11
$\begingroup$

Let $R$ be a DVR with uniformizer $\pi$, fraction field $K$ and residue field $k$. Let $X/K$ be a Severi-Brauer variety and $\mathscr X/R$ a smooth, projective model of it. Is it true that $\mathscr X_k/k$ is also a Severi-Brauer variety? What if we only assume that $\mathscr X/R$ is smooth and proper?

One way to try and prove this would be to show that if $X$ is trivial as a Severi-Brauer variety, then $\mathscr X \cong \mathbb P^n_R$ over $R$. Is this true?

(Note: I asked this a few hours ago on mathstackexchange but on the suggestion of someone, I am posting it here. I have deleted the stackexchange post.)

$\endgroup$
  • 1
    $\begingroup$ If the residue field is of characteristic $0$, you can prove this in many ways, e.g., by using the characterization of projective space by Cho, Miyaoka, and Shepherd-Barron. If the residue field has positive characteristic, my recollection is that this is still true, the proof is elementary but slightly involved. $\endgroup$ – Jason Starr Feb 3 '18 at 16:36
15
$\begingroup$

I wrote up some notes on this in 2004. There have been some developments since then that I will indicate below.

Denote the smooth, proper morphism as follows, $$\pi:\mathcal{X}\to \text{Spec}\ R.$$ Since $\pi$ is flat and proper, also the fiber product morphism, $$\mathbb{P}^n\times_{\text{Spec}\ R}\mathcal{X}\to \text{Spec}\ R,$$ is also flat and proper. For a flat, proper morphism, Michael Artin proved representability of the relative Hilbert functor by an algebraic space that is locally finitely presented and separated over the base. Thus, there exists a universal pair $$(\rho:I\to \text{Spec}\ R,\phi:I\times_{\text{Spec}\ R}\mathbb{P}^n \xrightarrow{\cong} I\times_{\text{Spec}\ R}\mathcal{X})$$ of a separated, locally finitely presented scheme $I$ over $\text{Spec}\ R$ and an isomorphism $\phi$ of $I$-schemes, and this is compatible with arbitrary base change of $\text{Spec}\ R$.

Since $\text{Ext}^2_{\mathcal{O}_{\mathbb{P}^n}}(\Omega_{\mathbb{P}^n_F/F},\mathcal{O}_{\mathbb{P}^n_F})$ equals $0$, the morphism $\rho$ is smooth. Thus, the image of $\rho$ is an open subset, and the restriction of $\rho$ over this open subset is a (smooth) torsor for the (smooth) automorphism group scheme $\text{Aut}(\mathbb{P}^n)=\textbf{PGL}_{n+1}$. In particular, if we restrict over the open image of $\rho$, then $\pi$ is étale locally isomorphic to $\mathbb{P}^n_R$.

Question 1. Is $\rho$ surjective on points?

This can be checked after surjective base change of $\text{Spec}\ R$. Thus, without loss of generality, assume that the generic fiber $X_K$ is $K$-isomorphic to $\mathbb{P}^n_K$. Let $H\subset X_K$ be an effective Cartier divisor whose associated invertible sheaf generates the Picard group. Since $\mathcal{X}$ is smooth over $\text{Spec}\ R$, also $\mathcal{X}$ is regular. Thus the closure $\mathcal{H}$ of $H$ in $\mathcal{X}$ is a Cartier divisor that is flat over $\text{Spec}\ R$. Denote by $\mathcal{L}$ the associated invertible sheaf (the dual of the ideal sheaf of $\mathcal{H}$). The restriction of $\mathcal{L}$ to $X_K$ is an ample generator of $\text{Pic}(X_K)$.

Lemma 2. Let $\pi:\mathcal{X}\to \text{Spec}\ R$ be a proper, flat morphism of schemes, and let $\mathcal{L}$ be an invertible sheaf on $\mathcal{X}$ whose restriction to $X_K$ is a generator of the Picard group. If the closed fiber $X_k$ is integral, then every invertible sheaf on $\mathcal{X}$ is isomorphic to $\mathcal{L}^{\otimes m}$ for some integer $m$.

Proof. Since $\pi$ is proper and flat with integral closed fiber, also $\mathcal{X}$ is integral. Thus, every invertible sheaf on $\mathcal{X}$ is $R$-flat. So the pushforward by $\pi$ is a finitely generated, torsion-free $R$-module, i.e., it is a finite, free $R$-module.

For every invertible sheaf $\mathcal{M}$ on $\mathcal{X}$, since $\mathcal{L}|_{X_K}$ generates $\text{Pic}(X_K)$, there exists an integer $m$ such that $\mathcal{M}|_{X_K}$ is isomorphic to $\mathcal{L}^{\otimes m}|_{X_K}$. The pushforward $\pi_*\textit{Hom}_{\mathcal{O}_X}(\mathcal{M},\mathcal{L}^{\otimes m})$ is a finite, free $R$-module whose $K$-fiber equals $H^0(X_K,\mathcal{O}_{X_K})$. Thus, there exists an element of this $R$-module whose $K$-germ is an invertible element of $H^0(X_K,\mathcal{O}_{X_K})$. This element corresponds to a homomorphism of invertible sheaves, $$\alpha:\mathcal{M} \to \mathcal{L}^{\otimes m},$$ that is an isomorphism on $X_K$. Thus, the support of the cokernel of $\alpha$ is an effective Cartier divisor on $\mathcal{X}$ that is disjoint from $X_K$.

Since the closed fiber $X_k$ is integral, the support of the cokernel equals $r\underline{X}_k$ for some integer $r\geq 0$. Thus $\mathcal{M}$ is isomorphic to $\mathcal{L}^{\otimes m}(-r\underline{X}_k)$ for some integer $r$. Since $\mathcal{O}(-\underline{X}_k)$ is the pullback by $\pi$ of the maximal ideal of $R$, and since this is a free $R$-module, also $\mathcal{O}(-\underline{X}_k)$ is a free $\mathcal{O}_{\mathcal{X}}$-module. Therefore, $\mathcal{M}$ is isomorphic to $\mathcal{L}^{\otimes m}$. QED

Proposition 3. With the same hypotheses as in Lemma 1, assume also that $\mathcal{X}$ is regular. Then either the invertible sheaf $\mathcal{L}$ is $\pi$-ample or the invertible sheaf $\mathcal{L}^\vee$ is $\pi$-ample.

Proof. This is automatic if $\pi$ is finite. Hence assume that $\pi$ has positive fiber dimension. By Chow's Lemma, a blowing up of $X_K$ is projective. This blowing up is covered by complete intersection curves whose pushforwards to $X_K$ are numerically equivalent. Up to replacing $\mathcal{L}$ by $\mathcal{L}^\vee$, assume that the degree of $\mathcal{L}$ on these curves is nonnegative. The claim is that for every integer $m$ that is sufficient positive and divisible, the invertible sheaf $\mathcal{L}^{\otimes m}$ is globally generated and the complete linear system defines a finite morphism to projective space.

For every open affine $U \subset \mathcal{X}$ whose closed fiber $U_k$ is nonempty, for every pair $p,q\in U_k$ of distinct closed points, there exists $f\in H^0(U,\mathcal{O}_{\mathcal{X}})$ that is zero on $p$ and that is nonzero on $q$. The zero scheme $\text{Zero}(f)$ is a Cartier divisor in $U$. The closure in $\mathcal{X}$ of $\text{Zero}(f)$ is a Cartier divisor $D$ that contains $p$, yet does not contain $q$. By the previous lemma, $\mathcal{O}(\underline{D})$ is isomorphic to $\mathcal{L}^{\otimes m}$ for some integer $m$. Since $D$ is effective, its restriction to $X_K$ is effective. Thus, the intersection number with the covering family of curves is positive. Therefore, the integer $m$ is also positive. Via this isomorphism, there exists a global section $s$ of $\mathcal{L}^{\otimes m}$ that vanishes at $p$ yet does not vanish at $q$. Note that for every positive integer $d$, also $s^d$ is a global section of $\mathcal{L}^{\otimes md}$ that vanishes at $p$ yet does not vanish at $q$.

Thus, for every integer $d\geq 1$, for the base locus $B_d$ of the complete linear system of $\mathcal{L}^{\otimes d}$, for every $q\in B_d$, there exists an integer $m\geq 1$ such that $B_{dm}$ is contained in $B_d$ yet does not contain $q$. By Noetherian induction, for all integers $m$ that are sufficiently positive and divisible, $B_m$ is empty. Moreover, for every closed point $p\in X_k$, for every open affine $U$ neighborhood of $p$, for every closed point $q\in U_k$, there exists an integer $m\geq 1$ such that the associated $R$-morphism $$\phi_m:\mathcal{X}\to \mathbb{P}H^0(\mathcal{X},\mathcal{L}^{\otimes m}),$$ does not contain $(p,q)$ in the fiber product $$\mathcal{X}\times_{\mathbb{P}H^0(\mathcal{L}^m)}\mathcal{X}\subset \mathcal{X}\times_{\text{Spec}\ R}\mathcal{X}.$$ Thus, by another Noetherian induction argument, for all integers $m$ that are sufficiently positive and divisible, every irreducible component of the fiber product projects finitely to $\mathcal{X}$ under projection on the first factor. Thus, $\phi_m$ is a finite morphism, and $\mathcal{L}^{\otimes m}$ is ample. QED

Nota bene. There is a vast generalization of Lemma 2 for specializations of rationally connected varieties in the following article of Gounelas and Javanpeykar.

Frank Gounelas, Ariyan Javanpeykar
Invariants of Fano varieties in families
https://arxiv.org/pdf/1703.05735.pdf

Corollary 4. With the same hypotheses as in Proposition 3, for an $R$-flat closed subscheme $\mathcal{Y}\subset \mathcal{X}$ of pure relative dimension $e$, if the $\mathcal{L}$-Hilbert polynomial of $\mathcal{Y}$ equals $P_{\mathcal{Y},\mathcal{L}}(t) = 1\cdot (t^e/e!) + O(t^{e-1})$, then the closed fiber $Y_k$ of the normalization of $\mathcal{Y}$ is geometrically integral.

Proof. Since $\mathcal{Y}$ is $R$-flat, the Hilbert polynomial of the closed fiber equals the Hilbert polynomial of the generic fiber. Since the normalization of $\mathcal{Y}$ is $S_2$, the closed fiber $Y_k$ is $S_1$. Thus, to prove that $Y_k$ is geometrically integral, it suffices to prove that it is geometrically irreducible and reduced at the generic point. The leading coefficient of the Hilbert polynomial is additive for irreducible components and multiplicative for multiplicity at a generic point. Since the coefficient equals $1$, and since $\mathcal{L}$ is ample (so that the leading coefficient of every irreducible component with its reduced structure is a positive integer), it follows that $Y_k$ is geometrically irreducible and has multiplicity $1$ at every generic point. QED

Corollary 5. With the same hypotheses as in Proposition 3, assume further that $(X_K,\mathcal{L}|_{X_K})$ is $K$-isomorphic to $(\mathbb{P}^n_K,\mathcal{O}(1))$ for $n\geq 1$. Let $\mathcal{Y}$ be an $R$-flat Cartier divisor in $\mathcal{X}$ in the linear system of $\mathcal{L}$. Then $\mathcal{Y}$ is smooth over $\text{Spec}\ R$.

Proof. Denote by $\mathcal{Y}_{\text{sm}}$ the maximal open subscheme of $\mathcal{Y}$ that is smooth over $\text{Spec}\ R$. Let $\mathcal{Y}_{\text{sing}}$ denote the closed complement of this open in $\mathcal{Y}$. The claim, to be proved by contradiction, is that $\mathcal{Y}_{\text{sing}}$ is empty. These are both compatible with arbitrary base change of $\text{Spec}\ R$. Thus, if $\mathcal{Y}_{\text{sing}}$ is nonempty, then after faithfully flat base change of $\text{Spec}\ R$ assume that there exists a $k$-point $p$ of $\mathcal{Y}_{\text{sing}}$ and a $k$-point $q$ of $X_k\setminus \mathcal{Y}_{\text{sing}}$. After further base change, assume that these are the specializations of $K$-points $p_K\in Y_K$ and $q_K \in X_K\setminus Y_K$.

Since $(X_K,\mathcal{L}|_{X_K})$ is isomorphic to $(\mathbb{P}^n_K,\mathcal{O}(1))$ there exists a reduced, closed $K$-curve $C_K$ in $X_K$ containing the $K$-points $p_K$ and $q_K$ having $\mathcal{L}$-degree equal to $1$. This curve is the image of a closed immersion of $K$-schemes, $$u_K:(\mathbb{P}^1_K,0,\infty)\mapsto (X_K,p_\eta,q_\eta),$$ with $u_K^*\mathcal{L}$ isomorphic to $\mathcal{O}(1)$.

Form the closure of $C_K$ in $\mathcal{X}$, and then form the normalization of this closed subscheme of $\mathcal{X}$. Denote this normalization as follows, $$(\rho:\mathcal{C}\to \text{Spec}\ R,u:\mathcal{C}\to \mathcal{X}).$$ By the previous corollary, the closed fiber $C_k$ is geometrically integral. As a geometrically integral specialization of a smooth curve of genus $0$, also $C_k$ is smooth of genus $0$ (this can fail in higher genus, e.g., a smooth plane cubic can specialize to an irreducible, nodal plane cubic). The inverse image of $\mathcal{Y}$ is an effective Cartier divisor in $\mathcal{C}$ that does not contain $q$, and thus it does not contain the entire closed fiber $C_k$. Thus, this Cartier divisor in $\mathcal{C}$ is $R$-flat. Since the degree of this Cartier divisor on $C_K$ equals $1$, also the degree on $C_k$ equals $1$. However, the intersection multiplicity at $p$ is at least as large as the multiplicity of the Cartier divisor $\mathcal{Y}$ at $p$. Thus, this Cartier divisor in the closed fiber $X_k$ has multiplicity $1$ at $p$, i.e., it is smooth at $p$. This contradicts that $p$ is contained in $\mathcal{Y}_{\text{sing}}$. This contradiction proves that $\mathcal{Y}$ is smooth over $\text{Spec}\ R$. QED

Let $\mathcal{Y}$ be the closure in $\mathcal{X}$ of a Cartier divisor in $X_K$ that is in the linear system of $\mathcal{L}|_{X_K}$.

Proposition 6. With hypotheses as in Corollary 5, the restriction homomorphism, $$r:H^0(\mathcal{X},\mathcal{L})\to H^0(\mathcal{Y},\mathcal{L}|_{\mathcal{Y}}),$$ is surjective, the complete linear system of $\mathcal{L}$ is globally generated, and the associated morphism of the complete linear system is an isomorphism to $\mathbb{P}^n_R$.

Proof. This is proved by induction on $n$. The base case is when $n$ equals $1$. Every smooth, proper curve of genus $0$ is geometrically isomorphic to $\mathbb{P}^1$. Thus, by way of induction, assume that $n\geq 2$ and assume that the result is true for smaller values of $n$. In particular, by Corollary 5, the pair $(\mathcal{Y},\mathcal{L}|_{\mathcal{Y}})$ satisfies the hypotheses for $n-1$. By the induction hypothesis, this pair is isomorphic to $(\mathbb{P}^{n-1}_R,\mathcal{O}(1))$. After a further finite, flat base change, assume that there exists a section $\sigma$ of $\pi$ whose image is disjoint from $\mathcal{Y}$, i.e., the $k$-fiber is not contained in $\mathcal{Y}_k$.

Let $H_k\subset \mathcal{Y}_k$ be a Cartier divisor in the linear system of $\mathcal{L}|_{\mathcal{Y}_k}$. Since $(\mathcal{Y},\mathcal{L}|_{\mathcal{Y}})$ is $R$-isomorphic to $(\mathbb{P}^{n-1}_R,\mathcal{O}(1))$, there exists a lift of this Cartier divisor to an $R$-flat Cartier divisor $\mathcal{H}$ in $\mathcal{Y}$. In the generic fiber $X_K$, the subset $\mathcal{H}_K$ is a codimension $2$ linear subvariety, and $\sigma_K$ is a $K$-point that is disjoint. Thus, there exists a unique Cartier divisor $D_K$ in the linear system of $\mathcal{L}|_{X_K}$ that contains $\mathcal{H}_K$ and $\sigma$. By the previous corollary, the closure $\mathcal{D}$ in $\mathcal{X}$ of $D_K$ is a smooth Cartier divisor in the linear system of $\mathcal{L}$ that contains $\mathcal{H}$ and that contains $\sigma$.

If the intersection of $\mathcal{D}$ with $\mathcal{Y}_k$ is strictly larger than $\mathcal{H}_k$, then it completely contains the Cartier divisor $\mathcal{Y}_k$. Since $\mathcal{D}_k$ is irreducible, this implies that $\mathcal{D}_k$ equals $\mathcal{Y}_k$. This contradicts that $\mathcal{D}$ contains $\sigma$, since $\sigma$ is disjoint from $\mathcal{Y}$. Therefore, the restriction of $\mathcal{D}$ to $\mathcal{Y}$ equals $\mathcal{H}$.

As the divisors $\mathcal{H}$ vary over a bases for the complete linear system of $\mathcal{L}|_{\mathcal{Y}}$, the divisors $\mathcal{D}$ give a subsystem of the complete linear system of $\mathcal{L}$ that restricts isomorphically to the complete linear system of $\mathcal{L}|_{\mathcal{Y}}$. In particular, the base locus of this linear subsystem of $\mathcal{L}$ is disjoint from $\mathcal{Y}$, since the base locus of $\mathcal{L}|_{\mathcal{Y}}$ is empty. Consider the linear subsystem of $\mathcal{L}$ generated by this linear system together with the Cartier divisor $\mathcal{Y}$. This has empty base locus, and thus defines an $R$-morphism, $$\phi:\mathcal{X}\to \mathbb{P}^n_R.$$ Since $\mathcal{L}$ is ample, this morphism is finite. Moreover, since $\phi$ is an isomorphism on $K$-fibers, $\phi$ is birational. Since $\mathbb{P}^n_R$ is normal (and even regular), it follows from Zariski's Main Theorem that $\phi$ is an isomorphism. QED

Remark. If the residue field has characteristic $0$, then this result, often called "deformation-in-the-large", also holds if the geometric generic fiber is a smooth quadric hypersurface in $\mathbb{P}^n$ (the Ph.D. thesis of Jun-Muk Hwang), or if the geometric generic fiber is a projective homogeneous variety of cominuscule type (joint result of Jun-Muk Hwang and Ngaiming Mok).

MR1608587 (99b:32027)
Hwang, Jun-Muk(KR-SNU); Mok, Ngaiming(PRC-HK)
Rigidity of irreducible Hermitian symmetric spaces of the compact type under Kähler deformation.
Invent. Math. 131 (1998), no. 2, 393–418.
https://arxiv.org/abs/math/9604227

This has been extended to positive characterstic and mixed characteristic by Jan Gutt (using totally different techniques that do not reduce to Kobayashi-Ochiai as in Hwang-Mok). Precisely, for a fixed cominuscule type $(G,P)$, for a fixed integer $m$, there exists an explicit integer $p_0(G,P,m)$ depending on certain Schubert calculus computations such that for every prime $p\geq p_0(G,P,m)$, if the residue characterstic of $k$ equals $p$ (or $0$) and if $\mathcal{L}^{\otimes m}|_{\mathcal{X}_k}$ is very ample, then also $\mathcal{X}_k$ is a projective homogeneous variety of the same cominuscule type $(G,P)$.

Jan Gutt
On the extension theorem of Hwang and Mok
Journal für die reine und angewandte Mathematik (Crelles Journal),
ISSN (Online) 1435-5345, ISSN (Print) 0075-4102,

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.