Set $\mathcal{F}_{m}:=\pi_{*}\mathcal{O}_{\mathcal{X}}(m\mathfrak{L})$, and pick a very ample line bundle $H$ on $B$. By the Castelnuovo-Mumford criterion [Laz04, Theorem 1.8.5] it is enough to show the existence of $m_{0}\in\mathbf{N}$ such that

(A.4)

$$H^{1}\left(B,\mathcal{O}_{B}(mm_{0}H)\otimes\mathcal{F}_{m}\right)=0$$

for all $m$ large and divisible.

Since $\mathfrak{L}$ is ample over the generic point of $B$, the $\mathcal{O}_{B}$-algebra $\bigoplus_{m\geq 0}\mathcal{F}_{m}$ is finitely generated at the generic point of $B$. After perhaps replacing $\mathfrak{L}$ by $d\mathfrak{L}$ for some $d\in\mathbf{N}$, we may further assume that the generators have degree $1$, so that $\mathcal{F}_{m}/\mathcal{F}_{1}^{m}$ has zero-dimensional support for all $m\geq 1$. As a consequence, the map

$$H^{1}\left(B,\mathcal{O}_{B}(mm_{0}H)\otimes\mathcal{F}_{1}^{m}\right)\to H^{1}\left(B,\mathcal{O}_{B}(mm_{0}H)\otimes\mathcal{F}_{m}\right)$$

is surjective for all $m\geq 1$. Upon replacing $\mathfrak{X}$ with $\Proj_{B}\left(\bigoplus_{m\geq 0}\mathcal{F}_{1}^{m}\right)$ we are thus reduced to proving (A.4) when $\mathfrak{L}$ is $\pi$-ample, i.e. ample on all fibers of $\pi$. In that case we have $R^{q}\pi_{*}\mathcal{O}_{\mathfrak{X}}(m\mathfrak{L})=0$ for $m\gg 1$ and $q>0$ by Serre vanishing, and the degeneration of the Leray spectral sequence yields

$$H^{1}\left(B,\mathcal{O}_{B}(mm_{0}H)\otimes\mathcal{F}_{m}\right)\simeq H^{1}\left(\mathfrak{X},\mathcal{O}_{\mathfrak{X}}(m(\mathfrak{L}+m_{0}\pi^{*}H)\right),$$

which vanishes for all $m\gg 1$ if we choose $m_{0}$ such that $\mathfrak{L}+m_{0}\pi^{*}H$ is ample on $\mathfrak{Y}$. ∎