= Solution
Let the degree-$d$ <projective hypersurface> $X_d$ be defined by the homogeneous polynomial $F$. Multiplication by $F$ identifies its ideal sheaf with $\mathcal O_{\mathbb P^n}(-d)$, giving the <ideal-sheaf sequence of a projective hypersurface>
$$
0\longrightarrow\mathcal O_{\mathbb P^n}(-d)
\xrightarrow{\cdot F}\mathcal O_{\mathbb P^n}
\longrightarrow i_*\mathcal O_{X_d}\longrightarrow0.
$$
In particular, the kernel in the question is $\mathcal O_{\mathbb P^n}(-d)$.
For $n=2$, additivity of the <Euler characteristic> in this short exact sequence and the line-bundle formula
$$
\chi(\mathbb P^2,\mathcal O(m))=\binom{m+2}{2}
$$
give
$$
h^0(X_d,\mathcal O_{X_d})-h^1(X_d,\mathcal O_{X_d})
=\chi(\mathcal O_{X_d})
=1-\binom{2-d}{2}
=1-\frac{(d-1)(d-2)}2.
$$
Here $H^2(X_d,\mathcal O_{X_d})=0$ by the preceding cohomological-dimension argument. Moreover $H^0(\mathbb P^2,\mathcal O(-d))=H^1(\mathbb P^2,\mathcal O(-d))=0$ for $d>0$, so the long exact sequence gives $H^0(X_d,\mathcal O_{X_d})\cong k$. Consequently
$$
h^1(X_d,\mathcal O_{X_d})=\frac{(d-1)(d-2)}2,
$$
which is the <genus-degree formula> for a projective plane curve.
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