ANNALES
DE LA FACULTÉ
DES SCIENCES
DE TOULOUSE
Mathématiques

JÁNOS KOLLÁR

Symmetric powers of Severi-Brauer varieties

Tome XXVII, n° 4 (2018), p. 849-862.

<http://afst.cedram.org/item?id=AFST_2018_6_27_4_849_0>

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# Symmetric powers of Severi-Brauer varieties (*)

JÁNOS KOLLÁR (1)

**ABSTRACT.** — We classify products of symmetric powers of a Severi-Brauer variety, up to stable birational equivalence. The description also includes Grassmannians, flag varieties and moduli spaces of genus 0 stable maps.

**RÉSUMÉ.** — Nous classons les produits de puissances symétriques d'une variété de Severi-Brauer, à équivalence binationnelle stable près. Notre classification concerne aussi les grassmanniennes, les variétés de drapeaux et les espaces de modules d'applications stables de genre 0.

## 1. Introduction

Let $P$ be a Severi-Brauer variety over a field $k$. That is,

$$ P_{k^s} := P \times_{\text{Spec } k} \text{Spec } k^s \cong \mathbb{P}^{\text{dim } P}, $$

where $k^s$ denotes a separable closure of $k$; see Definition 2.1 for related notions and basic properties of Severi-Brauer varieties.

There are several ways to associate other varieties to $P$. These include

* the Grassmannians $\text{Grass}(\mathbb{P}^{m-1}, P)$,

* the flag varieties $\text{Flag}(\mathbb{P}^{m_1-1}, \dots, \mathbb{P}^{m_r-1}, P)$,

* the symmetric powers $\text{Sym}^m(P)$ and

* the moduli spaces $\bar{M}_0(P, d)$ of genus 0 stable maps of degree $d$ to $P$;

see Definitions 2.2-2.5. While all these varieties are geometrically rational, they are usually not rational over the ground field and it is an interesting problem to understand their birational properties over $k$. The results of this note are partly weaker than birational classification, since we describe

<footnote>(*) Reçu le 10 juillet 2016, accepté le 7 décembre 2016.</footnote>

<footnote>(1) Princeton University, Princeton NJ 08544-1000, USA —
kollar@math.princeton.edu</footnote>

<footnote>Article proposé par Damian Rössler.</footnote>
only the stable birational equivalence classes, but partly stronger, since we
also describe the products of these varieties. Birational equivalence of two
varieties is denoted by $X \stackrel{\text{bir}}{\sim} Y$ and stable birational equivalence by $X \stackrel{\text{stab}}{\sim} Y$.
The latter holds iff $X \times \mathbb{P}^m \stackrel{\text{bir}}{\sim} Y \times \mathbb{P}^n$ for some $n, m \in \mathbb{N}$.

Thus let $\mathrm{MSym}(P)$ denote the multiplicative monoid generated by stable
birational equivalence classes of Grassmannians, flag varieties, symmetric
powers of $P$ and the moduli spaces $\bar{M}_0(P, d)$. We show that $\mathrm{MSym}(P)$ is
finite, identify its elements and also the multiplication rules.

**THEOREM 1.1.** — Let $P$ be a Severi–Brauer variety of index $i(P)$ (see Definition 2.1(2)) over a field $k$. Then

(1) $\mathrm{MSym}(P) = \{\mathrm{Grass}(\mathbb{P}^{d-1}, P) : d \mid i(P)\}$ and products are given by

(2) $\mathrm{Grass}(\mathbb{P}^{d-1}, P) \times \mathrm{Grass}(\mathbb{P}^{e-1}, P) \stackrel{\text{stab}}{\sim} \mathrm{Grass}(\mathbb{P}^{(d,e)-1}, P)$, where $(d,e)$
is the greatest common divisor. The identity is $\mathrm{Grass}(\mathbb{P}^{i(P)-1}, P) \stackrel{\text{stab}}{\sim} \mathbb{P}^0$.

*The class of a flag variety is given by the rule*

(3) $\mathrm{Flag}(\mathbb{P}^{m_1-1}, \dots, \mathbb{P}^{m_r-1}, P) \stackrel{\mathrm{stab}}{\sim} \mathrm{Grass}(\mathbb{P}^{e-1}, P)$ where $e := (m_1, \dots, m_r, i(P))$. In particular, $\mathrm{Grass}(\mathbb{P}^{d-1}, P) \stackrel{\mathrm{stab}}{\sim} \mathrm{Grass}(\mathbb{P}^{e-1}, P)$ where $e = (d, i(P))$.

*The class of a symmetric power is given by the rules*

(4) $\mathrm{Sym}^d(P) \stackrel{\mathrm{stab}}{\sim} \mathrm{Sym}^{(d,i(P))}(P)$ for every $d \ge 0$,

(5) $\mathrm{Sym}^d(P) \stackrel{\text{bir}}{\sim} \mathrm{Grass}(\mathbb{P}^{d-1}, P) \times \mathbb{P}^{d(d-1)}$ for $d \le n+1$ and

(6) $\mathrm{Sym}^d(P) \times \mathrm{Sym}^e(P) \stackrel{\text{stab}}{\sim} \mathrm{Sym}^{(d,e)}(P)$.

*The class of $\bar{M}_0(P, d)$ is determined by the parity of $d$.*

(7) $\bar{M}_0(P, 2e) \stackrel{\text{stab}}{\sim} P$ except when $\dim P = e = 1$ in which case
$\bar{M}_0(P, 2) \cong \mathbb{P}^2$.

(8) $\bar{M}_0(P, 2e + 1) \stackrel{\text{stab}}{\sim} \mathrm{Grass}(\mathbb{P}^1, P)$. Note that $\mathrm{Grass}(\mathbb{P}^1, P)$ is rational iff $i(P) \in \{1, 2\}$ and stably birational to $P$ iff $i(P)$ is odd.

**Remark 1.2.** — Several key special cases of these results have been known.
[18] shows that $\mathrm{Sym}^{\dim P+1}(P)$ is rational and the statement 1.1 (3) on flag
varieties is proved in [17, 4.2].

The most natural description seems to be in terms of symmetric powers,
so we start with them. The relationship with Grassmannians and flag vari-
eties is easy to establish. The moduli spaces $\bar{M}_0(P, d)$ end up birationally
the simplest but understanding them is more subtle.
The case $\dim P = e = 1$ is quite exceptional for the moduli space of genus 0 stable maps. $\bar{M}_0(P, 2)$ aims to classify double covers of $P$ ramified at 2 points. The coarse moduli space is $\mathrm{Sym}^2(P) \cong \mathbb{P}^2$. However, if $P \not\cong \mathbb{P}^1$ then there are no such double covers defined over $k$. The problem is that every double cover has an order 2 automorphism. In all other cases, a dense open subset of $\bar{M}_0(P, d)$ parametrizes maps without automorphisms, even embeddings if $\dim P \ge 3$.

It is possible that the stable birational equivalences in Theorem 1.1 can be replaced by birational equivalences. For instance, it is possible that

$$ \mathrm{Sym}^d(P) \stackrel{\text{bir}}{\sim} \mathrm{Grass}(\mathbb{P}^{(d,i(P))-1}, P) \times \mathbb{P}^m \quad \text{for suitable } m, $$
but I do not even know how to show that

$$ \mathrm{Sym}^{n+2}(P) \stackrel{\text{bir}}{\sim} P \times \mathbb{P}^{n(n+1)} \quad \text{for } n = \dim P. $$

Several steps in the proof naturally give only stable birational equivalences and the difference between stable birational equivalence and birational equivalence is not even understood for Severi-Brauer varieties.

**Acknowledgments.** I thank A. Auel, I. Coskun, P. Gille, D. Krashen, M. Lieblich, D. Saltman and T. Szamuly for helpful comments. I am grateful to C. Böhning and H.-C. von Bothmer for calling my attention to [5]. Partial financial support was provided by the NSF under grant number DMS-1362960.

## 2. Basic definitions

**DEFINITION 2.1** (Severi-Brauer varieties I). — Let $k$ be a field with separable closure $k^s$. A $k$-scheme $P$ is called a Severi-Brauer variety if $P_{k^s} := P \times_{\mathrm{Spec} k} \mathrm{Spec} k^s \cong \mathbb{P}^n$ for some $n$. The following basic results go back to Severi and Châtelet, see [7, Chap. 5] or [15] for modern treatments and references.

(1) $P$ is trivial, that is $P \cong \mathbb{P}^n$, iff $P(k) \neq \emptyset$.

(2) The index of $P$ is the $\gcd$ of the degrees of all $0$-cycles on $P$; it is denoted by $i(P)$. Its value divides $\dim P + 1$ and $P$ contains a reduced, effective $0$-cycle $Z$ of degree $i(P)$. Thus $P$ has a $k'$ point for some separable field extension $k'/k$ of degree $i(P)$.

(3) A subscheme $L \subset P$ is called twisted linear if $L_{k^s}$ is a linear subspace of $P_{k^s} \cong \mathbb{P}^n$. Thus $L$ is also a Severi-Brauer variety. The minimal twisted linear subvarieties have dimension $i(P) - 1$ and they are isomorphic to each other; call this isomorphism class $P^{\min}$.
(4) Given $P^{\min}$ and $r \ge 1$ there is a unique (up-to isomorphism) Severi-Brauer variety $P_r$ of dimension $r(\dim P^{\min} + 1) - 1$ such that $(P_r)^{\min} \cong P^{\min}$.

(5) $P_r \stackrel{\text{bir}}{\approx} P^{\min} \times \mathbb{P}^m$ for $m = (r-1)(\dim P^{\min} + 1)$.

(6) Two Severi-Brauer varieties $P_1, P_2$ are Brauer equivalent, denoted by $P_1 \sim P_2$, iff $P_1^{\min} \cong P_2^{\min}$. This holds iff the smaller dimensional one is isomorphic to a twisted linear subvariety of the other.

**DEFINITION 2.2** (Grassmannians and flag varieties). — Let $P$ be a Severi-Brauer variety of dimension $n$. Fix natural numbers $0 < m_1 < \dots < m_r < n$. The flag variety $\text{Flag}(\mathbb{P}^{m_1-1}, \dots, \mathbb{P}^{m_r-1}, P)$ is the $k$-scheme that represents the functor that associates to a scheme $S$ the set of all nested subschemes

$$L_1 \subset \dots \subset L_r \subset P_S$$

where, for every $i$, the projection $L_i \to S$ is flat and its geometric fibers are linear subspaces of $P$ of dimension $m_i$. For $r=1$ we set $\text{Grass}(\mathbb{P}^{m_1-1}, P) := \text{Flag}(\mathbb{P}^{m_1-1}, P)$.

The flag varieties $\text{Flag}(\mathbb{P}^{m_1-1}, \dots, \mathbb{P}^{m_r-1}, \mathbb{P}^n)$ are rational; in fact they can be written as finite, disjoint unions of affine spaces; see [9, Chap. XIV] or [3, 19].

**DEFINITION 2.3** (Symmetric powers). — Let $X$ be a quasi-projective $k$-scheme. Its $n$th symmetric power is the $k$-scheme $\text{Sym}^n X := X^n/S_n$, the quotient of the $n$th power $X^n$ by the action of the symmetric group $S_n$ permuting the coordinates.

It is easy to see that $\text{Sym}^n \mathbb{P}^1 \cong \mathbb{P}^n$. For $n, r \ge 2$ the symmetric powers $\text{Sym}^n \mathbb{P}^r$ are singular (hence not isomorphic to a projective space) but they are rational; see [20] for a short proof.

The following result says that stable birational equivalence is preserved by symmetric powers.

**LEMMA 2.4.** — Let $U$ be a positive dimensional, geometrically irreducible $k$-variety. Then $\text{Sym}^m(U \times \mathbb{P}^r) \stackrel{\text{bir}}{\approx} \text{Sym}^m(U) \times \mathbb{P}^{rm}$.

Therefore, if $U, V$ are positive dimensional, geometrically irreducible $k$-varieties such that $U \stackrel{\text{stab}}{\sim} V$ then $\text{Sym}^m(U) \stackrel{\text{stab}}{\sim} \text{Sym}^m(V)$ for every $m$.

*Proof*. — There is a natural projection map $\text{Sym}^m(U \times \mathbb{P}^r) \to \text{Sym}^m(U)$. We claim that its generic fiber $F_{\text{gen}}$ is rational. To construct it, set $L := k(\text{Sym}^m(U))$ and $K := k(\text{Sym}^{m-1}(U) \times U)$. Here we think of $\text{Sym}^m(U)$ as $U^m/S_m$ and $\text{Sym}^{m-1}(U) \times U$ as $U^m/S_{m-1}$ where $S_{m-1} \subset S_m$ are the permutations that fix the last factor. Thus $K/L$ is a degree $m$ field extension
and $F_{gen} \stackrel{\text{bir}}{\sim} \mathfrak{R}_{K/L}(\mathbb{P}^r)$, the Weil restriction of $\mathbb{P}^r$ from $K$ to $L$. Thus $F_{gen}$ is rational. $\square$

**DEFINITION 2.5 (Spaces of maps).** — Let $X, Y$ be k-schemes and $L_X, L_Y$ line bundles on them. Let $\text{Mor}((X, L_X), (Y, L_Y))$ denote the k-scheme that represents the functor that associates to a scheme $S$ the set of all morphisms $\phi: X_S \to Y_S$ such that $\phi^*(L_Y)_S \cong (L_X)_S$. See [13, Sec. I.1] for details where the notation $\text{Hom}(\cdot, \cdot)$ is used instead.

If $X \cong \mathbb{P}^1$ and we fix an ample line bundle $L_Y$ on $Y$ then $\text{Aut}(\mathbb{P}^1)$ acts on the space of maps by precomposition and

$$M_0^\circ(Y, d) := \text{Mor}((\mathbb{P}^1, \mathcal{O}_{\mathbb{P}^1}(d)), (Y, L_Y))/\text{Aut}(\mathbb{P}^1)$$

is called the open moduli space of genus 0 stable maps of degree $d$ to $Y$.

We refer to [6] for a general introduction to stable maps and the definition of the “true” moduli space $\bar{M}_g(Y, d)$ of genus $g$ stable maps of degree $d$.

For $m \ge 2$ the open moduli space $M_0^\circ(\mathbb{P}^m, d)$ is dense in the “true” moduli space $\bar{M}_0(\mathbb{P}^m, d)$; cf. [2, Sec. 4]. So we do not lose any birational information by working with $M_0^\circ(\mathbb{P}^m, d)$.

It is easy to see (we in fact prove this in Lemma 4.7) that the moduli spaces $\bar{M}_0(\mathbb{P}^m, d)$ are unirational. They are even rational but this is harder to prove; see [11, 8, 12].

### 3. Symmetric powers

A key step in understanding symmetric powers is the following.

**THEOREM 3.1 ([18]).** — Let $P$ be a Severi-Brauer variety of dimension $n$. Then $\text{Sym}^{n+1}(P)$ is rational.

The following is a geometric proof. The Euler number of $\mathbb{P}^n$ is $n+1$, thus a general section of the tangent bundle $T_{\mathbb{P}^n}$ vanishes at $n+1$ points. For any Severi-Brauer variety this gives a rational map $\pi : H^0(P, T_P) \dashrightarrow \text{Sym}^{n+1}(P)$.

Let $Z \subset P$ be a reduced 0-cycle of degree $n+1$ whose linear span equals $P$. Then $\pi^{-1}(Z)$ is the linear space $H^0(P, T_P(-Z)) \subset H^0(P, T_P)$ of dimension $n$. Let $V \subset H^0(P, T_P)$ be a general affine-linear subspace of codimension $n$. Then $\pi|_V : V \dashrightarrow \text{Sym}^{n+1}(P)$ is birational.

**COROLLARY 3.2.** — Let $P$ be a Severi-Brauer variety of index $i(P)$. Then $\text{Sym}^d(P)$ is stably rational iff $i(P) \mid d$.
*Proof.* — If $d$ is not divisible by $i(P)$ then $\mathrm{Sym}^d(P)(k) = \emptyset$ by 2.1(2), hence $\mathrm{Sym}^d(P)$ is not stably rational.

To see the converse, assume that $i(P) \mid d$. By 2.1(4)-(5) $P$ is stably birational to a Severi-Brauer variety $P'$ of dimension $d-1$. Furthermore, $\mathrm{Sym}^d(P')$ is rational by Theorem 3.1 and it is stably birational to $\mathrm{Sym}^d(P)$ by Lemma 2.4. $\square$

*Proof of Theorem 1.1(1)-(6).* — The easiest is 1.1(5). Given $d \le n+1$ points in linearly general position, they span a linear subspace of dimension $d-1$. This gives a natural map $\pi: \mathrm{Sym}^d(P) \dashrightarrow \mathrm{Grass}(\mathbb{P}^{d-1}, P)$. (A priori, this map is defined only over $k^s$. However, it is invariant under $\mathrm{Gal}(k^s/k)$, thus $\pi$ is defined over $k$. This can be seen by applying Weil's lemma on the field of definition of a subscheme to the graph of $\pi$; see [21, I.7.Lem. 2] or [16, Sec. 3.4] for proofs.)

Let $K$ be the function field of $\mathrm{Grass}(\mathbb{P}^{d-1}, P)$ and $L_K \subset P_K$ the linear subspace corresponding to the generic point. Thus $L_K$ is a Severi-Brauer subvariety of dimension $d-1$. The generic fiber of $\pi$ is $\mathrm{Sym}^d(L_K)$ which is rational by Theorem 3.1. Thus $\mathrm{Sym}^d(P) \stackrel{\mathrm{bir}}{\sim} \mathrm{Grass}(\mathbb{P}^{d-1}, P) \times \mathbb{P}^{d(d-1)}$.

Next we show the stable birational equivalences

$$ \begin{aligned} \mathrm{Sym}^d(P) \times \mathrm{Sym}^e(P) \times \mathbb{P}^{n(d,e)} &\stackrel{\mathrm{stab}}{\sim} \mathrm{Sym}^d(P) \times \mathrm{Sym}^e(P) \times \mathrm{Sym}^{(d,e)}(P) \\ &\stackrel{\mathrm{stab}}{\sim} \mathbb{P}^{nd} \times \mathbb{P}^{ne} \times \mathrm{Sym}^{(d,e)}(P). \end{aligned} $$

First let $K$ be the function field of $\mathrm{Sym}^d(P) \times \mathrm{Sym}^e(P)$. Then $P_K$ has 0-cycles of degrees $d$ and $e$, thus it also has a 0-cycle of degree $(d,e)$. Thus $\mathrm{Sym}^{(d,e)}(P_K)$ is stably rational by Corollary 3.2, proving the first part.

Similarly, let $L$ be the function field of $\mathrm{Sym}^{(d,e)}(P)$. Then $\mathrm{Sym}^d(P_L)$ and $\mathrm{Sym}^e(P_L)$ are stably rational by Lemma 3.2, proving the second part. This implies 1.1(6).

Using this for $e = i(P)$ gives that

$$ \mathrm{Sym}^d(P) \times \mathrm{Sym}^{i(P)}(P) \stackrel{\mathrm{stab}}{\sim} \mathrm{Sym}^{(d,i(P))}(P). $$

Since $\mathrm{Sym}^{i(P)}(P)$ is stably rational by Corollary 3.2, we get 1.1(4) and, together with 1.1(5), it implies 1.1(2).

Next we show, following [17, 4.2], that, with $e := (m_1, \dots, m_r, i(P))$,
$$ \begin{aligned} \mathrm{Flag}(\mathbb{P}^{m_1-1}, \dots, \mathbb{P}^{m_r-1}, P) &\stackrel{\mathrm{stab}}{\sim} \mathrm{Flag}(\mathbb{P}^{m_1-1}, \dots, \mathbb{P}^{m_r-1}, P) \times \mathrm{Grass}(\mathbb{P}^{e-1}, P) \\ &\stackrel{\mathrm{stab}}{\sim} \mathrm{Grass}(\mathbb{P}^{e-1}, P). \end{aligned} $$
Let $L$ be the function field of $\text{Flag}(\mathbb{P}^{m_1-1}, \dots, \mathbb{P}^{m_r-1}, P)$. Then $P_L$ has twisted linear subspaces of dimensions $m_1-1, \dots, m_r-1$, hence 0-cycles of degrees $m_1, \dots, m_r$. Therefore $i(P_L) | e := (m_1, \dots, m_r, i(P))$ and $\text{Sym}^e(P_L)$ is stably rational by Lemma 3.2. Thus $\text{Grass}(\mathbb{P}^{e-1}, P_L)$ is stably rational by 1.1 (5).

Conversely, let $K$ be the function field of $\text{Grass}(\mathbb{P}^{e-1}, P)$. We use induction on $r$ to show that $\text{Flag}(\mathbb{P}^{m_1-1}, \dots, \mathbb{P}^{m_r-1}, P_K)$ is stably rational. Since $e | m_1$, there is a twisted-linear subvariety $R_K \subset P_K$ of codimension $m_1$. Then the map

$$ (L_1 \subset L_2 \subset \dots \subset L_r) \mapsto (L_2 \cap R_K \subset \dots \subset L_r \cap R_K) \times (L_1) $$

determines a birational equivalence

$$ \text{Flag}(\mathbb{P}^{m_1-1}, \dots, \mathbb{P}^{m_r-1}, P_K) \stackrel{\text{bir}}{\sim} \text{Flag}(\mathbb{P}^{m'_2-1}, \dots, \mathbb{P}^{m'_r-1}, R_K) \times \text{Grass}(\mathbb{P}^{m_1-1}, P_K) $$

where $m'_i := m_i - m_1$. By induction $\text{Flag}(\mathbb{P}^{m'_2-1}, \dots, \mathbb{P}^{m'_r-1}, R_K)$ is stably rational and so is $\text{Grass}(\mathbb{P}^{m_1-1}, P_K)$ by Lemma 3.2 and 1.1 (5). This shows 1.1 (3).

We have proved that every flag variety of $P$ is stably birational to a symmetric power $\text{Sym}^d(P)$ for some $d | i(P)$. Next we show that these $\text{Sym}^d(P)$ are not stably birational to each other.

Let $d < e$ be different divisors of $i(P)$. There is thus a prime $p$ such that $d = p^a d'$, $e = p^c e'$ where $a < c$ and $d', e'$ are not divisible by $p$. Let $p^b$ be the largest $p$-power dividing $i(P)$.

By 2.1 (2) $P$ has a $k'$ point for some field extension $k'/k$ of degree $i(P)$. Let $k''/k$ be the Galois closure of $k'/k$ and $K$ the invariant subfield of a $p$-Sylow subgroup of $\text{Gal}(k''/k)$. Set $K' = k'K$. Note that $p$ does not divide $\text{deg}(K/k)$ and $\text{deg}(K'/K) = p^b$, hence $i(P_K) = p^b$.

Although $K'/K$ need not be Galois, the Galois group of its Galois closure is a $p$-group, hence nilpotent. Thus there is a subextension $K' \supset L \supset K$ of degree $p^{b-a}$. It is enough to show that $\text{Sym}^d(P_L)$ and $\text{Sym}^e(P_L)$ are not stably birational over $L$. By 1.1 (4),

$$ \text{Sym}^d(P_L) \stackrel{\text{stab}}{\sim} \text{Sym}^{p^a}(P_L) \quad \text{and} \quad \text{Sym}^e(P_L) \stackrel{\text{stab}}{\sim} \text{Sym}^{p^c}(P_L). $$

Note that $P_L$ has a point in $K'$ and $\text{deg}(K'/L) = p^a$, hence $i(P_L) = p^a$ and so $\text{Sym}^{p^a}(P_L)$ is stably rational by Corollary 3.2. By contrast $\text{Sym}^{p^c}(P_L)$ does not have any $L$-points. Indeed, an $L$-point on $\text{Sym}^{p^c}(P_L)$ would mean a 0-cycle of degree $p^c$ on $P_L$ hence a 0-cycle of degree $p^{b-a}p^c = p^{b-a+c}$ on
$P_K$. This is impossible since $i(P_K) = p^b$ and $b-a+c < b$. Thus $\mathrm{Sym}^{p^a}(P_L)$ and $\mathrm{Sym}^{p^c}(P_L)$ are not stably birational. $\square$

**4. Moduli of Severi-Brauer subvarieties**

We need some results on twisted line bundles and maps between Severi-Brauer varieties; see [7, Chap. 5] or [15] for proofs and further references.

**DEFINITION 4.1** (Twisted line bundles). — Let $X$ be a geometrically normal, proper $k$-variety. A twisted line bundle of $X$ is a line bundle $L$ on $X_{k^s}$ such that $L^\sigma \cong L$ for every $\sigma \in \mathrm{Gal}(k^s/k)$. For example, if $P$ is a Severi-Brauer variety then $\mathcal{O}_{P_{k^s}}(r)$ is a twisted line bundle for every $r$. Let $|L|$ denote the irreducible component of the Hilbert scheme (or Chow variety) of $X$ parametrizing subschemes $H \subset X$ such that $H_{k^s}$ is in the linear system $|L_{k^s}|$. (See [13, Chap. I.] for the Hilbert scheme or the Chow variety.) This is clearly a Severi-Brauer variety. There is a natural map $\iota_L : X \dasharrow |L|^\vee$ given by $x \mapsto \{H : H \ni x\}$.

The dual of a Severi-Brauer variety $P$ is defined as $P^\vee := |\mathcal{O}_{P_{k^s}}(1)|$.

Let $\phi : X \dasharrow Y$ be a map between geometrically normal, proper varieties and $L_Y$ a twisted line bundle on $Y$. Assume that either $\phi$ is a morphism or $X$ is smooth. Then $\phi^*L_Y$ is a twisted line bundle on $X$ and $|\phi^*L_Y| \sim |L_Y|$.

Let $X, Y$ be geometrically normal, proper varieties and $L_X, L_Y$ twisted line bundles on them. Definition 2.5 extends to give $\mathrm{Mor}((X, L_X), (Y, L_Y))$, the moduli space of all maps $\phi : X \dasharrow Y$ such that $\phi^*L_Y \cong L_X$. For Severi-Brauer varieties we write

$$ \mathrm{Mor}_d(Q, P) := \mathrm{Mor}((Q, \mathcal{O}_Q(d)), (P, \mathcal{O}_P(1))). $$

Composing with $\iota_L$ gives an isomorphism

$$ \mathrm{Mor}((X, L), (P, \mathcal{O}_P(1))) \cong \mathrm{Mor}_1(|L|^\vee, P). $$

**RESULTS 4.2** (Severi-Brauer varieties II). — Let $P, Q$ be Severi-Brauer varieties. We use the following results; see [7] or [15] for proofs and further references.

(1) Their product is defined as $|\mathcal{O}_{P^\vee \times Q^\vee}(1, 1)| \cong |\mathcal{O}_{P \times Q}(1, 1)|^\vee$. I denote this by $P \otimes Q$. It is better to think of this as defined on Brauer equivalence classes. The set of Brauer equivalence classes forms the Brauer group with product $P \otimes Q$, identity $\mathbb{P}^0 \sim \mathbb{P}^m$ and inverse $P \mapsto P^\vee$. The group is torsion; the order of $P$ is called the period of $P$ and denoted by $\mathrm{per}(P)$. The period divides the index.
(2) $\mathrm{Mor}_1(Q, P) \sim Q^\vee \otimes P$. The natural map is $\phi \mapsto \{(x, H) : \phi(x) \in H\} \in \mathcal{O}_{Q \times P}^\vee(1, 1)$.

(3) If $|L|$ is non-empty then $|L^m| \sim |L|^{\otimes m}$; this comes from identifying the symmetric power of a vector space $V$ with the subspace of symmetric tensors in $V^{\otimes m}$.

(4) $\mathrm{Mor}_d(Q, P) \sim (Q^\vee)^{\otimes d} \otimes P$; this follows from the previous two claims.

Putting together 4.2(1) and 4.2(4) we obtain the following.

**COROLLARY 4.3.** — Let $P \sim P'$, $Q \sim Q'$ be Severi-Brauer subvarieties and $d, d' \in \mathbb{N}$ such that $d \equiv d' \mod \mathrm{per}(Q)$. Then

$$ \mathrm{Mor}_d(Q, P) \sim \mathrm{Mor}_{d'}(Q', P'). $$

*Proof.* — This follows from the chain of Brauer equivalences

$$ \mathrm{Mor}_d(Q, P) \sim (Q^\vee)^{\otimes d} \otimes P \sim (Q'^\vee)^{\otimes d'} \otimes P' \sim \mathrm{Mor}_{d'}(Q', P'). \quad \square $$

We next define the spaces of Severi-Brauer subvarieties of a Severi-Brauer variety. That is, given a Severi-Brauer variety $P$ we look at the subset of the Chow variety $\mathrm{Chow}(P)$ parametrizing subvarieties $X \subset P$ whose normalization $\bar{X}$ is a Severi-Brauer variety. For technical reasons it is better to work with $\bar{X} \to P$.

**DEFINITION 4.4.** — Fix integers $0 \le m \le n$, $1 \le d$ and a Severi-Brauer variety $P$ of dimension $n$. Let $M_{\mathbb{P}^m}^\circ(P, d)$ denote the moduli space parametrizing morphisms $\phi : Q \to P$ satisfying the following assumptions.

(1) $Q$ is a Severi-Brauer variety of dimension $m$.

(2) $\phi^*\mathcal{O}_P(1) \cong \mathcal{O}_Q(d)$.

(3) Either $m < n$ and $\phi : Q \to \phi(Q)$ is birational or $m = n$ and every automorphism of the triple $(\phi : Q \to P)$ that is the identity on $P$ is also the identity on $Q$.

(4) Two such morphisms $\phi_i : Q_i \to P$ are identified if there is an isomorphism $\tau : Q_1 \cong Q_2$ such that $\phi_1 = \phi_2 \circ \tau$.

The spaces $M_{\mathbb{P}^m}^\circ(P, d)$ are quasi-projective. They should be thought of as open subschemes of the projective moduli spaces of stable maps $\bar{M}_{\mathbb{P}^m}(P, d)$ [1]. Since we are interested in their birational properties, these compactifications are not important to us.

(Comment on the notation. The moduli space of stable maps from a genus $g$ curve to $Y$ is usually denoted by $M_g(Y, \beta)$, where $\beta$ is the homology class of the image. If $Y = \mathbb{P}^n$ then $\beta$ is usually replaced by $\deg \beta$. Thus $M_{\mathbb{P}^m}^\circ(P, d)$ follows mostly the stable maps convention, except that the degree of $\phi(Q)$ is $d^m$.)
Note that if $\phi : Q \to \phi(Q)$ is birational then every automorphism of $\phi : Q \to P$ that is the identity on $P$ is also the identity on $Q$. This is why the most naive way of identifying two maps is adequate in (4):

$$ (\phi_1 : Q_1 \to P)_{\bar{k}} \cong (\phi_2 : Q_2 \to P)_{\bar{k}} \Leftrightarrow (\phi_1 : Q_1 \to P) \cong (\phi_2 : Q_2 \to P). $$

(As we discussed in Remark 1.2, failure of this is one of the problems with $\bar{M}_0(P, 2)$ if $\dim P = 1$.)

If $d = 1$ then we get $M_{\mathbb{P}^m}^{\circ}(P, 1) = \text{Grass}(\mathbb{P}^m, P)$ and if $m = 1$ then the $M_{\mathbb{P}^1}^{\circ}(P, d)$ are open subschemes of the space of genus 0 stable maps $\bar{M}_0(P, d)$.

These moduli spaces are related to the spaces of maps from Definition 4.1:

$$ M_{\mathbb{P}^m}^{\circ}(P, d) \stackrel{\text{bir}}{\sim} \text{Mor}_d(\mathbb{P}^m, P) / \text{Aut}(\mathbb{P}^m). $$

The resulting map $\Pi : \text{Mor}_d(\mathbb{P}^m, P) \dashrightarrow M_{\mathbb{P}^m}^{\circ}(P, d)$ is not a product, not even birationally. Indeed the fiber of $\Pi$ over a given $\phi : Q \to P$ is the space of isomorphisms $\text{Isom}(\mathbb{P}^m, Q)$. This is a principal homogeneous space under $\text{Aut}(\mathbb{P}^m)$ but it is not isomorphic to $\text{Aut}(\mathbb{P}^m)$ unless $Q$ is trivial since a $k$-point of $\text{Isom}(\mathbb{P}^m, Q)$ is exactly an isomorphism $\mathbb{P}^m \cong Q$.

Our aim is to understand the spaces $M_{\mathbb{P}^m}^{\circ}(P, d)$ for arbitrary ground fields. This is achieved only for $m = 1$ but we have the following general periodicity property.

**THEOREM 4.5.** — Let $P \sim P'$ be Brauer equivalent Severi-Brauer varieties of dimensions $n, n'$. Fix $0 \le m \le \min\{n, n'\}$ and $1 \le d, d'$. Assume that $d \equiv d' \mod (m+1)$. Then

$$ M_{\mathbb{P}^m}^{\circ}(P, d) \stackrel{\text{stab}}{\sim} M_{\mathbb{P}^m}^{\circ}(P', d'). $$

*Proof.* — The idea is similar to the “no-name method” explained in [4, Sec. 4], where it is attributed to Bogomolov and Lenstra.

Let $\text{Isom}_{\mathbb{P}^m}(d, P, d', P')$ denote the scheme parametrizing triples

$$ \{(\phi : Q \to P); (\phi' : Q' \to P'); \tau\} $$

where $(\phi : Q \to P) \in M_{\mathbb{P}^m}^{\circ}(P, d)$, $(\phi' : Q' \to P') \in M_{\mathbb{P}^m}^{\circ}(P', d')$, and $\tau : Q \to Q'$ is an isomorphism. (No further assumptions on $\phi$ and $\phi' \circ \tau$.) We prove that

$$ M_{\mathbb{P}^m}^{\circ}(P, d) \stackrel{\text{stab}}{\sim} \text{Isom}_{\mathbb{P}^m}(d, P, d', P') \stackrel{\text{stab}}{\sim} M_{\mathbb{P}^m}^{\circ}(P', d'), $$

using the natural projections

$$ \pi : \text{Isom}_{\mathbb{P}^m}(d, P, d', P') \to M_{\mathbb{P}^m}^{\circ}(P, d) $$

$$ \text{and } \pi' : \text{Isom}_{\mathbb{P}^m}(d, P, d', P') \to M_{\mathbb{P}^m}^{\circ}(P', d'). $$
It is sufficient to show that their generic fibers are rational. The roles of $d, d'$ are symmetrical, thus it is enough to consider $\pi : \text{Isom}_{\mathbb{P}^m}(d, P, d', P') \to \mathcal{M}_{\mathbb{P}^m}(P, d)$.

Note that the fiber of $\pi$ over $(\phi : Q \to P)$ consists of pairs $\{(\phi' : Q' \to P'); \tau\}$ where $(\phi' : Q' \to P') \in \mathcal{M}_{\mathbb{P}^m}(P', d')$ and $\tau : Q \to Q'$ is an isomorphism. Specifying such a pair is the same as giving $(\phi \circ \tau : Q \to P') \in \text{Mor}_{d'}(Q, P')$. We have thus proved that the fiber of $\pi$ over $(\phi : Q \to P)$ is isomorphic to $\text{Mor}_{d'}(Q, P')$.

We assumed that $d \equiv d' \mod (\dim Q + 1)$, thus

$$d \equiv d' \mod \text{per}(Q). \qquad (4.1)$$

Therefore $\text{Mor}_{d'}(Q, P') \sim \text{Mor}_d(Q, P)$ by Corollary 4.3.

Let now $K$ be the function field of $\mathcal{M}_{\mathbb{P}^m}(P, d)$ and $\phi_K : Q_K \to P_K$ the corresponding map. Then $\phi_K$ gives a $K$-point of $\text{Mor}_d(Q, P)$. Thus $\text{Mor}_{d'}(Q, P')$ also has a $K$-point hence it is rational by (4.1).

This shows that

$$\mathcal{M}_{\mathbb{P}^m}(P, d) \stackrel{\text{stab}}{\sim} \text{Isom}_{\mathbb{P}^m}(d, P, d', P')$$

$$\text{and } \text{Isom}_{\mathbb{P}^m}(d, P, d', P') \stackrel{\text{stab}}{\sim} \mathcal{M}_{\mathbb{P}^m}(P', d')$$

follows by interchanging $P$ and $P'$.

<figure><img src="image_1.png" /></figure>

**Remark 4.6.** — Note that the assumption $d \equiv d' \mod (m+1)$ is used only through its consequence $d \equiv d' \mod \text{per}(Q)$ in (4.1).

There are a few more cases when one can guarantee (4.1). For example, assume that $d, d'$ and $\text{per}(P)$ are all relatively prime to $m+1$. If there is a map $Q \to P$ of degree $d$ then $\text{per}(Q)$ divides $d \cdot \text{per}(P)$. Since $\text{per}(Q)$ also divides $\dim Q + 1$, $Q$ is in fact trivial. Using this observation for $d' = 1$ we obtain that

$$\mathcal{M}_{\mathbb{P}^m}(P, d) \stackrel{\text{stab}}{\sim} \mathcal{M}_{\mathbb{P}^m}(P', 1) \cong \text{Grass}(\mathbb{P}^m, P') \quad \text{if } (m+1, d \cdot \text{per}(P)) = 1.$$

As a consequence of Theorem 4.5, in order to describe the stable birational types of $\mathcal{M}_{\mathbb{P}^m}(P, d)$, it is sufficient to understand $\mathcal{M}_{\mathbb{P}^m}(P, d)$ for $d \le m+1$. There are two cases for which the answer is easy to derive.

**LEMMA 4.7.** — Let $P$ be a Severi-Brauer variety. Then

$$(1) \quad \mathcal{M}_{\mathbb{P}^m}(P, d) \stackrel{\text{stab}}{\sim} \text{Grass}(\mathbb{P}^m, P) \text{ if } d \equiv 1 \mod (m+1).$$

$$(2) \quad \mathcal{M}_{\mathbb{P}^m}(P, d) \stackrel{\text{stab}}{\sim} P \text{ if } d \equiv 0 \mod (m+1) \text{ and } (m+1) \mid 420.$$

*Proof.* — If $d \equiv 1 \mod (m+1)$ then $\mathcal{M}_{\mathbb{P}^m}(P, d) \stackrel{\text{stab}}{\sim} \mathcal{M}_{\mathbb{P}^m}(P, 1)$ by Theorem 4.5 and, essentially by definition, $\mathcal{M}_{\mathbb{P}^m}(P, 1) = \text{Grass}(\mathbb{P}^m, P)$.
For the second claim we check the stable birational isomorphisms

$$M_{\mathbb{P}^m}^{\circ}(P, m+1) \stackrel{\text{stab}}{\sim} M_{\mathbb{P}^m}^{\circ}(P, m+1) \times P \stackrel{\text{stab}}{\sim} P.$$

First let $K$ be the function field of $M_{\mathbb{P}^m}^{\circ}(P, m+1)$. We need to show that $P_K$ is trivial. By assumption, there is a $K$-map $\phi_K : Q_K \to P_K$ for some Severi-Brauer variety $Q_K$ of dimension $m$. By 4.2(4) this corresponds to a $K$-point of $(Q_K^{\vee})^{\otimes m+1} \otimes P_K$. By 4.2(1) $Q_K^{\otimes m+1}$ is trivial and so is $P_K$.

For the second part, let $L$ be the function field of $P$. Then $P_L$ is trivial, hence

$$M_{\mathbb{P}^m}^{\circ}(P_L, m+1) \\ \cong M_{\mathbb{P}^m}^{\circ}(\mathbb{P}_L^n, m+1) \stackrel{\text{bir}}{\sim} \mathbb{P}(H^0(\mathbb{P}_L^m, \mathcal{O}_{\mathbb{P}^m}(m+1))^{n+1}) / \text{PGL}_{m+1}.$$

It is conjectured that this quotient if always stably rational, but this seems to be known only when $(m+1) | 420$; see [5, p. 316] and the references there. $\square$

Next we give a geometric proof of Lemma 4.7.2 for $m = 1$. That is, we show that $\bar{M}_0(P, 2) \stackrel{\text{bir}}{\sim} P$ for $\dim P \ge 2$.

**RESULTS 4.8** (Conics in Severi-Brauer varieties). — We compute, in two different ways, the space $T$ parametrizing triples $(C, \ell_1, \ell_2)$ where $C \subset P$ is a conic and the $\ell_i$ are secant lines of $C$.

Forgetting the lines gives a map to $\bar{M}_0(P, 2)$. Let $C_K$ be the conic corresponding to the generic point of $\bar{M}_0(P, 2)$. The linear span of $C_K$ is a 2-dimensional Severi-Brauer variety that contains a conic, hence isomorphic to $\mathbb{P}_K^2$. A secant line of $C_K$ is determined by a point in the dual $(\mathbb{P}_K^2)^{\vee} \cong \mathbb{P}_K^2$. Thus $T \stackrel{\text{bir}}{\sim} \bar{M}_0(P, 2) \times \mathbb{P}^4$.

Generically the secants lines $\ell_i$ meet at a unique point; this gives a map $T \dasharrow P$. Given $p \in P$, the fiber is obtained by first picking 2 points in $\mathbb{P}(T_pP) \cong \mathbb{P}_{k(p)}^{n-1}$. Once we have 2 lines, they determine a plane $\langle \ell_1, \ell_2 \rangle$ and the 5-dimensional linear system $|\ell_1 + \ell_2|$ on the plane $\langle \ell_1, \ell_2 \rangle$ gives the conics. Thus $T \stackrel{\text{bir}}{\sim} P \times \mathbb{P}^{2\dim P+3}$ and hence $\bar{M}_0(P, 2) \stackrel{\text{stab}}{\sim} P$.

*Proof of Theorem 1.1 (7)-(8).* — If $d = 2e$ is even then

$$\bar{M}_0(P, 2e) \stackrel{\text{bir}}{\sim} M_{\mathbb{P}^1}(P, 2e) \stackrel{\text{stab}}{\sim} M_{\mathbb{P}^1}(P, 2) \stackrel{\text{bir}}{\sim} \bar{M}_0(P, 2),$$

where the birational equivalences are by definition and the stable birational equivalence holds by Theorem 4.5. Next $\bar{M}_0(P, 2) \stackrel{\text{stab}}{\sim} P$ follows either from Lemma 4.7(2) or from Paragraph 4.8. This gives 1.1(7).
Similarly, if $d = 2e + 1$ is odd then $1.1(8)$ follows from

$$ \bar{M}_0(P, 2e+1) \stackrel{\text{bir}}{\sim} M_{\mathbb{P}^1}(P, 2e+1) \stackrel{\text{stab}}{\sim} M_{\mathbb{P}^1}(P, 1) = \text{Grass}(\mathbb{P}^1, P). \quad \square $$

**Remark 4.9.** — So far we have worked with a fixed Severi-Brauer variety $P$, but it would be interesting to understand how the $\text{MSym}(P)$ for different Severi-Brauer varieties interact with each other.

For example, assume that $P, Q$ are Severi-Brauer varieties such that $\text{index}(P)$ and $\text{index}(Q)$ are relatively prime. We claim that $\text{Sym}^d(P)$ and $\text{Sym}^e(Q)$ are stably birational to each other iff they are both stably rational.

To see this, assume that $\text{Sym}^e(Q)$ is not stably rational. By Corollary 3.2 this holds iff $\text{index}(Q) \nmid e$. Set $K := k(P)$. Since $\text{index}(P)$ and $\text{index}(Q)$ are relatively prime, $\text{index}(Q_K) = \text{index}(Q)$, thus $\text{Sym}^e(Q_K)$ is not stably rational by Corollary 3.2. By contrast, $P_K$ is trivial hence $\text{Sym}^d(P_K)$ is stably rational; even rational by [20]. Thus $\text{Sym}^d(P)$ and $\text{Sym}^e(Q)$ are not stably birational to each other.

See also [14, 10] for related questions.

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