the special fibre. Such a property is only possible if ${{\mathscr{F}}}=\sum{{\mathscr{O}}}_Ys_i$. The proof is then complete once we note that, for a $y\in  Y$, the $n$ generators $\{s_{i,y}\}_{i=1}^n$ of ${{\mathscr{F}}}_y\simeq{{\mathscr{O}}}_y^{{\oplus}n}$ do not admit any non-trivial relation.

In organising our findings, we shall make repeated use of a certain class of morphisms of proper schemes. In order to avoid repetitions and to serve as a reference for the reader, we put forward:

\[26.06.2018–1\]Let $X$ be a connected, proper and flat $A$-scheme carrying an $A$-point $x_0$. Let ${\mathfrak{S}}(X,x_0)$ (or simply ${\mathfrak{S}}$ if context prevents any misunderstanding) be the set of all $X$-schemes ${\varphi}:Y\to X$ such that:

1.  ${\varphi}$ is proper and surjective.

2.  $Y$ is $H^0$-flat over $A$.

3.  The canonical arrow $Y\to{\mathrm{Spec}\,}{H^0({{\mathscr{O}}}_Y)}$ admits a section $y_0$ such that ${\varphi}y_0$ extends $x_0$. (There exists an $H^0({{\mathscr{O}}}_Y)$-point in $Y$ above $x_0$.)

We shall denote by ${\mathfrak{S}}^+(X,x_0)$ (or simply ${\mathfrak{S}}^+$) the subset of morphisms ${\varphi}:Y\to X$ in ${\mathfrak{S}}$ which, in addition, satisfy

1.  The canonical morphism $Y\to{\mathrm{Spec}\,}H^0({{\mathscr{O}}}_Y)$ is flat.

\[05.02.2019–1\]Let ${\varphi}:Y\to X$ belong to ${\mathfrak{S}}(X,x_0)$ as in Definition \[26.06.2018–1\]; it turns out that in this case, connectedness of $X$ is automatic because of ${\mathfrak{S}}3$. Indeed, let $y_0$ be the alluded $H^0({{\mathscr{O}}}_Y)$-point of $Y$ above 