Is every proper regular relative algebraic space curve over a Dedekind domain projective?

This question is in some sense a follow up to a related question Is a normal proper relative curve over a DVR projective?

Let $R$ be a Dedekind domain, let $S := \mathrm{Spec}(R)$, and let $X \rightarrow S$ be a proper flat morphism with $X$ a regular algebraic space whose $S$-fibers are purely of dimension $1$ (and hence schemes, e.g. by http://stacks.math.columbia.edu/tag/0ADD). Is $X \rightarrow S$ actually projective, so that $X$ is a scheme itself?

The projectivity in the case of a scheme $X$ is an old result of Lichtenbaum.

• Just below Cor. 2.9 in Ch. IV of Deligne-Rapoport it is asserted without proof or reference that any regular algebraic space "curve" (by which I presume is meant separated, flat, of finite type with 1-dimensional fibers) over $\mathbf{Z}$ is a (quasi-projective) scheme. Presumably whatever argument they had in mind works over any Dedekind base. The case of smooth generic fiber of genus 1 without a rational point seems most mysterious. Anyway, so the answer appears to be "yes", but an argument in complete generality may require a good trick. May 24 '15 at 5:19

Yes. The task is to show that $X$ is a scheme (as then Lichtenbaum's result may be applied). By standard "spreading out" arguments, we may assume $S = {\rm{Spec}}(R)$ for a discrete valuation ring $R$, say with fraction field $K$, residue field $k$, and maximal ideal $\mathfrak{m}$. The special fiber $X_k$ is a scheme. Let $\{C_1, \dots, C_n\}$ be the irreducible components of $X_k$ (say with reduced structure).

Let $x_i \in C_i$ be a closed point, and let $U_i \rightarrow X$ be a residually trivial etale affine neighborhood of $x_i$ that is a scheme, so $U_i$ is a regular affine $R$-curve (i.e., flat finite type over $R$ with fibers of pure dimension 1). We may shrink $U_i$ so that its special fiber misses each $k$-finite $C_i \cap C_j$ for $i \ne j$, which is to say that $(U_i)_k$ lands in $C_i$. By going-down for flat morphisms, applied to $R \rightarrow O_{U_i,x_i}$, we can choose a closed point $u_i \in (U_i)_K$ whose closure in $U_i$ contains $x_i$. Thus, the image $\xi_i \in X_K$ is a closed point whose closure in $X$ contains $x_i$.

Since $X$ is regular of pure relative dimension 1 over $R$, the closure $D_i$ of $\xi_i$ in $X$ has invertible ideal sheaf $\mathscr{I}_{D_i}$. (Indeed, such invertibility may be checked on an etale scheme cover of $X$, where it becomes the fact that a height-1 prime in a regular domain in invertible, as regular local rings are UFD's.)

Let $\mathscr{L}$ denote the tensor product of the inverse sheaves $\mathscr{I}_{D_i}^{-1}$. Since each $D_i$ is $R$-flat, the formation of $\mathscr{L}$ commutes with base change on $R$, such as passage to the special fiber. By the theory of algebraic curves, $\mathscr{L}_k$ is thereby seen to be ample on $X_k$. Hence, for any coherent sheaf $\mathscr{F}$ on $X$, for all sufficiently large $m$ we have that $\mathscr{F} \otimes \mathscr{L}^m$ has vanishing degree-1 cohomology on $X_k$.

If $\mathscr{F}$ is $R$-flat and $\pi \in R$ is a uniformizer then we have a short exact sequence $$0 \rightarrow \mathscr{F} \otimes \mathscr{L}^m \stackrel{\pi}{\rightarrow} \mathscr{F} \otimes \mathscr{L}^m \rightarrow \mathscr{F}_k \otimes \mathscr{L}_k^m \rightarrow 0$$ whose associated cohomology sequence gives the vanishing of ${\rm{H}}^1(X, \mathscr{F} \otimes \mathscr{L}^m)=0$ for such large $m$ (due to Nakayama and $R$-finiteness of this H$^1$). Applying this with $\mathscr{F}$ equal to the ideal sheaf on $X$ of a (varying) closed point of $X_k$, we see that for some large $m_0$ the line bundle $\mathscr{L}^{m_0}$ is generated by global sections. By replacing $m_0$ with a big multiple, the formation of the $R$-finite (free) module of global sections commutes with reduction modulo $\mathfrak{m}$.

Now we get a natural $R$-map $f:X \rightarrow {\rm{Proj}}(\Gamma(R, \mathscr{L}^{m_0}))$ whose formation commutes with reduction modulo $\mathfrak{m}$, and on the special fiber it is quasi-finite since $\mathscr{L}_k$ is ample on $X_k$ with $m_0$th-power generated by global sections. For any map of finite type between noetherian algebraic spaces, the quasi-finite locus is open on the source (as may be checked etale-locally, where it reduces to the known analogue for schemes), so the open quasi-finite locus $U \subset X$ of $f$ contains $X_k$. But $X$ is $R$-proper, so this forces $U=X$. Hence, $f$ exhibits $X$ as separated and quasi-finite over a scheme, so $X$ is a scheme (by a theorem of Knutson).

[The preceding is not optimal, since the conclusion that $X$ is a scheme, even quasi-projective, should be true with "proper" relaxed to "separated, flat, and finite type".

• Nice! I didn't quite understand the paragraph about generation by global sections, but since "$1$" $\in \Gamma(X, \mathscr{L})$ generates away from the $D_i$, only generation at the closed points of the $D_i$ needs to be inspected, where it follows from looking at the twists of the sequences defining the $D_i$ (similar flavor to your argument). As for the "separated, flat, and finite type" version, for such $X$, can one possibly find an embedding $X \subset \overline{X}$ with $\overline{X}$ regular, flat, and proper over $R$ to reduce to the proper case? May 24 '15 at 17:05
• @LisaS.: I had considered to mention Nagata compactification and resolution of singularities for proper algebraic space curves to remove properness (it does indeed work but as in the scheme case there are subtleties when $R$ isn't excellent; I prefer not to discuss it here). But I wasn't convinced that it should really be necessary to invoke Nagata for algebraic spaces, since a-priori $X$ is a scheme away from a finite closed subset of $X_k$ (so I thought some clever argument building on that might work, using the rich theory of regular proper models of curves). May 24 '15 at 17:38