Hilbert divisor charts and the Picard diagonal

Fix the very ample $\mathcal{H}$ on $A$. The relative effective-divisor functor is an open subscheme of the Hilbert scheme by [F1]: being cut out by a fibrewise regular element with invertible ideal is open, and for a fixed Hilbert polynomial the divisor scheme $D$ is finite type and quasi-projective. Impose a fixed Castelnuovo-Mumford regularity bound by finitely many higher-cohomology vanishings with respect to $\mathcal{H}$; by [F2] these vanishings propagate to all nonnegative twists and are computed by the universal finite cohomology complex, so they cut out an open positive chart $D^{+}$; do not define openness by an infinite intersection of vanishings. Serre vanishing ensures that after twisting by a sufficiently high power of $\mathcal{H}$ every individual test family lies locally on its base in such a chart. A line bundle satisfying these conditions has finite locally free sections of positive rank, compatibly with every base change.

Given $c\in {\mathcal{P}}_{A\mathrm{/}k}(T)$, choose an fppf covering $T^{\mathrm{\prime }}\to T$ on which $c$ has a line-bundle representative $\mathcal{L}$. On $T^{\mathrm{\prime }}$ its divisor fibre is $\mathbf{P}((p_{\ast }\mathcal{L}{)}^{\vee })$: fibrewise nonzero sections, including twists by base line bundles, give exactly the relative effective Cartier divisors, since the geometric fibres are integral. Positivity makes $p_{\ast }\mathcal{L}$ locally free of positive rank and compatible with base change by step 1.1. The two pullbacks of these projective bundles have canonical identifications, because both represent the same divisor-class fibre functor; they satisfy the cocycle condition. These projective spaces descend to a scheme: locally their common rank is $r$, the canonical relative anticanonical bundle is $\mathcal{O}(r)$ and its section algebra and homogeneous embedding equations descend by faithfully flat module/algebra descent. Taking the descended relative Proj gives a projective scheme whose pullback is the projective bundle; the canonical identifications glue these schemes on $T$. Its smoothness, properness and surjectivity follow from the projective-space description after the cover (flatness descends and the geometric fibres are projective spaces). A representative on $T$ itself gives the displayed projective bundle directly. If $e$ exists, normalize representatives along $e$; scalar global functions make every rigidified isomorphism unique, so their descent data satisfy the cocycle condition and descend a line bundle on $A_T$. Without $e$ this last conclusion is not asserted. This proves (b) on arbitrary tests.

The linear-equivalence relation $R\subseteq D^{+}{\times }_k{D}^{+}$ is represented: two divisor families are equivalent when their classes agree, whose projections are smooth projective bundles: the test $D^{+}$ carries the universal divisor line bundle, so the represented-case clause of step 2.1 applies. It is a monomorphism into the square. Hence the generic quotient theorem [F3] applies and yields a nonempty saturated open $W\subseteq D^{+}$ with a quotient representing the corresponding open subfunctor of the Picard sheaf; all-test statements reduce to finitely generated test algebras by finite-presentation spreading.

First suppose a point $e\in A(k)$ is supplied. On a Noetherian product $T$ of divisor charts normalize $\mathcal{M}={\mathcal{L}}_1^{-1}\otimes {\mathcal{L}}_2$ along $e$. Apply [F2] to $\mathcal{M}$ and ${\mathcal{M}}^{-1}$, locally choosing finite free complexes in nonnegative degrees. Their degree-zero kernels represent sections on every test, and evaluation at $e$ is represented by linear chain maps to ${\mathcal{O}}_T$ (a finite free complex permits such a representative). Impose the finitely many linear equations $d(s)=0$, $d(t)=0$, $e(s)=e(t)=1$ in the product of the two degree-zero affine vector bundles. The product $st$ is a global function, hence a scalar by [F4], and its value at $e$ is $1$, so $st=1$. Thus this closed affine scheme represents exactly the unique rigidified isomorphism ${\mathcal{L}}_1\to {\mathcal{L}}_2$ when one exists. It represents the diagonal and is of finite presentation. For general $A$, pass to a finite separable extension having a rational point by [F4]. The affine representations of the equality-of-classes functor carry canonical descent data, even when the chosen rational points differ on overlaps; affine descent [F3] makes the diagonal affine over $T$. Finite-presentation spreading and fppf-local representatives extend this conclusion from chart products to arbitrary tests.

To check separatedness once representability holds, apply the valuative criterion. After a faithfully flat field extension a rational section is available, so normalize a line bundle $\mathcal{M}$ on $A_V$ for a DVR $V$ whose class is generically zero. Choose inverse generic trivializing sections. Proper coherent cohomology makes their section modules finite over $V$, and flatness of the line bundles injects them into the generic section spaces. Rescale each generic section by a power of the uniformizer until it extends and its reduction is nonzero: a finite torsion-free module over a DVR is a lattice, and the exact sequence for multiplication by the uniformizer makes the reduction map on global sections injective. On the integral special fibre the product of these two nonzero sections is nonzero. Their global product is a scalar by [F4], so it is a unit of $V$. The extended sections are therefore inverse trivializations after rescaling by that unit; normalization at the section makes the generic isomorphism extend uniquely. This proves the valuative criterion, and hence separatedness of the locally finite-type Picard representative.

Finally the universal Hilbert ideal is flat over its Noetherian chart base, and on a fibre where it is a line bundle the Noetherian fibrewise-flatness criterion makes it flat and the finite-flat local-freeness criterion makes it rank one; the locus is open, and proper projection of its failure gives the divisor open in the Hilbert chart. Arbitrary test families descend locally to finitely generated $k$-algebras by finite-presentation spreading of the line bundle and its inverse, so the representing opens and linear-system universal properties apply to every test.