Blowing up the base ideal resolves a rational map to projective space

Trivialize $L$ by $e$ on an affine $U=\mathrm{Spec}A$. Write $s_i=h_{i}e$ and clear denominators by $a\in A\setminus \{0\}$, obtaining $p_i=a{h}_i\in A$. Then $I{\mathrm{\mid }}_U=a^{-1}J$ with $J=(p_i)$, so it is quasi-coherent of finite type. Its powers are also quasi-coherent. Degreewise multiplication by $a^d$ identifies its Rees algebra with $R(J)$. Changing $a$ or the frame rescales these maps in each degree, inducing the same degree-zero ratio maps on Proj. Thus the relative Proj is defined and locally the ordinary blowup of $J$; $J\mathrm{\ne }0$, so $B$ is integral.

The extended fractional ideal $I{\mathcal{O}}_B=a^{-1}(J{\mathcal{O}}_B)$ is invertible by [F3]. The sections ${\pi }^{\ast }{s}_i$ lie in $M={\pi }^{\ast }L\otimes (I{\mathcal{O}}_B)$ and generate it. If $b$ generates $J{\mathcal{O}}_B$ on a local chart, the frame of $M$ is ${\pi }^{\ast }e\otimes (b\mathrm{/}a)$, and the coefficients of these sections are $p_i\mathrm{/}b$, which are regular and generate the unit ideal. Therefore [F4] gives $\psi :B\to {\mathbb{P}}_k^n$, with coordinates $p_i\mathrm{/}{p}_j=h_i\mathrm{/}{h}_j$ on chart $j$, resolving $\phi$.

Locally on $X$, [F6] makes $(\pi ,\psi )$ a closed immersion with chart rings $A[J\mathrm{/}{p}_j]$. These immersions agree on overlaps because their coordinate ratios agree, hence glue. On the nonempty open where a nonzero $p_j$ is invertible, it is the graph of $\phi$; this open is dense in the corresponding domain chart by [F5]. Consequently $B$ is the schematic closure of that graph: a function on a domain chart vanishing on this dense principal open is zero. For any representative ${\phi }_D$ the graph over $D$ is closed in $D\times {\mathbb{P}}_k^n$ by separatedness, and its nonempty dense part is the same graph just considered. The closure restricted to $D$ is therefore exactly that graph, with its reduced scheme structure.

Let $f,\theta ,D$ satisfy the stated hypotheses. The nonempty open $f^{-1}(D)$ is dense in the integral scheme $Y$, and $(f,\theta )$ lands in $B$ there by step 3.1. The pullback of the ideal of the closed immersion $B\hookrightarrow X\times {\mathbb{P}}_k^n$ vanishes on this dense open. It vanishes everywhere: on each nonempty affine open of an integral scheme, a regular function zero on a dense open is zero in its domain coordinate ring. Thus $(f,\theta )$ factors through $B$. Any other $X$-lift has the same projective component on $f^{-1}(D)$, since $B$ over $D$ is the graph. Two maps from a reduced integral scheme to separated projective space agreeing on a dense open agree everywhere, by the same ideal-vanishing argument applied to the diagonal. Hence the two lifts agree, as $B$ is a closed subscheme of the product.