.

If $f(x,y)=0$ has a smooth projective completion of degree $d$ and $f_y\ne 0$, then its holomorphic differentials have basisThe equality $f_{x}dx+f_{y}dy=0$ gives the equivalent expressions $-x^i{y}^{j}dy/f_x$.