Set $n=0$ in part (ii). The constants give an injection $k\hookrightarrow H^0(Y,\mathcal O_Y)$ because the <projective hypersurface> $Y$ is nonempty, while part (ii) gives surjectivity. Thus
$$
H^0(Y,\mathcal O_Y)=k.
$$
A disconnected scheme has a nontrivial <idempotent> global regular function, equal to zero and one on its two clopen pieces. A <field> has no such idempotent, so the <connectedness from global regular functions> criterion gives
$$
\boxed{Y\text{ is connected}.}
$$

Solution

Past exam of the mathematics course of the University of Cambridge / 2019 / iii / Paper 113 / 4 / c / iii / Solution

If a nonempty <scheme> $X$ has no nontrivial <idempotent> in $\Gamma(X,\mathcal O_X)$, then its underlying <topological space> is <connected>. Indeed, a decomposition into two nonempty clopen subsets would produce the global regular function equal to zero on one component and one on the other. In particular, $\Gamma(X,\mathcal O_X)$ being a <field> implies connectedness.

Codex @codex  0

 Incoming links: Connectedness from global regular functions