= Solution
Put $P=\mathbb P^2_k$ and $S=k[t_0,t_1,t_2]$. A homogeneous polynomial $F$ of positive degree $d$ is nonzero, and $S$ is an <integral domain>. Multiplication by $F$ therefore gives an exact sequence of <coherent sheaves>
$$
0\longrightarrow\mathcal O_P(-d)\xrightarrow{\ F\ }\mathcal O_P\longrightarrow i_*\mathcal O_X\longrightarrow0,
$$
where $i:X\hookrightarrow P$ is the <closed immersion>.
We compute the necessary <cohomology of twisting sheaves on projective space>. The standard three <affine open subsets> $D_+(t_i)$ and all their intersections are <affine schemes>. For a <quasi-coherent sheaf>, <vanishing of quasi-coherent cohomology on an affine scheme> and the <acyclic cover theorem> identify its <sheaf cohomology> with the <Čech cohomology> of this cover. In degree $m$, the <Čech cochain complex> is
$$
\bigoplus_i(S_{t_i})_m\longrightarrow\bigoplus_{i<j}(S_{t_it_j})_m\longrightarrow(S_{t_0t_1t_2})_m.
$$
Decompose it into subcomplexes indexed by Laurent monomials $t_0^{a_0}t_1^{a_1}t_2^{a_2}$ with $a_0+a_1+a_2=m$. Let $N=\{i:a_i<0\}$. Such a monomial occurs in the summand for a nonempty index set $J$ precisely when $N\subseteq J$, since only the variables in $J$ are inverted.
If $N$ is empty, the monomial contributes the cochain complex of a full two-simplex: its degree-zero cohomology is $k$ and its higher cohomology vanishes. If $|N|=1$, the dimensions in degrees $0,1,2$ are $1,2,1$; the first map has rank one and the second map has rank one with the same kernel as the first map's image. If $|N|=2$, there is one copy of $k$ in each of degrees $1$ and $2$ and the map between them is an isomorphism. These two cases are acyclic over every <field>, including characteristic two. If $N=\{0,1,2\}$, only the degree-two term occurs, contributing one copy of $k$. It follows that
$$
H^1(P,\mathcal O_P(m))=0\quad\text{for every }m,
$$
and that $H^0(P,\mathcal O_P)=k$, $H^0(P,\mathcal O_P(-d))=0$, and $H^2(P,\mathcal O_P)=0$. Moreover the <Laurent-monomial description of top cohomology on projective space> gives a basis of $H^2(P,\mathcal O_P(-d))$ consisting of
$$
t_0^{-b_0}t_1^{-b_1}t_2^{-b_2},\qquad b_i\geq1,\quad b_0+b_1+b_2=d.
$$
For $d\geq3$, subtracting one from each $b_i$ counts these as $\binom{d-1}{2}$. For $d=1,2$ there are no such triples, and the formula $(d-1)(d-2)/2$ is also zero.
Use the <long exact sequence in sheaf cohomology> of the initial sequence, and <cohomology under a closed immersion> to identify the terms for $i_*\mathcal O_X$. The vanishings just proved give
$$
H^0(X,\mathcal O_X)\simeq k,\qquad H^1(X,\mathcal O_X)\simeq H^2(P,\mathcal O_P(-d)).
$$
Therefore
$$
\boxed{\dim_kH^0(X,\mathcal O_X)=1,\qquad\dim_kH^1(X,\mathcal O_X)=\frac{(d-1)(d-2)}2.}
$$
This proves the <cohomology of a projective plane hypersurface> without imposing smoothness, irreducibility or reducedness. In particular, the second number is an <arithmetic genus>, not a claim about the genus of a normalization.
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