= 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 $d\ge3$, its <holomorphic one-forms> have the <basis>
$$
x^iy^j\frac{dx}{f_y},
\qquad i,j\ge0,\quad i+j\le d-3.
$$
The 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 $f_y=0$, use
$$
\frac{dx}{f_y}=-\frac{dy}{f_x},
$$
which follows from $f_xdx+f_ydy=0$. 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 $dx/(4y^3)$, $x\,dx/(4y^3)$ and $y\,dx/(4y^3)$. 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>.
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