= Solution
Let the <smooth projective curve> be the <projective completion> of $f(x,y)=0$, of <degree of a polynomial>[degree] $d$, and suppose after interchanging $x$ and $y$ if necessary that $f_y=\partial f/\partial y$ is nonzero. A basis of its <holomorphic differential forms> is
$$
\left\{
x^iy^j\frac{dx}{f_y}:i,j\geq0,\ i+j\leq d-3
\right\}.
$$
Equivalently one may use $-x^iy^jdy/f_x$. These expressions agree on their common domain because differentiating $f(x,y)=0$ gives $f_xdx+f_ydy=0$. The allowed <monomials> are precisely the homogeneous polynomials of degree at most $d-3$, in accordance with the <adjunction formula> $K_C\cong\mathcal O_C(d-3)$.
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