= Cohomology of a projective plane hypersurface
{title2=$h^0(\mathcal O_X)=1,\quad h^1(\mathcal O_X)=(d-1)(d-2)/2$}
For a nonzero homogeneous polynomial of degree $d>0$, multiplication gives $0\to\mathcal O_{\mathbb P^2}(-d)\to\mathcal O_{\mathbb P^2}\to i_*\mathcal O_X\to0$. Intermediate cohomology of the twisting sheaves vanishes. The top cohomology of $\mathcal O(-d)$ has Laurent basis $t_0^{-b_0}t_1^{-b_1}t_2^{-b_2}$ with $b_i\geq1$ and $\sum b_i=d$. There are $(d-1)(d-2)/2$ such monomials, with zero basis for $d=1,2$. The long exact sequence gives the displayed dimensions. The calculation allows singular, reducible and nonreduced hypersurfaces; it is an arithmetic-genus calculation, not an assertion about a normalization's genus.
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