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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