Let k be a field with separable closure K, and \(S\subset \mathbb {P}^{2}(K)\) a finite subset of order n. The field of moduli \(k_{S}\) of S is the subfield of K of elements fixed by Galois automorphisms \(\sigma \in \text {Gal}(K/k)\) such that \(\sigma (S)\) is linearly equivalent to S, i.e., such that there exist \(g\in \textrm{PGL}_{3}(K)\) with \(g(\sigma (S))=S\). We study the problem of whether S descends to a 0-cycle on \(\mathbb {P}^{2}_{k(S)}\), or more generally on a Brauer–Severi surface over \(k_{S}\).

A. Marinatto [18] studied the analogous problem over \(\mathbb {P}^{1}\). He showed that, if n is odd or equal to 4, then S descends to a divisor over \(\mathbb {P}^{1}_{k_{S}}\). Furthermore, he has given counterexamples where S does not descend to \(\mathbb {P}^{1}_{k_{S}}\) for every \(n\ge 6\) even. All of his counterexamples descend to a Brauer–Severi curve, though. In [7], we have shown that if \(n=6\), then S always descends to some Brauer–Severi curve, while there are counterexamples for every \(n\ge 8\) even.

Fields of moduli of curves, possibly with marked points, received a lot of attention, see [11,12,13, 15,16,17]. Furthermore, there are results about abelian varieties, most famously Shimura’s result that a generic, principally polarized, odd dimensional abelian variety is defined over the field of moduli [21], and about fields of moduli of curves in \(\mathbb {P}^{2}\) [1,2,3,4,5, 20]. Here is our result.

FormalPara Theorem 1

Assume \({\text {char}}\, k\ne 2\). Let \(S\subset \mathbb {P}^{2}(K)\) be a finite set of n points with field of moduli \(k_{S}\). If \(n\le 5\), then S descends to a finite subscheme of \(\mathbb {P}^{2}_{k_{S}}\). For every \(n\ge 6\), there exists a subset \(S\subset \mathbb {P}^{2}(\mathbb {C})\) with field of moduli equal to \(\mathbb {R}\) which does not descend to \(\mathbb {P}^{2}_{\mathbb {R}}\).

Notice that \(\mathbb {P}^{2}_{\mathbb {R}}\) is the only Brauer–Severi surface over \(\mathbb {R}\), hence our counterexamples do not descend to any Brauer–Severi surface over \(\mathbb {R}\).

1 Notation and conventions

Given a field k, we write \(\mathbb {P}^{n}_{k}\) for the projective space as a scheme over k. If \(k'/k\) is an extension, then \(\mathbb {P}^{n}_{k'}=\mathbb {P}^{n}_{k}\times _{k}{\text {Spec}}\, k'\) and \(\mathbb {P}^{n}_{k}(k)\subset \mathbb {P}^{n}_{k}(k')=\mathbb {P}^{n}_{k'}(k')\). Because of this, with an abuse of notation, we sometimes drop the subscript and just write \(\mathbb {P}^{n}(k)\) and \(\mathbb {P}^{n}(k')\).

Let \(Z\subset \mathbb {P}^{2}\) be a closed subscheme, and \(g\in \textrm{PGL}_{3}(K)\) a projective linear map. We say that g stabilizes Z, or that Z is g-invariant, if \(g(Z)=Z\). We say that g fixes Z if \(g(Z)=Z\) and \(g|_{Z}:Z\rightarrow Z\) is the identity. If \(G\subset \textrm{PGL}_{3}(K)\) is a finite subgroup, we say that G stabilizes (resp. fixes) Z if every element \(g\in G\) stabilizes (resp. fixes) Z. The fixed locus of g (resp. G) is the subspace of points \(x\in \mathbb {P}^{2}\) with \(gx=x\) (resp. \(\forall g\in G:gx=x\)).

Let \(S\subset \mathbb {P}^{2}(K)\) be a finite subset with finite automorphism group. Up to replacing k with \(k_{S}\), we may assume that k is the field of moduli. We recall some definitions from [9].

A twisted form of \((\mathbb {P}^{2}_{K},S)\) over a k-scheme M is the datum of a projective bundle \(P\rightarrow M\) and a closed subscheme \(Z\subset P\) such that \((P_{K},Z_{K})\) is étale locally isomorphic to \((\mathbb {P}^{2}_{K},S)\times _{K} M_{K}\), i.e., there exists an étale cover \(M'\rightarrow M_{K}\) and an isomorphism

$$\begin{aligned} (\mathbb {P}^{2}_{K},S)\times _{K} M'\simeq (P,Z)\times _{M}M'=(P_{K},Z_{K})\times _{M_{K}}M' \end{aligned}$$

over \(M'\). Notice that if we do not assume that k is the field of moduli, this definition is not correct since \((\mathbb {P}^{2}_{K},S)\) would not define in general a twisted form of \((\mathbb {P}^{2}_{K},S)\).

The fibered category \(\mathscr {G}_{S}\) of twisted forms of \((\mathbb {P}^{2}_{K},S)\) is a finite gerbe over \({\text {Spec}}\,\,k\) called the residual gerbe of S, see [9]. Namely, for a scheme M over k, the objects of the groupoid \(\mathscr {G}_{S}(M)\) are twisted forms (P, Z) of \((\mathbb {P}^{2}_{K},S)\) over M, and arrows \((P,Z)\rightarrow (P',Z')\) are given by isomorphisms \(\phi :P\rightarrow P'\) over M with \(\phi (Z)=Z'\). The universal bundle \(\mathscr {P}_{S}\rightarrow \mathscr {G}_{S}\) is the fibered category defined as follows: the objects of \(\mathscr {P}_{S}(M)\) are triples (P, Z, s) where (P, Z) is a twisted form of \((\mathbb {P}^{2}_{K},S)\) over M, and s is a section \(M\rightarrow P\) of \(P\rightarrow M\), and arrows are defined analogously. The base change of \(\mathscr {P}_{S}\rightarrow \mathscr {G}_{S}\) to K are the quotient stacks \([\mathbb {P}^{2}/{\underline{\textrm{Aut}}}_{K}(S)]\rightarrow [{\text {Spec}}\, K/{\underline{\textrm{Aut}}}_{K}(S)]\). See [9] for more details.

Another way of constructing \(\mathscr {P}_{S}\rightarrow \mathscr {G}_{S}\) is the following. Let \(\mathscr {N}_{S}\subset \text {Aut}_{k}(\mathbb {P}^{2}_{K})\) be the subgroup of k-linear automorphisms \(\tau \) of \(\mathbb {P}^{2}_{K}\) such that \(\tau (S)=S\), the fact that k is the field of moduli implies that \(\mathscr {N}_{S}\) is an extension of \(\text {Gal}(K/k)\) by \(\mathscr {N}_{S}\cap \text {Aut}_{K}(\mathbb {P}^{2}_{K})=\text {Aut}_{K}(\mathbb {P}^{2},S)\) (see [6, §3] for details). We have an induced action of \(\mathscr {N}_{S}\) on \({\text {Spec}}\, K\) with the natural projection \(\mathscr {N}_{S}\subset \text {Aut}_{k}(\mathbb {P}^{2}_{K})\rightarrow \text {Gal}(K/k)\), and the finite étale gerbe \(\mathscr {G}_{S}\) is the quotient stack \([{\text {Spec}}\, K/\mathscr {N}_{S}]\): the natural map \({\text {Spec}}\, K\rightarrow \mathscr {G}_{S}\) associated with the trivial twist of S is a pro-étale, Galois covering with Galois group equal to \(\mathscr {N}_{S}\). Similarly, we can view \(\mathscr {P}_{S}\) as the quotient stack \([\mathbb {P}^{2}_{K}/\mathscr {N}_{S}]\).

Twisted forms of \((\mathbb {P}^{2}_{K},S)\) contained in Brauer–Severi surfaces over k correspond to rational points of \(\mathscr {G}_{S}\). If |S| is prime with 3 and S descends to a 0-cycle over some Brauer–Severi surface P over k, then \(P\simeq \mathbb {P}^{2}_{k}\). In fact, if D is a canonical divisor on P, then \(D\cdot D\) defines a 0-cycle of degree 9 on P, hence P has index 1 (recall that the index is the greatest common divisor of the degrees of 0-cycles on P). This implies that P has a rational point and \(P\simeq \mathbb {P}^{2}_{k}\), see e.g. [14, Corollary 5.3.6, Theorem 5.1.3].

Denote by \(\textbf{P}_{S}\) the coarse moduli space of \(\mathscr {P}_{S}\), i.e., \(\mathbb {P}^{2}_{K}/\mathscr {N}_{S}\), since the action of \(\text {Aut}_{K}(S)\) on \(\mathbb {P}^{2}_{K}\) is faithful, the natural map \(\mathscr {P}_{S}\rightarrow \textbf{P}_{S}\) has a birational inverse \(\textbf{P}_{S}\dashrightarrow \mathscr {P}_{S}\) which, by composition, gives us a rational map \(\textbf{P}_{S}\dashrightarrow \mathscr {G}_{S}\).

2 Case \(n\le 5\)

It is well known that any set of 4 points in \(\mathbb {P}^{2}\) in general position (i.e., such that no line contains 3 of them) can be mapped by a projective linear transformation in a subset of \(\{(1:0:0),(0:1:0),(0:0:1),(1:1:1)\}\). Similarly, if \(n\le 4\) and at most three points are aligned, the set can be mapped by a projective linear transformation in a subset of \(\{(1:0:0),(0:1:0),(1:1:0),(0:0:1)\}\). In both these cases, we thus get that S descends to a finite subscheme of \(\mathbb {P}^{2}_{k}\).

Assume that \(n=4\) and all points are contained in a line. Up to a change of coordinates, we may assume that S is contained in the line \(L=\{(x:y:0)\}\simeq \mathbb {P}^{1}_{K}\), and regard it as a divisor of degree 4 on \(\mathbb {P}^{1}_{K}\). Notice that the subgroup \(\textrm{GL}_{2}(K)\subset \textrm{PGL}_{3}(K)\) acting on the first two coordinates maps surjectively on the group \(\textrm{PGL}_{2}(K)\) of projective linear transformations of L, hence every linear transformation of L extends to a linear transformation of \(\mathbb {P}^{2}_{K}\). By [7, Proposition 13], we may thus find \(g\in \textrm{GL}_{2}(K)\subset \textrm{PGL}_{3}(K)\) such that \(g(S)\subset L=\mathbb {P}^{1}_{K}\subset \mathbb {P}^{2}_{K}\) is Galois invariant with respect to the standard Galois action of \(\text {Gal}(K/k)\) on \(\mathbb {P}^{1}_{K}(K)\subset \mathbb {P}^{2}_{K}(K)\); it follows that g(S) descends to a finite subscheme of \(\mathbb {P}^{2}_{k}\) in this case, too.

Assume \(n=5\). Let \(S\subset \mathbb {P}^{2}(K)\) be a finite subset of degree 5 with field of moduli k, since 5 is prime with 3, it is enough to show that \(\mathscr {G}_{S}(k)\ne \emptyset \). We split the analysis in three cases: either S contains 4 points in general position, or it is contained in the union of two lines each containing at least three points of S, or it is contained in the union of a line and a point.

2.1 S contains 4 points in general position

Since we are assuming that there are 4 points of S in general position, there are two possibilities: either all 5 points are in general position, i.e., there is no line containing 3 of them, or there is a unique line containing exactly 3 points of S. Denote by C the unique non-degenerate conic passing through all the points of S in the first case, while in the second case C is the unique line containing 3 points of S.

In any case, C is a rational curve uniquely determined by S. Because of this, \(\mathscr {N}_{S}\) stabilizes C, consider the quotient \(\mathscr {C}=[C/\mathscr {N}_{S}]\subset \mathscr {P}_{S}\) and let \(\textbf{C}=C/\mathscr {N}_{S}\) be the coarse moduli space of \(\mathscr {C}\). Notice that, since \(C\cap S\ge 3\), the subgroup of \(\text {Aut}_{K}(\mathbb {P}^{2},S)\) fixing C has at most 2 elements, hence the map \(\mathscr {C}\rightarrow \textbf{C}\) is either birational or generically a gerbe of degree 2. In any case, since \({\text {char}}\, k\ne 2\) by the Lang-Nishimura theorem for tame stacks [10, Theorem 4.1] applied to a birational inverse \(\textbf{P}_{S}\dashrightarrow \mathscr {P}_{S}\) and to the generic point of \(\textbf{C}\subset \textbf{P}_{S}\), we get a generic section \(\textbf{C}\dashrightarrow \mathscr {C}\subset \mathscr {P}_{S}\).

The curve \(\textbf{C}\) is a Brauer–Severi variety of dimension 1 over k, and any canonical divisor has degree \(-2\), hence the index of \(\textbf{C}\) is either 1 or 2. Since \(C\cap S\) has odd degree, there exists an odd d such that \(C\cap S\) contains an odd number of orbits of degree d, let \(O\subset C\cap S\) be their union. Clearly, O is stabilized by \(\mathscr {N}_{S}\), hence \(O/\text {Aut}(\mathbb {P}^{2},S)\subset C/\text {Aut}(\mathbb {P}^{2},S)\) descends to a divisor of odd degree of \(\textbf{C}\); this implies that \(\textbf{C}\) has index 1, which in turn implies that \(\textbf{C}\) has a rational point and \(\textbf{C}\simeq \mathbb {P}^{1}_{k}\), see e.g. [14, Corollary 5.3.6, Theorem 5.1.3]. Since we have a map \(\textbf{C}\dashrightarrow \mathscr {C}\rightarrow \mathscr {P}_{S}\rightarrow \mathscr {G}_{S}\), this implies that \(\mathscr {G}_{S}(k)\ne \emptyset \) if k is infinite. If k is finite, the statement follows from the fact that \(\mathscr {N}_{S}\rightarrow \text {Gal}(K/k)\simeq \hat{\mathbb {Z}}\) is split and hence \(\mathscr {G}_{S}(k)\ne \emptyset \).

2.2 S is contained in the union of two lines

Assume that S is contained in the union of two lines \(L,L'\) each containing at least 3 points. Up to changing coordinates, we may assume that \((0:0:1),(0:1:0),(1:0:0)\in S\). It is now clear that, up to permuting the coordinates and multiplying them by scalars, we might assume that

$$\begin{aligned} S=\{(0:0:1),(0:1:0),(1:0:0),(0:1:1),(1:0:1)\}, \end{aligned}$$

which is clearly defined over k.

2.3 S is contained in the union of a line and a point

Suppose that S is contained in the union of a line L and a point p, choose coordinates such that \(p=(0:0:1)\) and \(L=\mathbb {P}^{1}\) is the line \(\{(s:t:0)\}\).

The field of moduli of \((\mathbb {P}^{2}_{K},S)\) is equal to the field of moduli of \((\mathbb {P}^{1}_{K},S\cap \mathbb {P}^{1}_{K})\): given \(\sigma \in \text {Gal}(K/k)\), clearly \(\sigma ^{*}(\mathbb {P}^{2}_{K},S)\simeq (\mathbb {P}^{2}_{K},S)\) if and only if \(\sigma ^{*}(\mathbb {P}^{1}_{K},S\cap \mathbb {P}^{1}_{K})\simeq (\mathbb {P}^{1}_{K},S\cap \mathbb {P}^{1}_{K})\). By [7, Proposition 13], \(\mathbb {P}^{1}_{K}\cap S\) descends to a closed subset of \(\mathbb {P}^{1}_{k}\). It follows that S descends to a closed subset of \(\mathbb {P}^{2}_{k}\).

3 Case \(n\ge 6\)

Let us now construct a counterexample with \(k=\mathbb {R}\), \(K=\mathbb {C}\) for every \(n\ge 6\).

If \(n\ge 6\), then either \(n=2m+4\) or \(n=2m+5\) for some \(m\ge 1\). Given \(a_{1},\dots ,a_{m}\in \mathbb {C}^{*}\), \(|a_{i}|\ne 1\), define

$$\begin{aligned} F= & {} \left\{ (\pm 1:0:1),(0:\pm 1:1)\right\} ,\\ S= & {} \left\{ (a_{i}:1:0),(1:-\bar{a}_{i}:0)\right\} _{i}\cup F,\\ S'= & {} S\cup \left\{ (0:0:1)\right\} , \end{aligned}$$

then \(|S|=2m+4\), \(|S'|=2m+5\). The matrix gives a linear equivalence between S and its complex conjugate, since \(g(F)=F\), \(g(a_{i}:1:0)=(1:-a_{i}:0)\) and \(g(1:-\bar{a}_{i}:0)=(\bar{a}_{i}:1:0)\). Similarly, g maps \(S'\) to its complex conjugate. It follows that both S and \(S'\) have field of moduli equal to \(\mathbb {R}\). Let us show that S is not defined over \(\mathbb {R}\) (the case of \(S'\) is analogous).

Let \(M\in \textrm{PGL}_{3}(\mathbb {C})\) be the image of . We have that M is a non-trivial automorphism of both S and \(S'\): in fact, \(M(a_{i}:1:0)=(-a_{i}:-1:0)=(a_{i}:1:0)\), \(M(\pm 1:0:1)=(\mp 1:0:1)\), \(M(0:\pm 1:1)=(0:\mp 1:1)\), \(M(0:0:1)=(0:0:1)\).

Lemma 2

For a generic choice of \(a_{1},\dots ,a_{m}\in \mathbb {C}\), \(|a_{i}|\ne 1\), and \(m\ge 1\), \({\text {Aut}}_{\mathbb {C}}(\mathbb {P}^{2},S)={\text {Aut}}_{\mathbb {C}}(\mathbb {P}^{2},S')=\left\langle M\right\rangle \).

Proof

For a generic choice of \(a_{1},\dots ,a_{m}\), there are exactly two lines containing exactly three points of \(S'\), hence their point of intersection (0 : 0 : 1) is fixed by \(\text {Aut}_{\mathbb {C}}(\mathbb {P}^{2},S')\subset \text {Aut}_{\mathbb {C}}(\mathbb {P}^{2},S)\). Since \(M\in \text {Aut}(\mathbb {P}^{2},S')\), it is enough to show \(\text {Aut}(\mathbb {P}^{2},S)=\left<M\right>\).

Let \(L=\left\{ (s:t:0)\right\} \) be the line at infinity. We first show that it is stabilized by \(\text {Aut}_{\mathbb {C}}(\mathbb {P}^{2},S)\) for a generic choice of the \(a_{i}\). If \(m\ge 2\), this is obvious since it is the only line containing at least four points of S.

Assume \(m=1\), \(a_{1}=a\). Since the stabilizer of F in \(\textrm{GL}_{2}(\mathbb {C})\) is finite and acts \(\mathbb {C}\)-linearly on \(\mathbb {P}^{2}\), for a generic a, there is no element of \(\text {Aut}_{\mathbb {C}}(\mathbb {P}^{2},S)\) swapping (a : 1 : 0) and \((1:-\bar{a}:0)\). Assume by contradiction that L is not stabilized. We may then also assume that the orbit of \((1:-\bar{a}:0)\) intersects F (if this happens for (a : 1 : 0) but not \((1:-\bar{a}:0)\), we just change coordinates).

Since M is an element of order 2 of \(\text {Aut}_{\mathbb {C}}(\mathbb {P}^{2},S)\) acting as a double transposition of F and no element of \(\text {Aut}_{\mathbb {C}}(\mathbb {P}^{2},S)\) swaps (a : 1 : 0) and \((1:-\bar{a}:0)\), it follows that there exists an element \(g\in \text {Aut}_{\mathbb {C}}(\mathbb {P}^{2},S)\) swapping some \(p\in F\) and \((1:-\bar{a}:0)\). In particular, g permutes the other four points \(F\cup \{(a:1:0)\}\smallsetminus \{p\}\), we may thus think of g as an element of \(S_{4}\). Since the four points \(F\cup \{(a:1:0)\}\smallsetminus \{p\}\) are in general position, for each element \(\sigma \in S_{4}\), there exists \(\phi _{\sigma }\in \textrm{PGL}_{3}(\mathbb {C})\) acting as \(\sigma \) on \(F\cup \{(a:1:0)\}\smallsetminus \{p\}\), and we may write \(\phi _{\sigma }\) as a \(3\times 3\) matrix whose entries are algebraic functions of a. Since complex conjugation is not algebraic, for a generic choice of a, we have \(\phi _{\sigma }(p)\ne (1:-\bar{a}:0)\) for every \(\sigma \in S_{4}\). This implies that for a generic choice of a, the automorphism g cannot exist (since \(g(p)= (1:-\bar{a}:0)\)), and hence L is stabilized.

If L is stabilized, then F is stabilized, too. The point (0 : 0 : 1) is the only point of intersection in \(\mathbb {P}^{2}{\smallsetminus } (L\cup S)\) of two lines passing through two points of F, hence it is fixed by \(\text {Aut}_{\mathbb {C}}(\mathbb {P}^{2},S)\). This implies that \(\text {Aut}_{\mathbb {C}}(\mathbb {P}^{2},S)\subset \textrm{GL}_{2}(\mathbb {C})\subset \textrm{PGL}_{3}(\mathbb {C})\).

The subgroup of \(\textrm{GL}_{2}(\mathbb {C})\) stabilizing F is \(D_{4}=\left<r,s\mid r^{4}=s^{2}=rsrs=1\right>\) generated by and , hence \(\text {Aut}_{\mathbb {C}}(\mathbb {P}^{2},S)\subset D_{4}\). The center of \(D_{4}\) is \(\left<r^{2}=-1\right>=\left<M\right>\subset \textrm{GL}_{2}(\mathbb {C})\), which is also the kernel of \(D_{4}\rightarrow \textrm{PGL}_{2}(\mathbb {C})\). Since \(D_{4}/\left<M\right>\simeq \mathbb {Z}/2\times \mathbb {Z}/2\) is finite and acts by \(\mathbb {C}\)-linear automorphisms on L, for a generic choice of \(a_{1},\dots ,a_{m}\), the intersection \(S\cap L\) is not stabilized by any non-trivial element of \(D_{4}/\left<M\right>\subset \textrm{PGL}_{2}(\mathbb {C})\). It follows that \(\text {Aut}_{\mathbb {C}}(\mathbb {P}^{2},S)=\left<M\right>\). \(\square \)

Write \(C_{a}\) for the cyclic group of degree a. Consider the natural projection \(C_{4}\rightarrow \text {Gal}(\mathbb {C}/\mathbb {R})=C_{2}\) giving a non-faithful action of \(C_{4}\) on \(\mathbb {C}\) and the Galois equivariant action on \(\mathbb {P}^{2}_{\mathbb {C}}\) given by \((a:b:c)\mapsto (-\bar{b}:\bar{a}:\bar{c})\). Clearly, \(C_{4}\) stabilizes S, hence we get a homomorphism \(C_{4}\rightarrow \mathscr {N}_{S}\). Lemma 2 implies that \(\mathscr {N}_{S}\) is an extension of \(\text {Gal}(\mathbb {C}/\mathbb {R})=C_{2}\) by \(C_{2}\), hence \(C_{4}\rightarrow \mathscr {N}_{S}\) is an isomorphism and \(\mathscr {G}_{S}\simeq [{\text {Spec}}\, \mathbb {C}/C_{4}]\). To conclude, it is enough to show that \([{\text {Spec}}\, \mathbb {C}/C_{4}](\mathbb {R})=\emptyset \).

By definition, an \(\mathbb {R}\)-point of \([{\text {Spec}}\, \mathbb {C}/C_{4}]\) corresponds to a \(C_{4}\)-torsor over \(\mathbb {R}\) with a \(C_{4}\)-equivariant map to \({\text {Spec}} \,\mathbb {C}\), see e.g. [19, Example 8.1.12]. There are two \(C_{4}\)-torsors over \(\mathbb {R}\), the trivial one and \(T={\text {Spec}}\, \mathbb {C}\cup {\text {Spec}} \,\mathbb {C}\), and neither of them has \(C_{4}\)-equivariant morphisms to \({\text {Spec}}\, \mathbb {C}\). This is clear for the trivial torsor, while for T, it follows from the fact that \(C_{2}\subset C_{4}\) acts non-trivially on each copy of \({\text {Spec}}\, \mathbb {C}\subset T\).

Notice that \(\textbf{P}_{S}(\mathbb {R})=\mathbb {P}^{2}_{\mathbb {C}}/C_{4}(\mathbb {R})\) is non-empty: however, the only real point (0 : 0 : 1) is an \(A_{1}\)-singularity, and hence we cannot apply the Lang-Nishimura theorem for stacks [10, Theorem 4.1] to conclude that \(\mathscr {P}_{S}(\mathbb {R})\) is non-empty (in fact, \(\mathscr {P}_{S}(\mathbb {R})=\emptyset \)). We mention that, for most types of 2-dimensional quotient singularities (but not \(A_{1}\)-singularities), the Lang-Nishimura theorem is still valid, see [9, §6] and [8] for details.