= Solution
For $n\geq2$, the omitted point has codimension at least two in the normal integral scheme $\mathbb P_k^n$. Regular functions extend across such a subset, so
$$
\Gamma(X,\mathcal O_X)=\Gamma(\mathbb P_k^n,\mathcal O)=k.
$$
Thus
$$
\boxed{\pi_*\mathcal O_X\cong\widetilde{k}}
$$
on $\operatorname{Spec}k$, and it is a <coherent sheaf> because it corresponds to the one-dimensional $k$-vector space $k$.
The complement of a rational point in $\mathbb P_k^1$ is $\mathbb A_k^1$. Its regular functions are $k[t]$, which is infinite-dimensional over $k$. Its pushforward to $\operatorname{Spec}k$ is therefore quasi-coherent but not coherent.
For an example on $X$, choose a projective line $L\subset\mathbb P_k^n$ through $p$. Then
$$
C=L\setminus\{p\}\cong\mathbb A_k^1
$$
is closed in $X$. For the closed immersion $i:C\hookrightarrow X$, the sheaf $\mathcal F=i_*\mathcal O_C$ is coherent, but
$$
\Gamma(X,\mathcal F)=\Gamma(C,\mathcal O_C)=k[t].
$$
Consequently $\pi_*\mathcal F$ is not coherent on $\operatorname{Spec}k$.
Back to article page
