 Holomorphic differentials on a smooth plane curve

Let $C\subset {\mathbb{CP}}^2$ be a nonsingular projective plane curve of degree $d$, and let $f(x,y)=0$ be its equation in an affine chart. For $d\ge 3$, its holomorphic one-forms have the basisThe projective plane model itself must be nonsingular; a singular plane curve with a smooth normalization need not have this basis.

The adjunction formula gives $K_C\cong {\mathcal{O}}_C(d-3)$. Its global sections are restrictions of homogeneous polynomials of degree $d-3$, and the Poincaré residue turns their affine expressions into the displayed differentials. Their number is $(d-1)(d-2)/2$, the genus of the smooth degree-$d$ plane curve. Where $f_y=0$, usewhich follows from $f_{x}dx+f_{y}dy=0$. Nonsingularity makes one denominator nonzero; the projective residue construction also ensures regularity at infinity.

For the smooth quartic $x^4+y^4=1$, the three forms are $dx/(4y^3)$, $xdx/(4y^3)$ and $ydx/(4y^3)$. Their apparent singularities at $y=0$ disappear in the equivalent $-dy/(4x^3)$ expression. They give the expected genus-three space of holomorphic one-forms.