Moduli spaces of stable curves and R -equivalence

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Moduli spaces of stable curves and R -equivalence

Notes for the summer school, Grenoble 2010

Alena Pirutka July 2, 2010

In these notes we present some known results on R-equivalence on rationally con- nected varieties dened over function elds in one variable overCand, more generally, over C1 elds or over elds of cohomological dimension at most one. We will mostly focus on rationally simply connected varieties and explain in detail how to show that R-equivalence on such varieties is trivial, as soon as we are over a function eld in one variable overC(cf.[Pi]).

The notion of rationally simply connected varieties has been introduced by de Jong and Starr in [dJS]. They proved that a smooth complete intersection ofrhypersurfaces in Pⁿ_k of respective degrees d1, . . . , dr and of dimension at least 3 is rationally simply connected if P^r

i=1

d²_i ≤n+ 1. We sketch some of their arguments. Then we discuss how, using their ideas, one can get the triviality ofR-equivalence over a function eld in one variable overC.

The study of R-equivalence on rationally simply connected varieties requires some understanding of rational points on moduli space of stable curves and we discuss it in section 2. In section 3 we proceed to the study ofR-equivalence on RSC varieties. To nish we sketch some results onR-equivalence on other rationally connected varieties, over elds as above (cf.[CTSa], [CTSk], [Ma]).

1 Introduction

LetX be a (separably) rationally connected variety, dened over a eldk. Recall that two rational pointsx1, x2 of X are called directly R-equivalent if there is a morphism f : P¹_k → X such that x1 and x2 belong to the image of P¹_k(k). This generates an equivalence relation calledR-equivalence.

Assuming one of the following hypotheses : (i) k=C(C)a function eld of a complex curve;

(ii) k=C((t))a formal power series eld;

(iii) kis a C1 eld;

(iv) cd k≤1

one wonders ([CT], 10.11) if the set X(k)/R is trivial. In general, the answer is not expected to be positive, as pointed out in [CT], see also the remark below.

Nevertheless, it turns out that this is the case in all results we know :

1. X is a smooth compactication of a linear algebraic group andcd k≤1([CTSa]);

2. X is a surface bered in conics of degree4 over the projective line and cd k≤1 ([CTSk]);

3. Xis a smooth intersection of two quadrics inPⁿ_k withn≥5andcd k≤1([CTSaSD]);

4. X is a smooth cubic hypersurface inPⁿ_k withn≥5andk isC1 ([Ma]);

5. k=C(C)ork=C((t))andX is a smooth complete intersection ofrhypersurfaces inPⁿ_k of dergreesd1, . . . drsatisfyingP

d²_i ≤n+1. More generally, the same holds for ak-rationally simply connected variety ([Pi]).

Remark 1.1. The triviality ofR-equivalence for smooth projective geometrically rational surfaces overC(t)would imply the unirationality of (smooth projective) varieties of dimension3, bered in conics overP²_C, which is an open question.

In fact, let p : X → P²_C be a morphism from a smooth projective variety X of dimension3, such that the general ber ofpis a conic. AsP²_C is birational toP¹_C×P¹_C, we can replaceX byp⁰ :X⁰ →P¹_C×P¹_C with the same assumption : the general ber ofp⁰ is a conic. Let η: SpecC(t)→P¹_C be the generic point. Base change byη on the second factor gives a morphismp⁰_η : X_η⁰ →P¹_C(t). The variety X_η⁰ is a (geometrically) rational surface overk=C(t). If the R-equivalence onX_η⁰(k)is trivial, one can nd a rational curvef :C=P¹_C(t) →X_η⁰ joining two rational points in dierent bers ofp⁰_η. This means that the induced mapp⁰_η◦f :C→P¹_C(t)is surjective. Base change byp⁰_η◦f gives the following diagram :

Y =X_η⁰ ×_P¹

C(t)C −−−−→^g C'P¹_C(t)

 y X_η⁰

The general ber ofgis a conic, having a rational point as the mapghas a section by our construction. This means that Y is rational over C(t). The projection Y → X_η⁰ now shows thatX⁰ is unirational.

In what follows we explain the proof of 5 and related questions. We will also give a sketch of ideas for other results. First, we need to establish some facts on moduli spaces of stable curves.

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2 Rational points on moduli space of curves

In this section we work with the moduli space M¯0,2(X, d) of stable curves of genus zero. We analyse what one can say about an object representing a rational point of this space. In particular, we are interested in applications toR-equivalence.

2.1 Basic facts on M ¯

_0,n

(X, d)

Let us rst make precise some facts about the moduli spaceM¯0,n(X, d). LetX be a projective variety over a eldk. LetH be an ample divisor onX. Let¯kbe an algebraic closure of k. The space of rational curves of xed degree¹ on X is not compact in general. One way to compactify it, due to Kontsevich, is to use stable curves.

Denition 2.1. A stable curve over X of degree dwith n marked points is a datum (C, p1, . . . , pn, f)of

(i) a proper geometrically connected and geometrically reducedk-curveC with only nodal singularities,

(ii) an ordered collectionp1, . . . , pn of distinct smoothk-rational points ofC, (iii) ak-morphismf :C→X with deg_Cf^∗H =d,

such that the stability condition is satised :

(iv) C has only nitely many ¯k-automorphisms xing the pointsp1, . . . , pn and com- muting withf.

We say that two stable curves (C, p1, . . . , pn, f)and (C⁰, p⁰₁, . . . , p⁰_n, f⁰)are isomorphic if there exists an isomorphismφ:C →C⁰ such thatφ(pi) =p⁰_i, i= 1, . . . , nand f⁰◦φ=f.

The precise construction of the moduli space of stable curves in this sense can be found in the article of Araujo and Kollár [AK]. Here are some important points from [AK] which we will use in what follows :

1. There exists a coarse moduli space M¯g,n(X, d) for all stable curves over X of arithmetic genus g of degree d with n marked points, which is a projective k- scheme ([AK], Thm. 50).²

2. Saying thatM¯g,n(X, d)is a coarse moduli space means the following : (i) there is a bijection of sets :

Φ :









isomorphism classes of genus gstable curves overk¯ f :C→X¯k withnmarked points,

deg_Cf^∗H =d









→∼ M¯g,n(X, d)(¯k);

(ii) if C → S is a family of genus g stable curves of degree d with n marked points, parametrized by ak-schemeS, then there exists a unique morphism MS :S→M¯g,n(X, d)such that for everys∈S(¯k)we have

MS(s) = Φ(Cs).

1We x the degree of the curves we consider in order to have a space of nite type.

2Over Cthe construction was rst given in [FP]. In this paper, the authors x a curve class β∈H^2dimX−2(X,Z)rather than the degree. They consider stable curvesf :C→X such that the class off∗[C]isβ. They prove that there exists a coarse moduli spaceM¯g,n(X, β)parametrizing all genusgstable curves of class βwithnmarked points. The result in [AK] holds over arbitrary, not necessarily algebraically closed eld and, more generally, over a noetherian base.

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3. Note that over nonclosed elds we do not have a bijection between isomorphism classes of stable curves and rational points of the corresponding moduli space, see [AK] p.31. In particular, ak-point of M¯g,n(X, d) does not in general correspond to a stable curve dened overk.

We denote byMg,n(X, d)the open locus corresponding to irreducible curves and by evn: ¯Mg,n(X, d)→X| ×. . .{z ×X}

n

the evaluation morphism which sends a stable curve to the image of its marked points.

In what follows we will focus on stable curves of genus zero.

Let P and Q be two k-points of X. Suppose there exists a stable curve of genus zerof : C →Xk¯ over ¯k with two marked points mapping to P andQ, such that the corresponding pointΦ(f)is ak-point ofM¯0,2(X, d). Our goal is to deduce thatP andQ areR-equivalent overk. We will explain two ways how to proceed. The rst one, quite elementary, makes use of the combinatorics particular to stable curves of genus zero.

The second way requires more sophisticated tools and applies to elds of cohomological dimension at most one. Let us state the main result of this section.

Proposition 2.2. LetX be a projective variety over a eldkof characteristic zero. Let P andQbek-points of X. Letf :C→X¯k be a stable curve over¯kof genus zero with two marked points mapping to P andQ. Let H be a xed ample divisor on X and let d= deg_Cf^∗H. If the corresponding pointΦ(f)∈M¯0,2(X, d)is ak-point ofM¯0,2(X, d), then the pointsP andQareR-equivalent overk.

2.2 Combinatorial arguments

2.2.1 Notations

Let k be a eld of characteristic zero. Let us x an algebraic closure ¯k of k. Let L,→ⁱ ¯kbe a nite Galois extension ofk, and letG= Autk(L). For anyσ∈Gwe denote σ^∗: SpecL→SpecLthe induced morphism. If Y is anL-variety, denote ^σY the base change of Y by σ^∗ and ^σYk¯ the base change by (i◦σ)^∗. We denote the projection

σY → Y by σ^∗ too. If f : Z → Y is an L-morphism of L-varieties, then we denote

σf : ^σZ→ ^σY and ^σf¯k: ^σZ¯k → ^σY¯k the induced morphisms.

Note that if Y ⊂Pⁿ_L is a projective variety, then ^σY can be obtained by applying σto each coecient in the equations deningY. Thus, ifY is dened overk, then the subvarietiesY,^σY of Pⁿ_L are given by the same embedding for all σ∈G. In this case the collection of morphisms {σ^∗ : Y → Y}σ∈G denes a right action ofG on Y. By Galois descent ([BLR], 6.2), if a subvarietyZ⊂Y is stable under this action ofG, then Z also is dened overk.

2.2.2 Some lemmas on graphs

Let us rst give a proof of the following well-known lemma :

Lemma 2.3. LetC be a projective geometrically connected and geometrically reduced curve of arithmetic genuspa(C) =h¹(C, OC) = 0over a perfect eld k. AssumeC has only nodal singularities. Then any two smoothk-pointsa, bof C areR-equivalent.

Proof. For any eld extension F of k let us call an F-path joining a and b a closed F-subcurveC⁰⊂CF such that

(i) C⁰=C₁⁰ ∪C₂⁰ ∪. . .∪C_r⁰ whereC_i⁰,i= 1, . . . r are smoothF-rational curves;

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(ii) a∈C₁⁰,b∈C_r⁰;

(iii) if1 ≤i≤r−1, the intersection C_i⁰∩C_i+1⁰ is an F-point and the curves C_i⁰ and C_i+j⁰ do not intersect for j >1.

Since the arithmetic genus of C is zero, its geometric irreducible components are smooth rational curves over ¯k intersecting transversally. As C is geometrically connected, there exists ak¯-pathC⁰ joiningaandb. We may assume thatC⁰ is anL-path for some nite Galois extension L of k. Moreover, such a path is unique : if there were two dierent paths we would have a cycle formed by components ofCL, which is impossible aspa(C) = 0. We would like to nd ak-path, thus we will achieve the proof.

Let us writeC⁰ =C₁⁰ ∪C₂⁰ ∪. . .∪C_r⁰,a∈C₁⁰,b∈C_r⁰. We will show thatC⁰ comes from ak-path by base extension. Let us takeσ∈Autk(L). Then ^σC₁⁰, . . . ,^σC_r⁰ is an L-path joiningaandb. Since such a path is unique, for everyi= 1, . . . rthe components C_i⁰ and ^σC_i⁰ ofCLare equal and C_i⁰∩C_i+1⁰ = ^σC_i⁰∩^σC_i+1⁰ =σ(C_i⁰∩C_i+1⁰ ),i= 1, . . . r.

This means that every component of the path C_i⁰ ⊂ C_L is stable over the action of Autk(L) on CL, hence it is dened over k, that is C_i⁰ = Di×kL, for some k-curve Di⊂C. By the same argument, the intersection points ofDi andDi+1,i= 1, . . . r−1 arek-points. We deduce that {D1, . . . Dr} is ak-path joiningaandb, so the pointsa andbareR-equivalent overk.

The next lemma will be used in the proof of Proposition 2.2. The necessity of all the hypotheses, which amounts to saying that we have a rational point on the moduli space, will be clear from the context.

Lemma 2.4. LetX be a projective variety over a perfect eldk. LetLbe a nite Galois extension ofk. Denote G= Autk(L). LetP andQ bek-points ofX. Suppose we can nd an L-stable curve of genus zero f :C →XL with two marked points a, b∈C(L), satisfying the following conditions :

(i) f(a) =P,f(b) =Q;

(ii) for everyσ∈Gthere exists a¯k-morphismφσ:Ck¯→ ^σC¯k such that φσ(a) =σ(a), φσ(b) =σ(b)and ^σf¯k◦φσ =f¯k. Then the pointsP andQ areR-equivalent overk.

Proof. By Lemma 2.3, we have a uniqueL-path{C1, . . . , Cm} joining a∈C1(L)and b ∈ Cm(L), where the Ci are irreducible components of C. We will use the curves f(C1), . . . f(Cm)to show thatP andQareR-equivalent overk. Let us rst show that these curves are dened overkand not only over L.

For every σ ∈G we have an L-path {^σC1, . . . ,^σCm} joining σ(a) ∈ ^σC1(L) and σ(b)∈ ^σCm(L). On the other hand,{φσ(C_1,¯k), . . . , φσ(C_m,¯k)}is a¯k-path joining σ(a) andσ(b). Since the arithmetic genus of^σC¯k is zero, such a path is unique. That is, it coincides with the path{^σC_1,¯k, . . . ,^σC_m,¯k}. So we have

φσ(C_i,¯k) = ^σC_i,¯k, i= 1, . . . , m.

Let us x 1 ≤i ≤ m. Denote the image f(Ci) of Ci in XL byZi. As ^σf is a base change byσ^∗, we have the following commutative diagram :

σCi

σf

−−−−→ ^σXL

 y

 y Ci f

−−−−→ XL

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We thus see that ^σf(^σCi) = ^σZi. Using base change by i : L → ¯k in the rst line of the diagram above we obtain that ^σf¯k(^σC_i,¯k) = ^σZ_i,¯k. On the other hand, since φσ(C_i,¯k) = ^σC_i,k¯ and ^σf¯k◦φσ=fk¯, we have ^σZ_i,k¯= ^σf¯k(^σC_i,¯k) = ^σfk¯(φσ(C_i,¯k)) = f¯k(C_i,¯k) =Z_i,¯k. Since ^σZi andZi areL-subvarieties of XL, we deduce that ^σZi =Zi

for allσ∈G. By Galois descent, this means that the curveZi is dened over k, that is, there exists ak-curveDi⊂X such thatZi=Di×kL.

To conclude, we will show that the curve Di is a k-rational curve on X, that is, it is the image of some morphism fromP¹_k to X, and that the pointf(Ci∩Ci+1)is the image of ak-point. Note that it may be not so obvious iff(Ci∩Ci+1) is a singular point ofDi.

LetD˜i →Di be the normalisation morphism. It induces an isomorphism over the smooth locusD^sm_i . Since Ci is smooth, the morphismf|Ci :Ci→Di×kLextends to a morphismfi:Ci →D˜i×kL:

D˜i×kL

²²C_i ^f|^Ci//

fi

;;v

vv vv vv vv

D_i×_kL.

This implies that D˜i is an L-rational curve. We have ^σf_i,¯k ◦φσ = f_i,¯k, as this is true over a Zariski open subset D^sm_i . Moreover, for every 1 ≤ i ≤ m−1 we have φσ(Ci∩Ci+1) = ^σCi∩ ^σCi+1. Using the same argument as above, we deduce that the pointfi(Ci∩Ci+1)is ak-point of D˜i. This implies thatD˜i is ak-rational curve as it isL-rational and has ak-point. Moreover, the pointf(Ci∩Ci+1)is ak-point ofX as the image offi(Ci∩Ci+1). HenceP isR-equivalent tof(C1∩C2)as there is a rational curveD˜1→X connecting them. By the same argument,f(Ci−1∩Ci)isR-equivalent tof(Ci∩Ci+1)for all1< i < m−1andf(Cm−1∩Cm)isR-equivalent toQ. Therefore P andQareR-equivalent.

2.2.3 Proof of Proposition 2.2.

We will show that the hypotheses of Lemma 2.4 are satised. We call aand b the marked points ofC. We may assume thatC,f,aandbare dened over a nite Galois extensionL,→ⁱ k¯ ofk. That is, we may assume that C is anL-curve,a, b∈C(L)and that we have anL-morphismf :C→XL. Let us denoteT = SpecL. We viewLas a k-scheme andf : C →X×kT as a family of stable curves parametrized by T. Thus we have a moduli mapMT :T = SpecL→M¯0,2(X, d)dened overkand such that for everyt∈T(¯k)we have

MT(t) = Φ(Ct) wheref_t:C_t→Xk¯ is the bre off :C→X×_kT overt.

Note that T×kk¯ = Q

G

Spec ¯k where the product is indexed by G= Autk(L) and the morphismQ

G

¯k→T = SpecLis given by(i◦σ)^∗ on the corresponding component.

This implies that a¯k-pointt∈T(¯k)corresponds to someσ∈Gand the morphismftis the base change by(i◦σ)^∗. Hence the morphism ftis the morphism ^σf¯k : ^σC¯k→X¯k

and the marked points of^σCk¯ areσ(a)andσ(b).

Since the curvef¯k:Ck¯→X¯k corresponds to a k-point ofM¯0,2(X, d), we can factor MT as

T = SpecL→Speck^Φ(f→^¯^k⁾M¯0,2(X, d).

We thus see that for every t ∈ T(¯k) the point MT(t) is the same point Φ(fk¯) of M¯_0,2(X, d). Hence for every σ ∈ G the curves ^σC¯k and C¯k are isomorphic as stable

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curves. This means that there exists ak¯-morphismφσ :C¯k→ ^σCk¯, such that φσ(a) =σ(a), φσ(b) =σ(b)and^σf¯k◦φσ =f¯k.

Now the proposition follows from lemma 2.4.

2.3 Stack-theoretical arguments

We will give a second proof of Proposition 2.2 in the case when the base eldk is of cohomological dimension at most1. In fact, using this assumption and the fact that the automorphism group a stable curve is nite, we will show that any rational point of a moduli space corresponds to an object dened over the base eld. We will present a point of view of [DDE], all the arguments can be found there.

Denition 2.5. Let k be a perfect eld, ¯k an algebraic closure and G = Gal(¯k/k). We say that k is of cohomological dimension at most 1 if for any (continuous) nite G-moduleM and any integeri≥2we have Hⁱ(G, M) = 0.

Example 2.6. AnyC1 eld is of cohomological dimension at most 1. Note that the converse is not true ([A]). Thus nite elds, function elds in one variable over an algebraically closed eld and formal series elds in one variable over an algebraically closed eld give examples of a eld ofcd≤1.

Let us give a short sketch of the argument. It is a general fact that, given a pointx on a moduli space, corresponding to an object over¯kwith the automorphisms groupG, the obstruction to liftxto an object over klives in a certain (non-abelian, asGis not necessarily abelian)2-cohomology set (and not a group in general), in the sense of [Gi].

Now, the fact that G is nite, allows us to reduce to the abelian case and to deduce thatH²vanishes under the hypothesis cd k≤1.

We rst give some notions from non-abelian cohomology.

2.3.1 Gerbes and non-abelian cohomology LetS be an étale site.

Denition 2.7. AnS-gerbe is a stack³ satisfying the following conditions :

(i) any two sections over an open set U are locally isomorphic, i.e. there exists an open subset V ⊂U such that the restrictions to V of the two given section to G(V)are isomorphic;

(ii) locally each ber is nonempty : each open setU admits an open subset V such that the ber aboveV is nonempty.

Example 2.8. Let us consider the following stackGf over the étale siteSpecket´ : (i) the objects ofGf over an extensionLofkare stable curvesD→XLoverLwhich

become isomorphic overk¯to thek-curve¯ C→X¯kcorresponding to the pointΦ(f) as in Proposition 2.2.

(ii) the morphisms between two stable curves overL areL-isomorphisms.

3Let us briey recall the denition of a stack. It is a categoryX bered on groupoids overS(i.e.

for each openU⊂Sthe berX(U)is a groupoid), such that for each openU⊂Sandx, y∈X(U), Hom(x, y)is a sheaf and the following glueing condition is satised : given an openU⊂S, an open covering(Ui)i∈I of U and elementsxi ∈X(Ui), i∈I, if for alli, j there exists an isomorphismφij

between the restrictions ofxiandxjtoUi∩Ujsuch thatφij=φikφkjoverUi∩Uj∩Uk, then there existsx∈X(U)restricting toxioverUi.

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One can also view Gf as the bre of the morphism M0,2(X, d) → M¯0,2(X, d) over the point Φ(f), where M0,2(X, d) is the stack of all genus zero stable curves over X of degree d with two marked points. By this description, the objects of Gf exist locally (that is, over some nite extension of k). Moreover, any two such objects are locally isomorphic (that is, after taking some nite extension ofk). What we want to prove is the existence of objects overk, that is, that the curveC→X¯kis dened overk. Denition 2.9. A gerbe which has objects overS is called neutral.

Remark 2.10. The following picture may illustrate the terminology of gerbes and stacks (champ in french, which means eld) :

COARSE MODULI

K−POINT

NEUTRAL GERBE GERBE

L −−−

EXTENSION OF K

OBJECTS OVER L

By considering the automorphism groups of objects of G one can associate a band L = L(G) to the gerbe G, see [Gi] Ch.IV for details. For example, if S = Speck_ét, then anS-band corresponds to a groupAendowed with a homomorphismGal(¯k/k)→ Aut(A)/Inn(A). Next, one can dene a cohomological set H²(S,L)parametrizing all classes of gerbes whose associated band isL. Note that in the case ofGf the automorphism group of objects is locally (that is, starting from some extension ofk) a nite constant group consisting of the automorphisms ofC→Xk¯ over¯k.

A morphismu:L → M of bands induces a relation H²(S,L)(H²(S,M)

wherep(q means that there are gerbesP andQof classespandqrespectively and anu-morphismP →Q. Ifpis neutral and ifp(q thanqis neutral.

2.3.2 Reduction to the abelian case

Let us give a sketch of the proof of the following result of Dèbes, Douai and Emsalem ([DDE], cor. 1.3) :

Theorem 2.11. If the cohomological dimension ofk is at most one, then any gerbe G over the étale siteS= Speck_étwhose associate bandLis locally a constant nite group C is neutral.

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Proof. As the authors point out, their idea goes back to the work of Springer [Sp]. Let us rst suppose that C is not nilpotent. Then one can nd a prime pand a p-Sylow subgroupH of C which is not normal. One veries that the following data denes a gerbeG⁰ overS :

(i) the objects ofG⁰over a separable extensionLofkare the couples(x, T),x∈ G(L) andT is a subsheaf inp-Sylow ofAutL(x);

(ii) theL-morphisms(x, T)→(x⁰, T⁰)are theL-morphismsa:x→x⁰ such that the induced morphisma∗: AutL(x)→AutL(x⁰)mapsT toT⁰.

The band associated to the gerbe G⁰ is locally the normalizer B = NC(H) and by contruction B is a proper subgroup of C. The morphism of bandsLB → LC induced by the inclusion gives a relationH²(S,LB)(H²(S,LC). Thus ifG⁰is neural, then we have the same forG.

Thus we may reduce to the caseC is nilpotent. Let 0→C⁰→C→C⁰⁰→0

be an exact sequence withC⁰⁰ abelian. The fact that π: C→ C⁰⁰ is surjective allows us to dene a band π∗L and the fact that C⁰⁰ is abelian implies that we have a well dened morphism π∗ : H²(S,L) → H²(S, π∗L). As cd k ≤ 1 and C⁰⁰ is abelian, the imageπ∗([G])is zero. As one can check, this implies that[G]comes from an element of H²(S,LC⁰). Thus we conclude the proof by an induction argument.

Now we can apply the previous theorem to Gf. We conclude that Gf is neutral and thus there is a genus zero k-stable curve f⁰ : C⁰ → X with two marked points a, b∈C⁰(k)such that the images of aand b are respectively the pointsP and Q. By lemma 2.3 we deduce thataand b areR-equivalent in C⁰, thus their images P andQ inX are alsoR-equivalent.

3 R-equivalence over function elds in one variable

3.1 Case of RSC varieties

One can view rationally connected varieties as an analogue of path connected spaces in topology. From this point of view, de Jong and Starr introduce the notion of rationally simply connected varieties as an algebro-geometric analogue of that of simply connected spaces. We use here the following denition⁴:

Denition 3.1. Let k be a eld of characteristic zero. A projective geometrically integral varietyX over k is called k-rationally simply connected if for any suciently large integer e there exists a geometrically irreducible component Me,2 ⊂ M¯0,2(X, e) intersecting the open locus of irreducible curvesM0,2(X, e), such that the restriction of the evaluation morphism

ev2:Me,2→X×X is dominant with rationally connected general ber.

Note that a k-rationally simply connected variety X over a eld k is rationally connected asX×X is dominated byMe,2∩M0,2(X, e)from the denition above. This implies that two general points ofX over any algebraically closed eld Ω⊃k can be connected by a rational curve.

The following result is essentially contained in [dJS] and gives an example of RSC varieties in the sense above (essentially the only one we know). We explain some points

4The denition in [dJS] is given overC. Here we emphasize that the distinguished componentMe,2

should be dened overk. Thus we use the notion ofk-rational simple connectedness.

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of the proof, having in mind that we are interested in applications to not algebraically closed elds.

Proposition 3.2. Let k be a eld of characteristic zero. LetX be a smooth complete intersection ofrhypersurfaces in Pⁿ_k of respective degreesd1, . . . , drwith P^r

i=1

d²_i ≤n+ 1.

Suppose thatdimX ≥3. Then for everye≥2 there exists a geometrically irreducible k-component Me,2⊂M¯0,2(X, e)such that the evaluation morphism

ev2:Me,2→X×X is dominant with rationally connected generic bre.

Proof. Let us rst recall the construction of [dJS] in the case k =C. In that paper, the authors work with the spaceM¯0,2(X, β)of [FP] which parametrizes stable curves of genus zero over X of classβ ∈H^{2dim X−2}(X,Z) with two marked points. Hovewer, asdimX ≥3, we know thatH^{2dim X−2}(X,Z) =Zαwhere the degree ofαequals to 1 ([V], 13.25). Thus we can replaceβ by its degreeeand work with the spaceM¯0,2(X, e) as in [AK].

In [dJS], de Jong and Starr prove that for every integer e ≥ 2 there exists an irreducible component M_e,2 ⊂ M¯_0,2(X, e) such that the restriction of the evaluation morphismev2:Me,2→X×X is dominant with rationally connected generic bre. We will specify more precisely how they get the componentMe,2. It will follow from their construction thatMe,2 is in fact the unique component satisfying the above property.

The construction ofMe,2is as follows :

1. One rst shows that there exists a unique irreducible componentM1,1⊂M¯0,1(X,1) such that the restriction of the evaluationev1|M1,1 :M1,1→X is dominant ([dJS], 1.7).

2. The component M1,0 ⊂ M¯0,0(X,1) is constructed as the image of M1,1 under the morphism M¯0,1(X,1) → M¯0,0(X,1) forgetting the marked point. Then one constructs the component of higher degree Me,0 as the unique component of M¯0,0(X, e) which intersects the subvariety of M¯0,0(X, e) parametrizing degree ecovers of smooth, free curves parametrized byM1,0 ([dJS], 3.3).

3. The componentM_e,2 ⊂M¯_0,2(X, e)is the unique component such that its image under the morphism M¯0,2(X, e)→ M¯0,0(X, e), which forgets about the marked points, isMe,0.

The proof of the fact that the general ber of the evaluation morphismev2:Me,2→ X ×X is rationally connected uses elaborated arguments, in particular, it uses the techniques of twisting surfaces. We will not discuss it here, see [dJS] for details.

Let us now consider the general case. Let ¯k be an algebraic closure of k. As k is of nite type over Q, we may assume that ¯k ⊂ C. Since the decomposition into geometrically irreducible components does not depend on which algebraically closed eld we choose, by the rst step above there exists a unique irreducible component M1,1⊂M¯0,1(Xk¯,1)such that the restriction of the evaluationev1|M1,1 is dominant. As this component is unique, it is dened overk. Hence, from the construction above, the componentMe,2is also dened over k, which completes the proof.

Next, we will give a proof of the following theorem :

Theorem 3.3. Letkbe either a function eld in one variable overCor the eldC((t)). LetX be a k-rationally simply connected variety overk. Then X(k)/R= 1.

Combined with the theorem of de Jong and Starr, this gives :

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Corollary 3.4. Let k be either a function eld in one variable over C or the eld C((t)). Let X be a smooth complete intersection of rhypersurfaces in Pⁿ_k of respective degrees d1, . . . , dr. Assume that P^r

i=1

d²_i ≤n+ 1.ThenX(k)/R= 1. 3.1.1 Sketch of the proof

The hypothesis thatX is rationally simply connected implies that there exists a (suf- ciently large) integereand a geometrically irreducible component Me,2⊂M¯0,2(X, e) such that the evaluation morphism ev2 : Me,2 → X×X is dominant with rationally connected general bre.

Let P and Q be two k-points of X. Let us suppose that k is a function eld in one variable over C. A general strategy is the following. We would like to apply the theorem of Graber, Harris and Starr [GHS] to deduce that there is a rational point in the bre over(P, Q). If it is so, we can use Proposition 2.2 to deduce thatP andQare R-equivalent. But we only know that a general bre ofev2 is rationally connected. We will use two methods to solve this problem. We will also show how to apply the same argument to the casek=C((t)).

3.1.2 Specialisation arguments

First method. The rst possibility is to use the following theorem of Hogadi and Xu [HX] :

Theorem 3.5. Let k be a eld of characteristic zero. Let h: Y → Z be a dominant proper morphism ofk-varieties such that Z is smooth and the generic bre of his rationally connected. Then for every pointz∈Z there exists a subvariety of the breYz, dened overk(z), which is geometrically irreducible and rationally connected.

Let X be as in Theorem 3.3. Let us, as before, take two k-points P and Q of X. By the theorem above, we can nd a rationally connected k-subvarietyV in the bre ev⁻¹₂ (P, Q). Ifk=C(C), there is a rational point inV by [GHS], hence inev₂⁻¹(P, Q). By Proposition 2.2, the pointsP andQareR-equivalent. So we obtainX(k)/R= 1. Second method. The next argument shows that every ber in a family with rationally connected general ber has a rational point, as soon as we are over the function eld of a complex curve. See also [Sta] p.25.

Lemma 3.6. Letk=C(C) be the function eld of a (smooth) complex curve C. Let Z andT be projective k-varieties, with T smooth. Letf :Z →T be a morphism with rationally connected general bre. Then for everyt∈T(k)there exists a rational point in the breZt.

Proof. One can choose proper modelsT →C and F :Z → T ofT andZ respectively withT smooth. We know that any bre of F over some open set U ⊂ T is rationally connected.

The pointt∈T(k)corresponds to a sections:C → T. What we want is to nd a sectionC→ Z ×T C. One can view the images(C)inT as a component of a complete intersectionC⁰ of hyperplane sections of T for some projective embedding. In fact, it is sucient to takedimT −1 functions in the ideal ofs(C)inT generating this ideal over some open subset of s(C). Moreover, one may assume that C⁰ is a special bre of a family C of hyperplan sections with general bre a smooth curve intersecting U. After localization, we may also assume thatC is parametrized byC[[t]]. LetAbe any ane open subset in C containing the generic point ξof s(C). We have the following

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diagram:

Z ×T SpecA

FA

²² //Z

F

²²ξ //SpecA⊗_C[[t]]C //

²²

SpecA

²² //T

SpecC //SpecC[[t]].

LetK =C((t))and let K¯ be an algebraic closure ofK. By construction, the generic bre ofFK¯ :Z ×T K¯ →SpecA⊗C[[t]]K¯ is rationally connected. By [GHS] we obtain a rational section ofFK¯. AsK¯ is the union of the extensionsC((t^1/N))forN ∈N, we have a rational section for the morphismZ ×TC[[t^1/N]]→SpecA⊗C[[t]]C[[t^1/N]]for someN. By properness, this section extends to all codimension1points ofSpecA⊗C[[t]]C[[t^1/N]], in particular, to the pointξ on the special ber. This extends again to give a section C→ Z ×_T C as desired.

3.1.3 Casek=C((t))

The theorem of Graber, Harris and Starr admits the following corollary over the power series elds :

Theorem 3.7. LetX be a projective rationally connected variety overC((t)). ThenX has a rational point.

Proof. We present here a proof from [CT], 7.5. We will use the following two facts : Fact 1. (Theorem of Greenberg, [Gr]) LetZbe a variety over a henselian discrete valuation

ringO. Lett be a generator of the maximal ideal. Suppose that the residue eld ofOis of characteristic zero. Then Z(O)6=∅ ⇔Z(O/tⁿ)6=∅for alln >0. Fact 2. (cf.[BLR] p.82) Let K be the eld of fractions of a henselian discrete valuation

ring. LetKˆ be a completion ofK. LetZ be a smoothK-variety. ThenZ(K)is dense inZ( ˆK).

Let us x some notations.

1. We denote Rthe henselisation ofC[t]in t= 0and K the eld of fractions ofR.

Note that Kˆ can be identied toC((t)). We also haveR/tⁿ'C[t]/tⁿ.

2. LetX be the closure ofX ⊂Pⁿ_C((t)) in Pⁿ_C[[t]]. There exists aC-algebra i:A ,→ C[[t]]of nite type and a projective and at A-scheme X⁰ such thatX_C[[t]]⁰ ' X (takeAto be generated by the coecients of equations dening X in Pⁿ_C[[t]].) 3. Let us denoteS= SpecAand letξ∈S(C[[t]])be the point corresponding toi.

4. LetU ⊂S be an open set such thatX_u⁰ is rationally connected for allu∈U. One may assume thatU is smooth.

Now we proceed to the proof of the theorem. By fact2,U(K)⊂U(C((t)))is dense.

On the other hand, the mapS(C[[t]])→S(C[t]/tⁿ)has open bres. Thus there exists a pointξn∈U(K)∩S(C[[t]])having the same image inS(C[t]/tⁿ) =S(R/tⁿ)asξ. By considering the valuation attwe observe thatξ_n is in fact anR-point.

Let us denoteXn the base change ofX⁰ byξn. We viewXn as an R-scheme. Note that the generic breXn/K ofXn is rationally connected by the choice of U and that Xn×RR/tⁿ ' X⁰×C[t]/tⁿ by the choice ofξn. As the fraction eld K ofR=C[t]^h_(t)

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is a union of function elds of curves,Xn(K)is not empty by [GHS]. By properness, Xn(R)is not empty. ThusX⁰×C[t]/tⁿ ' Xn(R/tⁿ)is not empty. By fact1, we deduce thatX(C((t)))6=∅.

3.1.4 Proof of 3.3

The theorem now follows by combining the previous results. Let, as before,Me,2 ⊂ M¯0,2(X, e)be a geometrically irreducible component, such that the restriction of the evaluation morphismev₂:M_e,₂→X×Xis dominant with rationally connected general bre.Let P and Q be two k-points ofX. We know that a general bre of ev2 is rationally connected. Ifk=C((t)), one concludes using 3.7 and the fact thatR-equivalence classes are Zariski dense in this case by [Ko99]. Ifk is a function eld in one variable overC,ev₂⁻¹(P, Q)(k)is not empty by 3.6. By Proposition 2.2, the pointsP andQare R-equivalent. So we obtainX(k)/R= 1in both cases.

Note that the methods used in the proof of the theorem apply more generally over a eld kof characteristic zero such that any rationally connected variety over k has a rational point.

As for the corollary, there is in fact a much simpler proof for anyC1eld in the case Pd²_i ≤n. The argument is due to Jason Starr.

Proposition 3.8. Letkbe aC1eld. LetX ,→ⁱ Pⁿ_k be the vanishing set ofrpolynomials f1, . . . fr of respective degreesd1, . . . dr. IfP

d²_i ≤nthen any two pointsx1, x2∈X(k) can be joined by two lines dened overk: there is a pointx∈X(k)such thatl(x, xi)⊂ X,i= 1,2, wherel(x, xi)denote the line through xandxi.

Proof. We may assume thatx1 = (1 : 0 :. . . : 0)and x2 = (0 : 1 : 0 :. . . : 0)via the embeddingi. The question is thus to nd a pointx= (x0:. . . :xn)with coordinates inksuch that ½

fi(tx0+s, tx1, . . . txn) = 0

fi(tx0, tx1+s, . . . txn) = 0, i= 1, . . . r.

Asx1, x2are inX(k)these conditions are satised fort= 0. Thus we may assumet= 1.

Writingfi(x0+s, x1, . . . xn) = P^dⁱ

j=0

P_jⁱ(x0, . . . xn)s^jwithP_jⁱhomogeneous of degreedi−j we see that each equationfi(x0+s, x1, . . . xn) = 0gives usdi conditions onx0, . . . xn

of degrees1, . . . di. By the same argument, each equationfi(x0, x1+s, . . . xn) = 0gives di−1conditions of degrees1, . . . di−1as we know from the previous equation that we have no term of degree zero. The sum of the degrees of all these conditions onx0, . . . xn

is P^r

i=0

d²_i. As P^r

i=0

d²_i ≤ n by Tsen-Lang theorem we can nd a solution over k, which nishes the proof.

3.2 Case of cubic hypersurfaces

The case of R-equivalence on cubic hypersurfaces was studied by Madore. Let us sketch the proof of the folowing result ([Ma]) :

Theorem 3.9. Letk be aC1eld. LetX ⊂Pⁿ_k be a cubic hypersurface. Ifn≥5, then X(k)/R= 1.

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Proof. LetP, Qbe two rational points ofX. Let us rst assume thatX is a singular at at least one of these points. We may thus assume thatP = (1 : 0 :. . .: 0)is a singular point ofX. ThusX is dened by an equationx0q(x1, . . . xn) +c(x1, . . . xn)withqand c respectively a quadratic and a cubic forms. LetQ= (y0 :. . .:yn)∈X(k). We will show thatP isR-equivalent toQ.

IfP =Qor if the lineP Qis contained inX, it is clear. Otherwiseq(y1, . . . yn)6= 0. LetS = (z1:. . .:zn)be a zero ofq, which exists inkby hypothesis thatkisC1.

Consider the rational map

φ:Pⁿ⁻¹_k 99KX,(x1:. . .:xn)7→(−c(x1, . . . xn) :x1q(x1, . . . xn) :. . .:xnq(x1, . . . xn)).

Lethbe the restriction ofφto any rational curveC⊂Pⁿ⁻¹_k joiningQ⁰= (y1:. . .:yn) toS. We have h(Q⁰) =Q. Thusφis dened at Q⁰, so it is dened on an open subset ofC. This implies thathis a well-dened morphism. Ifc(S)6= 0then his dened at S andh(S) =P. Thus the pointsP andQareR-equivalent.

Otherwise, the line l= (u:vz1 :. . .:vzn)joining P to (0 :z1. . .:zn)is contained inX. Let us show thath(S)is on this line. This will imply thatP isR-equivalent to Qby a chainP h(S)Q. In fact, consider the composite map

Pⁿ⁻¹_k 99K^φ X 99K^p Pⁿ⁻¹_k

where pis given by(x0 : x1 :. . . : xn)7→ (x1 : . . . : xn). Note that the map p◦φ is identity on its domain of denition.

The map pis dened atQ andp(Q) = (y1: . . .:yn). The map φis dened atQ⁰ andφ(Q⁰) =Q. Thus the compositep◦φis dened atQ⁰. This means thatp◦φinduces the identity map onC. Thus the image ofh(S) = (z₁:. . .:z_n)byφisS, which means thath(S) = (u:z1:. . .:zn)is on the linel, as desired.

Let us now suppose thatX is smooth atP and at Q. We want to prove that any two K-points P and Q of X are R-equivalent. Let T(P) and T(Q) by the tangent hyperplans toX atP and atQrespectively. The cubic hypersurfaceC(P) =X∩T(P) in T(P) ' Pⁿ⁻¹_k has a singular point P. Let us dene C(Q) similarly. Then either Q∈T(P)and the result follows from the singular case or T(P)is distinct fromT(Q). In the latter case,X∩T(P)∩T(Q)is dened by a cubic form inn−1 variables in the projective spaceT(P)∩T(Q)of dimensionn−2, thus it has a non trivial zeroM. Thus P(resp. Q) isR-equivalent toM, again by the singular case. This nishes the proof.

Remark 3.10. Note that in the theorem above one can replace the hypothesis thatk isC1 by that any quadratic and any cubic form overkin at leastn−1variables has a zero.

3.3 Some other cases

The triviality ofR-equivalence in cases1−3p. 2, follows from the explicit description of the set ofR-equivalence classes as some cohomology group H¹(Gal(¯k/k), M) : one uses the fact thatcd k≤1to establish that this group vanishes. Let us consider, as an example, the case X is a smooth compactication of an algebraic torus T. We know from [CTSa] Th.2 and Prop.13, that X(k)/R →^∼ H¹(G,S)ˆ where Sˆ is the character group of some particular torus, coming from so-called asque resolution ofT.

LetG=Gal(¯k/k). Note thatSˆis not a niteG-module, so we can not simply use the denition of a eld of cohomological dimension at most1. Consider a nite Galois extensionK/k trivialising Sˆ. LetH be an invariant subgroup of Gacting trivially on Sˆ and letL/k be the corresponding extension. The restriction-ination sequence gives an isomorphismH¹(G/H, S(L))→^∼ H¹(G, S(K)). Ascd k≤1, the rst group is zero, so is the second.

Moduli spaces of stable curves and R -equivalence