PomeLemma 11.15.2.927EFEB9D586 For a vector bundle $\mathcal E$ on $X\times T$, the parameters $t$ for which $\mathcal E_ t$ is ample form a Zariski-open subset of $T$.
Lemma 11.15.2927EFEB9D586
Proof. Put $Y=\mathbb P_{X\times T}(\mathcal E^*)$, with the line convention, and let $p:Y\to T$ be the projection, a projective flat morphism. Write $\mathcal L=\mathcal O_ Y(1)$. If $\mathcal E_{t_0}$ is ample, then $\mathcal L_{t_0}$ is ample. Choose $k$ so that its $k$th power is very ample and has vanishing higher cohomology. By Serre vanishing, semicontinuity, and cohomology and base change, after shrinking $T$ around $t_0$ the sheaf $\mathcal V=p_*\mathcal L^ k$ is locally free and commutes with restriction to fibers. After a further shrinking, the evaluation map $p^*\mathcal V\to \mathcal L^ k$ is surjective: its cokernel vanishes on the central fiber, and the image of its support under the proper map $p$ is closed and misses $t_0$.
Evaluation gives a proper morphism
\[ g:Y\longrightarrow \mathbb P_ T(\mathcal V^*), \qquad g^*\mathcal O(1)=\mathcal L^ k, \]
whose restriction to the central fiber is a closed immersion. The locus where $g$ is quasi-finite is open; its complement has closed image in $T$ by properness and misses $t_0$, so after a further shrinking $g$ is proper and quasi-finite, hence finite. For a finite morphism, pushforward commutes with base change. The cokernel of
\[ \mathcal O_{\mathbb P_ T(\mathcal V^*)}\longrightarrow g_*\mathcal O_ Y \]
therefore vanishes on the central fiber, and its support projects to a closed subset of $T$ missing $t_0$. After removing that subset, the map is surjective, so $g$ is a closed immersion. Consequently $\mathcal L^ k$ is very ample on every fiber, and $\mathcal E_ t$ is ample there. $\square$
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