Picard representation by generic quotient and translates

Let $V$ be the image of the saturated open $W$ of the divisor chart in $\mathcal{P}$; by [F2] $V$ is an open subfunctor, not merely a set of geometric classes. For a test $T\to \mathcal{P}$, restrict to the open $T^{+}$ on which the pullback has the fixed Hilbert polynomial and the finite positive-regularity conditions of the chart; polynomial local constancy and the finite-cohomology vanishing conditions make $T^{+}$ open, and finite-presentation descent handles arbitrary tests. Over $T^{+}$ the divisor map is the faithfully flat open projective-space bundle of sections of [F2], and saturation makes the preimage of $W$ invariant under its kernel pair, so it descends to an open of $T^{+}$ and hence of $T$; on that open the pullback is represented by the quotient $Y{\times }_{\mathcal{P}}T$. The fppf sheaf quotient equality identifies $V$ with $Y$, giving an open immersion of represented functors $V\hookrightarrow \mathcal{P}$.

Translate $V$ by every rigidified class over $k$, i.e. consider the subfunctors $V\cdot x$ for $x\in \mathcal{P}(k)$; every $k$-valued class lies in such a translate because choosing $v\in V(k)$ (nonempty since $V$ is a nonempty locally finite-type open) gives $x=(x-v)+v$. For an arbitrary test, pull the union of the translates back to each finite-type positive divisor chart of [F2] for every polynomial and sufficiently high twist: this pullback is open and contains every closed point of the chart, since closed points have residue field $k$; its closed complement is therefore empty by the Nullstellensatz [F3]. The positive divisor charts, with twists reversed, cover $\mathcal{P}$ fppf-locally on every test: locally a sufficiently positive twist has locally free nonzero sections, and a fibrewise nonzero section exists after the projective-space cover of [F2]. Hence every test pulls back to the union of the translates.

The represented open overlaps of the translates with identity transition maps glue along the open cover of step 2.1 to a scheme representing the full functor $\mathcal{P}$ on all tests, including nonreduced and non-Noetherian tests, by the gluing and descent statements of [F3]; the universal rigidified invertible sheaf is obtained by gluing the universal bundles of the chart quotients, and separatedness was proved on the divisor charts in [F2]. This chart argument does not assume that every geometric class descends to $k$.