Arithmetic and birational properties of linear spaces on intersections of two quadrics

In the latter part of the paper, we focus on rationality over the field $\mathbb{R}$ of real numbers. For a smooth complete intersection of two quadrics defined over $\mathbb{R}$, Theorem 1.1 implies that its Fano schemes of non-maximal linear spaces are $\mathbb{C}$-rational. One may further ask when these Fano schemes are rational over $\mathbb{R}$.

The locus of real points encodes additional obstructions to rationality over $\mathbb{R}$: if $Y$ is an $\mathbb{R}$-rational smooth projective variety, then $Y(\mathbb{R})$ is necessarily connected and nonempty. In dimensions $1$ and $2$, this topological obstruction characterizes rationality for $\mathbb{C}$-rational varieties [Com13]. In higher dimensions, however, this fails in general: in dimension $\geq 3$, there are $\mathbb{C}$-rational varieties, with nonempty connected real loci, that are irrational over $\mathbb{R}$ [BW20, Theorem 5.7]. Among complete intersections of quadrics $X\subset\mathbb{P}^{5}$, Hassett–Tschinkel showed that there exist examples that are irrational over $\mathbb{R}$ despite $X(\mathbb{R})$ being non-empty and connected [HT21b]. (See also [FJS⁺22, JJ24] for other examples in dimension $3$.)

Hassett–Kollár–Tschinkel studied $\mathbb{R}$-rationality for even-dimensional complete intersections of quadrics. In particular, they showed that a $4$-fold $X\subset\mathbb{P}^{6}$ is $\mathbb{R}$-rational if and only if its real locus is non-empty and connected. We prove an analogous result for the second maximal linear spaces on $X\subset\mathbb{P}^{2g}$:

Thus, in this case, the aforementioned necessary condition for $\mathbb{R}$-rationality is in fact sufficient. Moreover, we show that this property is determined by the real isotopy class of $X$ (Section 5.3). One can apply Theorem 1.5 to construct examples in any even dimension where $\mathcal{Q}^{(g-2)}$ is $\mathbb{R}$-rational but $F_{g-1}(X)(\mathbb{R})=\emptyset$ (see Corollary 5.7(2)), contrasting the odd-dimensional case as shown by Theorem 1.2. Furthermore, the analogue of Theorem 1.5 fails in every odd dimension (see Example 5.10).

As an application, in the case of $4$-dimensional $X\subset\mathbb{P}^{6}$, combining Theorems 1.4 and 1.5 with earlier results in [CTSSD87, HKT22], we completely determine $\mathbb{R}$-rationality and $\mathbb{R}$-unirationality of the Fano schemes of non-maximal linear spaces on a $4$-fold complete intersection $X$ of two real quadrics using an isotopy invariant of Krasnov [Kra18] (see Section 5.2). This invariant, which was used in the rationality classifications in [HT21b, HKT22], is defined using the signatures of the quadrics in the associated pencil $\cal Q\to\mathbb{P}^{1}$. In particular, we show that these $\mathbb{R}$-(uni)rationality properties of $F_{r}(X)$ for $r=0,1$ are controlled solely by the real isotopy class of $X$, extending the results of [HKT22] on $\mathbb{R}$-rationality of $X$:

In Section 2, we recall preliminary results on Fano schemes of linear subspaces on complete intersections of quadrics, the definition and properties of hyperbolic reductions, lemmas on pencils of quadrics from Reid’s thesis [Rei72], and Benoist–Wittenberg’s codimension 2 Chow scheme for threefolds. In Section 3, we describe the hyperbolic reduction of a pencil of quadrics. We prove Theorems 1.1, 1.3, and 1.4 in this section, and as consequences, we derive rationality results over certain fields in Section 3.3. Section 4, which is the most technical part of the paper, is devoted to the proof of Theorem 1.2 on odd-dimensional complete intersections. Finally, in Section 5 we turn to the case of even-dimensional complete intersections over $\mathbb{R}$. Here we prove Theorem 1.5 and Corollary 1.6, and we give examples contrasting the behavior in the even- and odd-dimensional cases.

Throughout $k$ is a field of characteristic $\neq 2$. For a variety $X$, we use $x\in X$ to denote a scheme-theoretic point. For $i\geq 0$, we let $\operatorname{CH}^{i}(X)$ denote the Chow group of codimension $i$ cycles on $X$. We denote the subgroup of algebraically trivial cycles by $\operatorname{CH}^{i}(X)_{\operatorname{alg}}\subset\operatorname{CH}^{i}(X)$, and we denote the quotient by $\operatorname{NS}^{i}(X)=\operatorname{CH}^{i}(X)/\operatorname{CH}^{i}(X)_{\operatorname{alg}}$. For non-negative integers $r$, $\operatorname{Sym}^{r}X$ denotes the $r$th symmetric power. We say a variety $X$ over $k$ is $k$-rational to emphasize that the rationality construction is defined over $k$; when we say rational without specifying the ground field, we usually mean ${\overline{k}}$-rational over an algebraically closed field. For a curve $C$ over $k$, we use $\operatorname{Pic}(C)$ to denote the Picard group over $k$, and $\operatorname{\mathbf{Pic}}_{C/k}$ to denote the relative Picard scheme over $k$. If $\ell_{1},\ldots,\ell_{r}\subset\mathbb{P}^{n}$ are linear subspaces, we denote their span by $\langle\ell_{1},\ldots,\ell_{r}\rangle$.

We thank Brendan Hassett and Jerry Wang for interesting discussions. We thank Olivier Debarre for feedback on an earlier version of the paper, and Jean-Louis Colliot-Thélène for comments, in particular for pointing out Remark 3.8 and for noting that Theorem 4.13 does not require the assumption that $F_{g-2}(X)(k)\neq\emptyset$. This work started during a visit of the first author to the University of California, Los Angeles, and she thanks Joaquín Moraga and Burt Totaro for their hospitality and for providing a welcoming environment.

Let $X\subset\mathbb{P}^{N}$ be a smooth complete intersection of two quadrics. For non-negative integers $r\geq 0$, $F_{r}(X)$ denotes the Fano scheme of $r$-planes on $X$. In this section, we recall some preliminary results about these Fano schemes $F_{r}(X)$.

For the sake of completeness, we add the following results, which will not be used in the rest of the paper.

In this section, we recall the definition and basic properties of quadric fibrations and of the hyperbolic reductions of a quadric fibrations; see, e.g., [Kuz21, Section 2], [ABB14, Section 1.3] for more details.

Let $S$ be an integral separated Noetherian scheme over a field $k$ of characteristic $\neq 2$. A quadric fibration over $S$ is a morphism $\cal Q\to S$ that can be written as a composition $\cal Q\hookrightarrow\mathbb{P}_{S}(\cal E)\coloneqq\underline{\operatorname{Proj}}_{S}\operatorname{Sym}^{\bullet}(\cal E^{\vee})\to S$ where $\cal E$ is a vector bundle and $\cal Q\hookrightarrow\mathbb{P}_{S}(\cal E)$ is a divisor of relative degree 2 over $S$. A quadric fibration is determined by a quadratic form $q\colon\operatorname{Sym}^{2}\cal E\to\cal L^{\vee}$ with values in a line bundle $\cal L$.

A subbundle $\cal F\subset\cal E$ is isotropic if $q|_{\cal F}=0$ (or, equivalently, if $\mathbb{P}_{S}(\cal F)\subset\cal Q$). An isotropic subbundle $\cal F\subset\cal E$ is regular if, for every (closed) point $s\in S$, the fiber $\mathbb{P}_{S}(\cal F)_{s}$ is contained in the smooth locus of the fiber $\cal Q_{s}$. If $\cal F$ is regular isotropic, then $\cal F$ is contained in the subbundle

of $\cal E$. Moreover, $\cal F$ is in the kernel of the restriction $q|_{\cal F^{\perp}}$, so we have an induced quadratic form on $\cal F^{\perp}/\cal F$.

The process of hyperbolic reduction along a regular isotropic subbundle preserves the degeneration divisor of a quadric fibration [ABB14, Corollary 1.3.9]. (More generally, it preserves the locus of corank $\geq i$ fibers for each $i$ [KS18, Lemma 2.4].)

Hyperbolic reduction can be described geometrically in terms of the linear projection of $\cal Q\subset\mathbb{P}_{S}(\cal E)$ from the linear subbundle $\mathbb{P}_{S}(\cal F)\subset\cal Q\subset\mathbb{P}_{S}(\cal E)$ [KS18, Proposition 2.5].

Next, over algebraically closed fields, we recall several results proven by Reid [Rei72] that we will use in the proof of Theorem 1.3.

For the proof, we need the following result stated in [Rei72, page 44]:

Throughout this section, let $X$ be a smooth, proper, geometrically connected, geometrically rational threefold over $k$. We recall several key properties of Benoist–Wittenberg’s codimension 2 Chow scheme of $X$ [BW23], which we will use in the proof of Theorem 1.2.

For such a threefold $X$, Benoist and Wittenberg use K-theory to define a functor $\operatorname{CH}^{2}_{X/k,\operatorname{fppf}}$ for codimension 2 cycles that is analogous to the Picard functor $\operatorname{Pic}_{X/k,\operatorname{fppf}}$. They show that this functor is represented by a smooth group scheme $\operatorname{\mathbf{CH}}^{2}_{X/k}$ over $k$ with the following properties [BW23, Theorem 3.1]:

1. (1)

   The identity component $(\operatorname{\mathbf{CH}}^{2}_{X/k})^{0}$ is an abelian variety, which we refer to as the intermediate Jacobian of $X$. Its base change to the perfect closure of $k$ agrees with the intermediate Jacobian as defined by Murre [Mur85, ACMV17].

2. (2)

   There is a $G_{k}$-equivariant isomorphism $\operatorname{CH}^{2}(X_{{\overline{k}}})\cong\operatorname{\mathbf{CH}}^{2}_{X/k}({\overline{k}})$.

3. (3)

   The component group $\operatorname{\mathbf{CH}}^{2}_{X/k}/(\operatorname{\mathbf{CH}}^{2}_{X/k})^{0}$ is identified with the $G_{k}$-module $\operatorname{NS}^{2}(X_{\overline{k}})$.

4. (4)

   If $X$ is $k$-rational, then there is a smooth projective (not necessarily connected) curve $B$ over $k$ such that $\operatorname{\mathbf{CH}}^{2}_{X/k}$ is a principally polarized direct factor of $\operatorname{\mathbf{Pic}}_{B/k}$.

For each class $\gamma\in\operatorname{NS}^{2}(X_{{\overline{k}}})^{G_{k}}=(\operatorname{\mathbf{CH}}^{2}_{X/k}/(\operatorname{\mathbf{CH}}^{2}_{X/k})^{0})(k)$, its inverse image $(\operatorname{\mathbf{CH}}^{2}_{X/k})^{\gamma}$ in $\operatorname{\mathbf{CH}}^{2}_{X/k}$ is a(n étale) $(\operatorname{\mathbf{CH}}^{2}_{X/k})^{0}$-torsor. The quotient map is a group homomorphism, so we have an equality of $(\operatorname{\mathbf{CH}}^{2}_{X/k})^{0}$-torsors

for any $\gamma,\gamma^{\prime}\in\operatorname{NS}^{2}(X_{{\overline{k}}})^{G_{k}}$.

Hassett–Tschinkel, over $k=\mathbb{R}$ [HT21b], and Benoist–Wittenberg, over arbitary fields [BW23], (see also [HT21a] for $k\subset\mathbb{C}$) observed that these intermediate Jacobian torsors can be used to refine the rationality obstruction (4). We call their refined obstruction the intermediate Jacobian torsor obstruction to rationality. For simplicity, we state a special case of this obstruction, which is enough for our application:

Throughout Section 3.1, we will work in the following setting. Fix an arbitrary field $k$ of characteristic $\neq 2$ and an integer $N\geq 2$. Let $X$ be a smooth complete intersection of two quadrics in $\mathbb{P}^{N}$, and let $\varphi\colon\mathcal{Q}\rightarrow\mathbb{P}^{1}$ be the associated pencil of quadrics. Let $0\leq r\leq\lfloor\frac{N}{2}\rfloor-1$. By Lemma 2.1, the Fano scheme $F_{r}(X)$ of $r$-planes on $X$ is non-empty. We will assume $F_{r}(X)(k)\neq\emptyset$ in what follows.

Choose $\ell\in F_{r}(X)(k)$. The projection $\pi_{\ell}\colon\mathbb{P}^{N}\dashrightarrow\mathbb{P}^{N-r-1}$ away from $\ell$ induces diagrams

To be more precise, choose homogeneous coordinates $x_{0},\dots,x_{N}$ for $\mathbb{P}^{N}$ so that

Then there exist forms $l_{ij},q_{i}\in k[x_{r+1},\dots,x_{N}]$ with $\deg l_{ij}=1,\deg q_{i}=2$ such that

Choosing a section $z$ whose zero set equals the divisor $\mathbb{P}_{\mathbb{P}^{N-r-1}}(\mathcal{O}^{\oplus r+1})\subset\mathbb{P}_{\mathbb{P}^{N-r-1}}(\mathcal{O}^{\oplus r+1}\oplus\mathcal{O}(1))$, there exist homogeneous coordinates $y_{r+1},\dots,y_{N}$ for $\mathbb{P}^{N-r-1}$ such that $x_{l}=y_{l}z$, and

where $l_{ij},q_{i}$ are in $y_{r+1},\dots,y_{N}$. We also have

We now define

The first projection defines a $\mathbb{P}^{N-2r-2}$-bundle $\mathcal{P}^{(r)}\rightarrow\mathbb{P}^{1}$, which restricts to a quadric hypersurface fibration $\varphi^{(r)}\colon\mathcal{Q}^{(r)}\rightarrow\mathbb{P}^{1}$. (By convention, $\mathcal{Q}^{(-1)}=\mathcal{Q}$ and $\varphi^{(-1)}=\varphi$.) Furthermore, define

and let $\pi^{(r)}\colon E^{(r)}\rightarrow\mathcal{Q}^{(r)}$ be the projection.