Relative very ampleness in the finite projective-space convention

Let $S$ be a scheme and let $n\ge 0$. Denote by $\pi :{\mathbb{P}}_S^n⟶S$ the relative projective space with its standard charts $U_0,\dots ,U_n$ and transition isomorphisms ${\theta }_{ij}:U_i\cap U_j\to U_j\cap U_i$ carrying the coordinates by $x_{\mathrm{\ell }}^{(i)}\mapsto x_{\mathrm{\ell }}^{(j)}\mathrm{/}{x}_i^{(j)}$ and $x_j^{(i)}\mapsto 1\mathrm{/}{x}_i^{(j)}$ (Relative projective space from standard charts). Write $x_j^{(i)}$ for the coordinate $t_j\mathrm{/}{t}_i$ on $U_i$.

The twisting sheaf on relative projective space. On each chart $U_i$ let ${\mathcal{O}}_{U_i}{e}_i$ be a free rank-one ${\mathcal{O}}_{U_i}$-module with basis $e_i$. On the overlap $U_i\cap U_j$, identify $({\mathcal{O}}_{U_i}{e}_i){\mathrm{\mid }}_{U_i\cap U_j}⟶({\mathcal{O}}_{U_j}{e}_j){\mathrm{\mid }}_{U_i\cap U_j},e_i⟼x_i^{(j)}{e}_j,$ that is, $e_j=x_j^{(i)}{e}_i$ on $U_i\cap U_j$; this is the standard normalization under which the coordinate section $x_j$ restricts to $x_j^{(i)}{e}_i$ on $U_i$ and to $e_j$ on $U_j$. This is an isomorphism of invertible sheaves: $x_i^{(j)}$ is a unit on $U_i\cap U_j$ by construction. The transition data are compatible on triple overlaps, $x_i^{(j)}\cdot x_j^{(k)}=x_i^{(k)}\text{on }{U}_i\cap U_j\cap U_k,$ because $x_i^{(k)}=(t_i\mathrm{/}{t}_k)$ and $x_i^{(j)}{x}_j^{(k)}=(t_i\mathrm{/}{t}_j)(t_j\mathrm{/}{t}_k)$; equivalently, the identifications obtained from $e_i\mapsto x_i^{(j)}{e}_j$, $e_j\mapsto x_j^{(k)}{e}_k$ and $e_i\mapsto x_i^{(k)}{e}_k$ agree on the triple overlap. Hence the trivial invertible sheaves on the $U_i$ glue to an invertible sheaf on ${\mathbb{P}}_S^n$, denoted ${\mathcal{O}}_{{\mathbb{P}}_S^n}(1)\text{or}\mathcal{O}(1)$ when the ambient space is clear, with $e_i$ a frame on $U_i$. For integers $d$ put $\mathcal{O}(d)=\mathcal{O}(1{)}^{\otimes d}$ for $d\ge 0$ and $\mathcal{O}(d)=(\mathcal{O}(-d){)}^{\vee }$ for $d<0$.

Definition. Let $f:X\to S$ be a quasi-compact morphism (Quasi-compact and quasi-separated morphisms) and let $L$ be an invertible ${\mathcal{O}}_X$-module (Invertible sheaves). Then $L$ is H-very ample relative to $S$ if there exist an integer $n\ge 0$ and a quasi-compact $S$-immersion $i:X⟶{\mathbb{P}}_S^n$ of schemes (Immersion of schemes) such that $L\cong i^{\ast }{\mathcal{O}}_{{\mathbb{P}}_S^n}(1)\mathrm{.}$ The value $n=0$ is allowed, with ${\mathbb{P}}_S^0\cong S$. If in addition $i$ can be chosen to be a closed immersion, then $L$ is closed H-very ample relative to $S$.

• The comparisons in these remarks assume the Axiom of Choice (The Axiom of Choice) where they invoke the AC-qualified projective space, Proj twist, and properness suppliers. The chart-based definition above makes no additional choice.

• Proper sources force closedness. Suppose that $X\to S$ is proper (Proper morphisms). Then every quasi-compact $S$-immersion $i:X\to {\mathbb{P}}_S^n$ is a closed immersion, so on a proper source the two notions coincide. Indeed, by Morphisms from a proper scheme to a separated one are proper the morphism $i$ is proper (here ${\mathbb{P}}_S^n\to S$ is separated, which holds because Finite-dimensional projective space is proper over every base shows it is proper), so its image $i(X)$ is closed in ${\mathbb{P}}_S^n$ by Proper morphisms are closed; an immersion with closed image is a closed immersion by An immersion with closed image is a closed immersion.

• Quasi-compactness of $X$. If $S$ is quasi-compact, then a quasi-compact $S$-morphism has quasi-compact source. Thus every morphism to which this definition applies has quasi-compact source when $S$ is quasi-compact; quasi-compactness of $X$ alone does not imply quasi-compactness of $X\to S$. Over a non-quasi-compact base the morphism $f:X\to S$ can be quasi-compact while $X$ is not quasi-compact, and the definition is stated for morphisms to cover that case.

• Relation to the twist on $\mathrm{Proj}$. Under the identification ${\mathbb{P}}_A^n=\mathrm{Proj}A[x_0,\dots ,x_n]$ of Projective space is Proj of a polynomial ring, the sheaf $\mathcal{O}(1)$ defined here agrees with the twist $\widetilde{A[x_0,\dots ,x_n](1)}$ of Twisting sheaf on Proj; the frames $e_i$ correspond to the degree-$1$ elements $x_i$. To verify the sheaf comparison directly, put $B=A[x_0,\dots ,x_n]$. On $D_{+}(x_i)$, multiplication by $x_i$ is an isomorphism $B_{(x_i)}\to B(1{)}_{(x_i)}$: its inverse divides a degree-one element of $B[x_i^{-1}]$ by $x_i$. On overlaps these frames satisfy $x_i=(x_i\mathrm{/}{x}_j)x_j$, exactly the transition $e_i=x_i^{(j)}{e}_j$ above. The chartwise identifications therefore glue to the claimed comparison, carrying each coordinate section to the corresponding homogeneous element.