Lemma 37.46.1. Consider a diagram of morphisms of schemes
\[ \xymatrix{ Z \ar[r]_{\sigma } \ar[rd] & X \ar[d] \\ & Y } \]
an a point $y \in Y$. Assume
$X \to Y$ is of finite presentation and flat,
$Z \to Y$ is finite locally free,
$Z_ y \not= \emptyset $,
all fibres of $X \to Y$ are geometrically reduced, and
$X_ y$ is geometrically connected over $\kappa (y)$.
Then there exists a quasi-compact open $X^0 \subset X$ such that $X^0_ y = X_ y$ and such that all nonempty fibres of $X^0 \to Y$ are geometrically connected.
Proof.
In this proof we will use that flat, finite presentation, finite locally free are properties that are preserved under base change and composition. We will also use that a finite locally free morphism is both open and closed. You can find these facts as Morphisms, Lemmas 29.25.8, 29.21.4, 29.48.4, 29.25.6, 29.21.3, 29.48.3, 29.25.10, and 29.44.11.
Note that $X_ Z \to Z$ is flat morphism of finite presentation which has a section $s$ coming from $\sigma $. Let $X_ Z^0$ denote the subset of $X_ Z$ defined in Situation 37.29.1. By Lemma 37.29.6 it is an open subset of $X_ Z$.
The pullback $X_{Z \times _ Y Z}$ of $X$ to $Z \times _ Y Z$ comes equipped with two sections $s_0, s_1$, namely the base changes of $s$ by $\text{pr}_0, \text{pr}_1 : Z \times _ Y Z \to Z$. The construction of Situation 37.29.1 gives two subsets $(X_{Z \times _ Y Z})_{s_0}^0$ and $(X_{Z \times _ Y Z})_{s_1}^0$. By Lemma 37.29.2 these are the inverse images of $X_ Z^0$ under the morphisms $1_ X \times \text{pr}_0, 1_ X \times \text{pr}_1 : X_{Z \times _ Y Z} \to X_ Z$. In particular these subsets are open.
Let $(Z \times _ Y Z)_ y = \{ z_1, \ldots , z_ n\} $. As $X_ y$ is geometrically connected, we see that the fibres of $(X_{Z \times _ Y Z})_{s_0}^0$ and $(X_{Z \times _ Y Z})_{s_1}^0$ over each $z_ i$ agree (being equal to the whole fibre). Another way to say this is that
\[ s_0(z_ i) \in (X_{Z \times _ Y Z})_{s_1}^0 \quad \text{and}\quad s_1(z_ i) \in (X_{Z \times _ Y Z})_{s_0}^0. \]
Since the sets $(X_{Z \times _ Y Z})_{s_0}^0$ and $(X_{Z \times _ Y Z})_{s_1}^0$ are open in $X_{Z \times _ Y Z}$ there exists an open neighbourhood $W \subset Z \times _ Y Z$ of $(Z \times _ Y Z)_ y$ such that
\[ s_0(W) \subset (X_{Z \times _ Y Z})_{s_1}^0 \quad \text{and}\quad s_1(W) \subset (X_{Z \times _ Y Z})_{s_0}^0. \]
Then it follows directly from the construction in Situation 37.29.1 that
\[ p^{-1}(W) \cap (X_{Z \times _ Y Z})_{s_0}^0 = p^{-1}(W) \cap (X_{Z \times _ Y Z})_{s_1}^0 \]
where $p : X_{Z \times _ Y Z} \to Z \times _ W Z$ is the projection. Because $Z \times _ Y Z \to Y$ is finite locally free, hence open and closed, there exists an affine open neighbourhood $V \subset Y$ of $y$ such that $q^{-1}(V) \subset W$, where $q : Z \times _ Y Z \to Y$ is the structure morphism. To prove the lemma we may replace $Y$ by $V$. After we do this we see that $X_ Z^0 \subset Y_ Z$ is an open such that
\[ (1_ X \times \text{pr}_0)^{-1}(X_ Z^0) = (1_ X \times \text{pr}_1)^{-1}(X_ Z^0). \]
This means that the image $X^0 \subset X$ of $X_ Z^0$ is an open such that $(X_ Z \to X)^{-1}(X^0) = X_ Z^0$, see Descent, Lemma 35.13.6. Finally, $X^0$ is quasi-compact because $X_ Z^0$ is quasi-compact by Lemma 37.29.4 (use that at this point $Y$ is affine, hence $X$ is quasi-compact and quasi-separated, hence locally constructible is the same as constructible and in particular quasi-compact; details omitted). In this way we see that $X^0$ has all the desired properties.
$\square$
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