Connected algebraic groups acting on Fano fibrations over P 1
aa r X i v : . [ m a t h . AG ] N ov CONNECTED ALGEBRAIC GROUPS ACTING ON FANOFIBRATIONS OVER P JÉRÉMY BLANC AND ENRICA FLORIS
Abstract.
Let X/ P be a Mori fibre space with general fibre of Picard rankat least two. We prove that there is a proper closed subset S ( X , invariant bythe connected component of the identity Aut ◦ ( X ) of the automorphism groupof X , which is moreover the orbit of a section s and whose intersection with afibre is an orbit of the subgroup of Aut ◦ ( X ) acting trivially on P .Such result is a tool to describe equivariant birational maps from X/ P to other Mori fibre spaces and therefore finds its applications in the study ofconnected algebraic subgroups of Aut ◦ ( X ) . This represents a first reductionstep towards a possible classification of maximal connected algebraic subgroupsof the Cremona group of rank . Contents
1. Introduction 12. Preliminaries 43. Existence of an invariant horizontal closed subspace 74. The group
Aut ◦ ( X ) P ( P ) or P ( T P ) ( P ) or P ( T P ) . 34References 391. Introduction
In this text, we work over the field of complex numbers. By a classical result,every maximal connected algebraic subgroup of
Bir( P ) is conjugate to either theconnected group Aut( P ) , or Aut ◦ ( P × P ) , or Aut( F n ) for some n ≥ . This result,essentially due to Enriques [Enr93], can be now be seen easily using modern tools, byfinding a smooth projective rational surface where the subgroup acts, then runninga minimal model program, which in the case of surfaces is a sequence of contractionsof ( − -curves, and which for rational varieties gives as an outcome a Mori fibration X → B . A Mori fibration is a fibration with ρ ( X/B ) = 1 and whose fibres are Fanovarieties. Therefore, if X is a rational surface the only possibilities are that either X = P and B is a point, or X is a Hirzebruch surface and B = P . The fact Date : November 11, 2020.2010
Mathematics Subject Classification. that
Aut( P ) , Aut ◦ ( P × P ) and Aut( F n ) , n ≥ are maximal connected algebraicsubgroups of Bir( P ) (via a birational map to P ) is then a direct consequenceof the fact that those groups act on the respective surfaces without fixed points.The study of (maximal) connected algebraic subgroups of Bir( X ) for a non-rationalsurface X can be done with the same strategy, see [Fon20].In dimension , the classification of the maximal connected algebraic subgroupsof the Cremona group was started by Enriques and Fano in [EF98] and achievedby Umemura in a series of four papers [Ume80, Ume82a, Ume82b, Ume85]. Itwas recovered in [BFT17, BFT19] using the minimal model program and studyingthe possible Mori fibrations and their automorphisms groups. In dimension 3, inmany cases, the automorphism group of a Mori fibre space is very small. Hence, themaximal connected algebraic subgroups of Bir( P ) correspond to Aut ◦ ( X ) for some Mori fibrations X → B . These are some very natural ones which are in some sense“symmetric enough” as they have a group Aut ◦ ( X ) large enough. It is interestingto determine which are the Mori fibrations realising the maximal subgroups asautomorphism groups. In particular, if B is a curve and X is a rational threefoldwith terminal singularities such that the group Aut ◦ ( X ) is a maximal connectedsubgroup of Bir( X ) (or equivalently of Bir( P ) via a birational map X P ), theneither X → B is a P -bundle or a Mori fibration with general fibres isomorphic to P × P but a generic fibre which is a smooth quadric of Picard rank (see [BFT19,Theorems D and E]). Moreover, in this latter case, there are plenty of examplesof maximal algebraic groups (essentially parametrised by classes of hyperellipticcurves) and each is conjugate to the group of automorphisms of infinitely manyMori fibrations X → P whose generic fibre is a smooth quadric of Picard rank ,but is not conjugate to a subgroup of automorphisms of any Mori fibration Y → B with dim B = 1 .For the moment, the study of maximal connected algebraic subgroups of Bir( P n ) for n ≥ (or more generally of Bir( X ) for some rationally connected varieties X ofdimension ≥ ) seems out of reach in its full generality, due to the incredibly largenumber of possible cases.In this text, we focus on the case of Mori fibre spaces X → P , with X a terminal Q -factorial variety and where the general fibres are smooth Fano varieties of Picardrank ≥ . If X has dimension , this corresponds to the quadric fibrations describedabove, whose general fibres are then isomorphic to P × P . If X has dimension , the general fibre is a Fano variety of dimension 3. There are deformationfamilies of smooth Fano threefolds of Picard rank ≥ [MM82, MM03] and amongthese, exactly occur as general fibres of klt Mori fibre spaces X → P [CFST16,Theorem 1.4]. If X has dimension ≥ , the possible classes for the general fibresare not fully classified (see [CFST16, CFST18] for partial results).In the study of connected algebraic groups acting on projective varieties, it isnatural to look for invariant subsets, as these can be used to construct equivariantbirational maps from one Mori fibre space to another. In particular, if π : X → B isa Mori fibre space and Aut ◦ ( X ) acts transitively on X , every Aut ◦ ( X ) -equivariantbirational map from X to any other Mori fibre space is an isomorphism. This ex-plains the importance played by the next result in the study of connected algebraicgroups acting on Mori fibres space over P , which is our main result: ONNECTED ALGEBRAIC GROUPS ACTING ON FANO FIBRATIONS OVER P Theorem A.
Let π : X → P be a Q -factorial terminal Mori fibre space such thata general fibre F satisfies ρ ( F ) ≥ . Then, the action of Aut ◦ ( X ) P = { g ∈ Aut ◦ ( X ) | πg = π } on a general fibre is not transitive. Moreover, there is a section s ⊂ X of π suchthat the following holds: (1) The set S = Aut ◦ ( X ) · s = Aut ◦ ( X ) P · s = (Aut ◦ ( X ) P ) ◦ · s is a proper closed subset of X ; (2) For each b ∈ P , the fibre π − ( b ) ∩ S of π | S : S → P is equal to π − ( b ) ∩ S = (Aut ◦ ( X ) P ) ◦ · p, where p ∈ s is the point such that π ( p ) = b . The proof of Theorem A is done in Section 3, by studying sets of sections ofthe Mori fibre space X → P , applying an equivariant version of Bend and Break(Proposition 3.1.4), and looking at actions of the different subgroups Aut ◦ ( X ) P and (Aut ◦ ( X ) P ) ◦ of Aut ◦ ( X ) on the set of minimal sections.One motivation for studying the group Aut ◦ ( X ) P comes from the following twoobservations, proven in Section 4: Proposition B.
Let π : X → P be a Mori fibre space such that a general fibre F satisfies ρ ( F ) ≥ . Assume that Aut ◦ ( X ) P = { g ∈ Aut ◦ ( X ) | πg = π } is either finite or a torus. Then Aut ◦ ( X ) is a torus of dimension r ∈ { dim(Aut ◦ ( X ) P ) , dim(Aut ◦ ( X ) P ) + 1 } . Moreover, if r ≥ , there is a smooth projective vari-ety C , a trivial P r -bundle Y → C and a birational map ψ : X Y such that ψ Aut ◦ ( X ) ψ − ( Aut ◦ ( Y ) . Proposition C.
Let π : X → B be a Mori fibre space such that Aut ◦ ( X ) B = { g ∈ Aut ◦ ( X ) | π ◦ g = π } is a linear group of positive dimension and that no orbit of Aut ◦ ( X ) B is dense in afibre of π . Then, k = max { dim((Aut ◦ ( X ) B ) ◦ · x ) | x ∈ X } > and there is a Morifibre space Y → C with dim C ≥ dim X − k > dim B , and an Aut ◦ ( X ) -equivariantbirational map X Y . As an application, we obtain the following result on Mori fibre spaces of dimen-sion . Theorem D.
Let π : X → P be a Q -factorial terminal Mori fibre space such thata general fibre F is a smooth threefold of Picard rank ≥ , and such that Aut ◦ ( X ) is not trivial. Then, one of the following holds: (1) There is a Mori fibre space π B : Y → B with general fibres isomorphic toeither P , or P , or a smooth quadric Q ⊂ P and an Aut ◦ ( X ) -equivariant birationalmap ϕ : X Y . (2) A general fibre F of π is isomorphic to one of the following two smooth Fanothreefolds of Picard rank ≥ with Aut ◦ ( F ) ≃ PGL ( C ) : ( i ) The blow-up of the quadric Q ⊂ P given by x x − x x + 3 x = 0 alongthe image of the Veronese embedding of degree of P . JÉRÉMY BLANC AND ENRICA FLORIS ( ii ) The threefold ( ( x, y, z ) ∈ ( P ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) X i =0 x i y i = X i =0 x i z i = X i =0 y i z i = 0 ) . Theorem D is proved as follows. In Section 5 (see Table 5.1) we recall [CFST16,Table 1], which lists all the smooth threefolds F with Picard rank at least thatare fibres of klt Mori fibre spaces. These varieties are “symmetric”, in the sensethat there is a finite group G ⊆ Aut( F ) such that Pic( F ) G has rank [Pro13,Theorem 1.2]. We then show (Proposition 5.2.2) that among the families listed,only the following four have infinite automorphism group: ( P ) , a smooth divisorof bidegree (1 , in P × P (isomorphic to P ( T P ) , see Lemma 5.3.2), the blow-ups of a smooth quadric along a Veronese curve of degree and the blow-up of P ( T P ) ⊂ P × P along a curve of bidegree (2 , whose projection on both factorsis an embedding. We moreover determine in which cases the automorphism groupis not a torus.In Section 6, we describe some symmetric birational maps from ( P ) or P ( T P ) ⊂ P × P (which blow-up curves balanced with respect to the gradings) and usethen these maps in Section 7, together with Theorem A, to get some Aut ◦ ( X ) -equivariant birational maps from Mori fibre spaces X → P having general fibresbeing isomorphic to ( P ) or P ( T P ) .The proof of Theorem D is then given at the end of Section 7.The authors thank Andrea Fanelli, Ronan Terpereau, Andreas Höring, VladimirLazić and Christopher Hacon for helpful discussions during the preparation of thistext. 2. Preliminaries
Mori fibre spaces and algebraic groups acting on them.
We work overthe complex numbers. We refer to [KM98] for the basic notions in birational geome-try and minimal model program. We recall that a fibration is a surjective morphismwith connected fibres.
Definition 2.1.1.
Let f : X → Y be a dominant projective morphism of normalvarieties. Then f is called a Mori fibre space if the following conditions are satisfied: • f has connected fibres, with dim Y < dim X ; • X is terminal Q -factorial with at most terminal singularities; • the relative Picard number of f is one and − K X is f -ample (i.e. there is anelement [ C ] ∈ N S ( X ) with − K X · C > , and such that each curve contracted by f is numerically equivalent to an element of R > · [ C ] ).We recall first the statement of the Blanchard’s lemma , which will be of funda-mental importance for us. This result is due to Blanchard [Bla56] in the settingof complex geometry, and the proof has been adapted to the setting of algebraicgeometry.
Lemma 2.1.2. [BSU13, Proposition 4.2.1]
Let f : X → Y be a proper morphismbetween varieties such that f ∗ ( O X ) = O Y . If a connected algebraic group G actsregularly on X , then there exists a unique regular action of G on Y such that f is G -equivariant. ONNECTED ALGEBRAIC GROUPS ACTING ON FANO FIBRATIONS OVER P We also recall the following classical fact, which follows from [Bri96, Proposi-tion 2, page 8] or from [New78, Lemma 3.7].
Lemma 2.1.3.
Let G be an algebraic group acting regularly on a projective vari-ety X . Let n = max { dim( G · x ) | x ∈ X } be the maximal dimension of an orbit of G . Then, the set { x ∈ X | dim( G · x ) < n } is a closed subset of X . In particular,the union of orbits of dimension n is a dense open G -invariant subset of X . The following theorem is a relative version of the so-called relative Base pointfree theorem [KMM87, Theorem 3-1-1] and follows from [Fuj11a, Theorem 2.1]. Aswe will use it many times in this note, we recall the statement.
Theorem 2.1.4.
Let X be a variety with terminal singularities, let π : X → S bea proper surjective morphism of normal varieties, and D a π -nef Cartier divisoron X . Assume that rD − K X is nef and big over S for some positive integer r .Then D is relatively semiample. More precisely, there exists a positive integer m such that for every m ≥ m the natural homomorphism π ∗ π ∗ O X ( mD ) → O X ( mD ) is surjective.Proof. We apply [Fuj11a, Theorem 2.1] to the pair ( X, . As the pair is terminal,the second hypothesis of [Fuj11a, Theorem 2.1] is verified. (cid:3) Rational Maps between Mori fibrations.
The following lemma is knownto experts (see [HX13]), we recall the proof here for the reader’s convenience.
Lemma 2.2.1.
Let π ′ U : Y U → U be a smooth projective fibration over a quasi-projective variety U such that ρ ( Y U /U ) = 1 , K Y U is π ′ U -antiample and there is aconnected group G acting on Y U . Let B ⊇ U be a G -equivariant compactification.Then there is a G -equivariant compactification Y ⊇ Y U and a morphism π ′ : Y → B such that π ′ | Y U = π ′ U , and Y → B is a Mori fibre space.Proof. Let Y be a G -equivariant compactification of Y U such that there is a mor-phism Y → B and let η : b Y → Y be a G -equivariant resolution of singularitieswhich is a composition of blow ups whose centers are contained in the singularlocus of Y . In particular η is an isomorphism over Y U . We run a K b Y -MMP over B with scaling of a relatively ample divisor. This MMP terminates by [Fuj11b,Theorem 2.3] (applied with ∆ = 0 ) with a Mori fibre space over B that we denoteby π ′ : Y → T . Since ρ ( Y U /U ) = 1 the MMP induces an isomorphism on Y U and B = T . (cid:3) Lemma 2.2.2.
Let X be a terminal variety, π : X → B be a fibration and let ε : X ′ → X be a Aut ◦ ( X ) -equivariant birational morphism.Assume that there are a non-empty open subset U ⊆ B , an integer a and adivisor L on X ′ such that for every b ∈ U (1) the divisor (( a − K X ′ + L ) | X ′ b is nef and big (2) the divisor ( aK X ′ + L ) | X ′ b is nef, not numerically zero and not big. JÉRÉMY BLANC AND ENRICA FLORIS where X ′ b ⊂ X ′ is the fibre over b . Then there is a commutative diagram X ′ ε (cid:15) (cid:15) ψ / / ❴❴❴ W f (cid:15) (cid:15) X π (cid:15) (cid:15) S } } ⑤⑤⑤⑤⑤⑤⑤⑤ B with Aut ◦ ( X ) -equivariant arrows, such that ψ is birational and dim S > dim B .Moreover, for each m > big enough, the map X ′ b → S b is given by | m ( aK X ′ + L ) | for each b ∈ U ( where S b ⊂ S denotes the fibre over b ) .Proof. As the group
Aut ◦ ( X ) acts on the fibres satisfying conditions (1)-(2), wecan assume that U is Aut ◦ ( X ) -invariant. Let X U = π − U and X ′ U = ( π ◦ ε ) − U .The divisor (( a − K X ′ + L ) | X ′ U is relatively nef and big by (1) and ( aK X ′ + L ) | X ′ U is nef by (2). Therefore by Theorem 2.1.4 the divisor ( aK X ′ + L ) | X ′ U is relativelysemiample. Therefore there is a diagram X ′ U (cid:15) (cid:15) h ! ! ❇❇❇❇❇❇❇❇ X U (cid:15) (cid:15) S U } } ③③③③③③③③ U where h is the morphism induced by | m ( aK X ′ + L ) | X ′ U | for some m big enough.As the divisor is not relatively big nor numerically zero by (2), dim U < dim S U < dim X ′ U .Then we consider an Aut ◦ ( X ) -equivariant compactification S of S U such thatthere is a morphism S → U and an Aut ◦ ( X ) -equivariant compactification W of X ′ U such that there are morphisms W → X and W → S and we proved our claim. (cid:3) The following lemma is an immediate consequence of [Fuj11b, Theorem 2.3] butwe add the proof for the sake of completeness.
Lemma 2.2.3.
Let W , S be quasi-projective varieties such that W is terminal.Let f : W → S be a fibration whose general fibre is uniruled. Then there is afactorisation W f (cid:23) (cid:23) ✵✵✵✵✵✵✵✵✵✵✵✵✵✵ φ / / ❴❴❴❴❴❴❴ Y π ′ (cid:15) (cid:15) T (cid:127) (cid:127) ⑧⑧⑧⑧⑧⑧⑧⑧ S where φ is a birational map and π ′ : Y → T is a Mori fibre space. Moreover, all themaps appearing in the diagram are Aut ◦ ( X ) -equivariant. ONNECTED ALGEBRAIC GROUPS ACTING ON FANO FIBRATIONS OVER P Proof.
We run a K W -MMP over S with scaling of a relatively ample divisor. ThisMMP terminates by [Fuj11b, Theorem 2.3] (applied with ∆ = 0 ) with a Mori fibrespace over S that we denote by π ′ : Y → T . (cid:3) Lemma 2.2.4.
Let π : X → P be a Mori fibre space such that a general fibresatisfies ρ ( F ) ≥ . Then π has at least two singular fibres. In particular, the actionof Aut ◦ ( X ) on P given by the Blanchard’s lemma fixes at least two points.Proof. Let U ⊆ P be the set { b ∈ P | π − b is terminal } . By [CFST16, Theorems2.2 and 2.5] and [KM92] for every b ∈ U the restriction r : N ( X ) Q → N ( F ) π ( U ) Q is a surjective map. We notice that since ρ ( X ) = 2 we have N ( X ) Q ∼ = Q .Moreover, the class of F is in the kernel of r , therefore there is a surjective map N ( X ) Q / Q [ F ] ∼ = Q → N ( F ) π ( U ) Q . If ρ ( F ) ≥ , then the fundamental group of U must be non-trivial. Therefore the locus of non-terminal fibres contains at leasttwo points. The last sentence follows from the fact that Aut ◦ ( X ) is a connectedgroup preserving the non-terminal locus. (cid:3) Finite morphisms and Fano manifolds.Lemma 2.3.1.
Let f : W → Z be a quasi-finite surjective morphism such that all itsfibres have the same cardinality, Z is normal projective and W is quasi-projective.Then W is projective and f is finite.Proof. By the existence of compactifications and resolution of indeterminacies wecan factor f as ¯ f ◦ ι where ι : W → W is an open immersion, W is projectiveand ¯ f : W → Z is projective and generically finite. We consider then the Steinfactorisation ¯ f = η ◦ g of ¯ f , where η : W → Z ′ has connected fibres and g : Z ′ → Z is finite. As ¯ f is quasi-finite, η is birational. Then the morphism η ◦ ι : W → Z ′ isbirational and all its fibres are finite; it is thus an open embedding by the Zariskimain theorem [Sta19, Lemma 37.38.1]. Therefore, we can take Z ′ = W and thenview W as an open subset of Z ′ , and f as the restriction of the finite morphism g : Z ′ → Z .Since all the fibres of f have the same cardinality, by semicontinuity of thecardinality of the fibres, all the fibres of g have the same cardinality. Since f issurjective, we get W = W which finishes the proof. (cid:3) We recall the following lemma for the readers’ convenience
Lemma 2.3.2.
Let Z be a Fano manifold. Then any finite étale map f : W → Z is an isomorphism.Proof. Since f is étale, the variety W is a Fano manifold. Then by the Kawamata-Viehweg vanishing theorem χ ( W ) = 1 = χ ( Z ) . On the other hand, if f is finiteétale, then χ ( W ) = deg f χ ( Z ) . Therefore deg f = 1 and f is an isomorphism. (cid:3) Existence of an invariant horizontal closed subspace
Defining some sets of sections and a bend and break result.Lemma 3.1.1.
Let X be a Q -factorial variety and let π : X → B be a Mori fibrationover a smooth irreducible curve B . Then, the set K = {− K X · s | s is a section of π } is a non-empty discrete subset of Q which is bounded from below. In particular, itadmits a minimum. JÉRÉMY BLANC AND ENRICA FLORIS
Proof.
The fact that K is non-empty follows from [GHS03, Theorem 1.1]. Let H be an ample divisor on X and let F be a fibre. Since ρ ( X/B ) = 2 , the divisor − K X is numerically equivalent to αH + βF for some α, β ∈ R . As π is a Mori fibration,the restriction of − K X to a general fibre is ample, so α > . For each section s ,we have H · s > , so − K X · s = αH · s + β ≥ β . This shows that K is boundedfrom below. Since X is Q -factorial, there exists r ∈ N such that rK X is Cartier.Therefore K ⊆ r Z is discrete. (cid:3) Definition 3.1.2.
Let X be a Q -factorial varieties and let π : X → B be a Morifibration over a smooth irreducible curve B . We say that a section s ⊂ X is a minimal section if − K X · s ≤ − K X · s ′ for each section s ′ ⊂ X . We say that asection s ⊂ X is an aut-minimal section if − K X · s ≤ − K X · s ′ for each section s ′ ⊂ Aut ◦ ( X ) · s . Lemma 3.1.1 shows that minimal and aut-minimal sections alwaysexist. Remark . If π : X → B is a P -bundle over a smooth irreducible curve B ,for each section s ⊂ X , the adjunction formula gives s = − s · K X − g ( B ) ,so Lemma 3.1.1 generalises the classical fact that s is bounded from below, andminimal sections correspond here to the sections of minimal self-intersection.We present now a Bend and Break result. The proof follows [Deb01, Proposi-tion 3.2]. Proposition 3.1.4.
Let X be a projective variety together with a fibration π : X → P , let s ⊂ X be a section of π and x ∈ s . Suppose that there is an irreducible curve Γ ⊆ Aut ◦ ( X ) such that g ( x ) = x and g ( s ) = s for a general g ∈ Γ . Then the -cycle s is numerically equivalent to a non-integral effective rational -cycle passingthrough x and contained in Aut ◦ ( X ) · s . In particular, s is not an aut-minimalsection.Proof. Let ν : C → Γ be the normalisation of Γ . By the Blanchard’s Lemma 2.1.2there is a morphism ν ′ : C → Aut( P ) = PGL ( C ) such that π ◦ ν ( g ) = ν ′ ( g ) ◦ π for each g ∈ C .Let ϕ : P → s be the morphism such that π ◦ ϕ = id P . We now prove that themorphism F : P × C → X × C ( p, g ) ( ν ( g )( ϕ ( p )) , g ) is finite. This is implied by the fact that for each g ∈ Γ , the morphism ψ g : P → X, t ν ( g )( ϕ ( p )) is injective. This last claim follows from the fact that π ◦ ψ g = ν ′ ( g ) ∈ Aut( P ) , as π ◦ ψ g ( p ) = ( π ◦ ν ( g ))( ϕ ( p )) = ( ν ′ ( g ) ◦ π )( ϕ ( p )) = ν ′ ( g )( p ) foreach p ∈ P .As F is finite, F ( P × C ) has dimension . We then follow the proof of [Deb01,Proposition 3.2]. Let C be a smooth compactification of C . Let S be the nor-malisation in C ( P × C ) of the closure in X × C of the image of F , with finitecanonical morphism F : S → X × C . Since P × C is normal, by uniqueness of the ONNECTED ALGEBRAIC GROUPS ACTING ON FANO FIBRATIONS OVER P normalisation we have F − ( X × C ) = P × C . We obtain the commutative diagram P × C (cid:15) (cid:15) / / S κ (cid:31) (cid:31) F (cid:15) (cid:15) e / / XX × C p < < ②②②②②②②② p (cid:15) (cid:15) C / / C No component of a fibre of κ is contracted by e because it would otherwise becontracted by F and F : S → F ( S ) is finite as it is a normalisation. The morphism κ : S → C is flat as C is a smooth curve [Har77, III, prop.9.7]. Therefore each fibreis 1-dimensional, with no embedded component, of genus 0 [Har77, III, cor.9.10].An integral fibre is a rational curve and a singular one is a tree of rational curves.Denoting by p = π ( x ) ∈ P the image of x , we have ϕ ( p ) = x . As everyelement of Γ fixes x , the image by F of { p } × C is { x } × C . In particular, theclosure C of { p } × C is contracted to x by e .We now prove that there is a non-integral fibre of κ . This part is different from[Deb01, Proposition 3.2]. Since C is contracted by e and e ( S ) is a surface, wehave C < . We then consider the morphism τ : S → C × P that is given by thetwo morphisms κ : S → C and π ◦ e : S → P . Observe that τ is birational: for anelement c ∈ C , the preimage in S of the curve P × { c } ⊆ P × C is sent to a sectionof π by F . As τ is a birational morphism and as C < , the morphism τ contractssome irreducible components contained in the fibres of C \ C , which are thereforenot all integral.Let z ∈ C be a point such that κ ∗ z is a non integral fibre. The curve s isequal to e ∗ κ ∗ z , where some z ∈ C is sent to the identity in Γ ⊂ Aut ◦ ( X ) P . Then e ∗ π ∗ z is an effective -cycle numerically equivalent to s and passes through x as C is a section of κ that is contracted to x . Moreover, e ∗ π ∗ z is not integral as π ∗ z is not integral and e does not contract any irreducible component of a fibre.It remains to see that s is not a minimal section. We write e ∗ π ∗ z = P ri =1 ℓ i where the ℓ , . . . , ℓ r are irreducible and reduced curves on X . As e ∗ π ∗ z is numeri-cally equivalent to a section, exactly one of the ℓ i , say ℓ , is a section, and ℓ , . . . , ℓ r are contained in fibres. As − K X is ample on the fibres, we have − K X · ℓ i > for i ≥ . This gives − K X · s = − K X · ℓ + r X i =1 − K X · ℓ i > − K X · ℓ and implies that s is not a minimal section. (cid:3) Transitivity on the fibres.Lemma 3.2.1.
Let π : X → P be a Mori fibre space such that the action of Aut ◦ ( X ) P = { g ∈ Aut ◦ ( X ) | πg = π } on a general fibre is transitive.Then, there is a dense open subset U ⊆ P , a smooth Fano variety F of Picardrank and an isomorphism θ : U × F ≃ −→ π − ( U ) such that π ◦ θ is the first projection U × F → U . Proof.
Let s ⊂ X be a minimal section (which exists by Lemma 3.1.1) and let ϕ : P → s be the morphism such that π ◦ ϕ = id P . Let G = Aut ◦ ( X ) P and G ◦ the connected component of the identity. As the finite group G/G ◦ acts on the setof orbits of G ◦ , the action of G ◦ on a general fibre is transitive.Let H = { g ∈ G ◦ | g ( s ) = s } ⊆ G ◦ . The quotient V = G ◦ /H is homogeneousfor the action of G ◦ and is thus smooth. Let x ∈ X be such that { x } = s ∩ F . Weobtain a surjective G ◦ -equivariant morphism Φ : V → F [ g ] g ( x ) , where [ g ] ∈ V = G ◦ /H denotes the class of g ∈ G ◦ .We prove that dim V = dim F . Indeed, otherwise dim V > dim F and there is anirreducible curve Γ ⊆ G ◦ such that g ( x ) = x and g ◦ ϕ = ϕ for a general g ∈ Γ . Thefact that g ∈ G ◦ and g ◦ ϕ = ϕ implies that g ( s ) = s , impossible by Lemma 3.1.4.We now consider the Stein factorisation of Φ is given by V ˜Φ → e F ν → F . Since dim V = dim F , the morphism ˜Φ is birational; it is moreover G ◦ -equivariant byBlanchard’s lemma (Lemma 2.1.2). Hence, ν is also G ◦ -equivariant.As ν is finite and G ◦ -equivariant and as G ◦ acts transitively on F , it is étale (bythe generic smoothness). Lemma 2.3.2 implies that ν is an isomorphism. We justproved that for a general fibre F , for any y ∈ F there is a unique g ∈ G ◦ such that y ∈ g ( s ) (or equivalently such that y = g ( ϕ ( π ( y ))) ). Therefore there is an open set U ⊆ P such that the morphism θ : V × U → π − ( U )([ g ] , t ) g ( ϕ ( t )) . is bijective. Restricting U we may assume that π − ( U ) is smooth, and thus by theMain theorem of Zariski, the above morphism is an isomorphism.This proves that π is a trivial V -bundle over U , so in particular V is isomorphicto a general fibre F . It remains to see that the Picard rank of V is equal to .Suppose for a contradiction that there exist two prime divisors D and D on F having classes in NS( F ) which are Q -independent (which exist as soon as ρ ( F ) ≥ )and D i the Zariski closure of θ ( D i × U ) in X . Then the classes D and D are notnumerically equivalent over the base, even after some multiple, so ρ ( X/ P ) ≥ ,contradicting that π is a Mori fibre space. (cid:3) Existence of invariant subsets and the proof of Theorem A.Lemma 3.3.1.
Let π : X → P be a Mori fibre space and let G = Aut ◦ ( X ) P = { g ∈ Aut ◦ ( X ) | πg = π } and let G ◦ the connected component of the identity. Then, the following hold (1) For each aut-minimal section s ⊂ X and for any two points p, q ∈ s , we have dim G · p = dim G ◦ · p = dim G ◦ · q = dim G · q. (2) For each p ∈ s for each point q ∈ G ◦ · p \ G ◦ · p there is an aut-minimalsection s ′ ⊂ X that contains q . (3) For each aut-minimal section s ⊂ X such that G · p is closed for some p ∈ s ,the stabilisers ( G ◦ ) q and ( G ◦ ) s are equal for all q ∈ s and the quotient G ◦ / ( G ◦ ) s ONNECTED ALGEBRAIC GROUPS ACTING ON FANO FIBRATIONS OVER P is a Fano variety. Moreover, denoting by ϕ : P → s the morphism such that π ◦ ϕ = id P , the morphism κ : G ◦ / ( G ◦ ) s × P → X ([ g ] , b ) g ( ϕ ( b )) . is a closed embedding.Proof. Lemma 3.1.1 implies that there exists a minimal section of X , which istherefore also aut-minimal.(1): As the finite group G/G ◦ acts on the set of orbits of G ◦ , we have dim G · p =dim G ◦ · p for each p ∈ s . It then suffices to show that dim G ◦ · p = dim G ◦ · q for any two points p, q ∈ s . Up to exchanging p and q we may assume that dim G ◦ · p < dim G ◦ · q , in order to derive a contradiction. Then the stabiliserssatisfy dim( G ◦ ) p > dim( G ◦ ) q . As the stabiliser ( G ◦ ) s ⊆ G ◦ of s is contained inboth ( G ◦ ) p and ( G ◦ ) q , we obtain dim( G ◦ ) p / ( G ◦ ) s > . There is thus an irreduciblecurve Γ ⊆ G ◦ ⊆ G = Aut ◦ ( X ) P , such that g ( p ) = p and g ( s ) = s for a general g ∈ Γ . This contradicts the minimality of s by Lemma 3.1.4.(2): Assume that G ◦ · p is not closed and let q ∈ G ◦ · p \ G ◦ · p . Set a = π ( q ) = π ( p ) ∈ P and consider the morphism κ : G ◦ × P → X ( g, b ) g ( ϕ ( b )) , where ϕ : P → s is the morphism such that π ◦ ϕ = id P . There exists an irreduciblecurve C ⊂ G ◦ × P such that q belongs to the closure of κ ( C ) and such that (id , a ) ∈ C . The closure Γ ⊆ G ◦ of the projection of C in G ◦ is an irreducible curvesuch that id ∈ Γ and such that q belongs to the closure of κ (Γ × P ) .Let Γ be a compactification of a normalisation of Γ . The morphism κ | Γ × P : Γ × P → X yields a rational map θ : Γ × P X . Let S be the Zariski closure of theimage of θ (or equivalently of κ (Γ × P ) ). Then S has dimension 2 and q ∈ S . Wetake a resolution of the indeterminacies of θ ˆ S ν | | ②②②②②②②② µ (cid:30) (cid:30) ❂❂❂❂❂❂❂❂ Γ × P θ / / ❴❴❴❴❴❴❴ S and find z ∈ Γ \ Γ such that q ∈ µ ∗ ν ∗ ( z × P ) . As id ∈ Γ , we obtain µ ∗ ν ∗ (id × P ) = s .Hence, µ ∗ ν ∗ ( z × P ) is a cycle numerically equivalent to s , that we can write as µ ∗ ν ∗ ( z × P ) ≡ s ′ + l where s ′ is a section and l is an effective -cycle contained in fibres of π . As − K X · l ≥ and s ′ ⊆ Aut ◦ ( X ) · s , the minimality of s implies that l = 0 and that s ′ ≡ s . The section s ′ is aut-minimal as Aut ◦ ( X ) · s ′ ⊆ Aut ◦ ( X ) · s . Moreover q belongs to µ ∗ ν ∗ ( z × P ) = s ′ , so q ∈ s ′ . This achieves the proof of (2).(3): By (1) for every p ∈ s we have dim( G ◦ ) p = dim( G ◦ ) s . Hence, the morphism τ : G ◦ / ( G ◦ ) s → G ◦ / ( G ◦ ) p is a quasi finite morphism. The morphism τ being G ◦ -equivariant, all of its fibres have the same cardinality. As the orbit G ◦ · p is closedin X , it is projective, hence the variety G ◦ / ( G ◦ ) p is projective, and thus Fano as itis homogeneous for the action of the linear connected group G ◦ by [IP99, Corollary G ◦ / ( G ◦ ) s is projective. Therefore τ is étale andby Lemma 2.3.2 it is an isomorphism. This gives ( G ◦ ) p = ( G ◦ ) s for every p ∈ s .The morphism κ : G ◦ / ( G ◦ ) s × P → X ([ g ] , b ) g ( ϕ ( b )) . is closed as G ◦ / ( G ◦ ) s is projective. It is moreover an isomorphism onto its imageas for a fixed b ∈ P it induces the isomorphism G ◦ / ( G ◦ ) ϕ ( b ) → G ◦ · ϕ ( b ) . (cid:3) Lemma 3.3.2.
Let π : X → P be a Mori fibre space such that a general fibre F hasPicard ρ ( F ) ≥ , and let s ⊂ X be an aut-minimal section such that Aut ◦ ( X ) P · p is closed for some p ∈ s .If Aut ◦ ( X ) P · s = Aut ◦ ( X ) · s , then there is an aut-minimal section s ′ ⊂ X suchthat dim Aut ◦ ( X ) · s ′ < dim Aut ◦ ( X ) · s. Proof.
Let us write G = Aut ◦ ( X ) P and denote by G ◦ the connected componentof the identity. By assumption, G · p is closed for some p ∈ s , so we can applyLemma 3.3.1(3). This implies that the the stabilisers ( G ◦ ) q and ( G ◦ ) s are equal foreach q ∈ s and that κ : G ◦ / ( G ◦ ) s × P → X ([ g ] , b ) g ( ϕ ( b )) . is a closed embedding, where ϕ : P → s is the morphism such that π ◦ ϕ = id P .In particular, G ◦ · q = G ◦ / ( G ◦ ) s · q is closed in X for each q ∈ s .Since G · s = Aut ◦ ( X ) · s , we have G ( Aut ◦ ( X ) . As the action of Aut ◦ ( X ) on P fixes at least two points (Lemma 2.2.4), we may change coordinates such thatthe action of Aut ◦ ( X ) on P is exactly the group, isomorphic to C ∗ , fixing [1 : 0] and [0 : 1] . We denote by α : Aut ◦ ( X ) → C ∗ the corresponding surjective grouphomomorphism.We consider the Aut ◦ ( X ) -equivariant morphism Φ : Aut ◦ ( X ) / ( G ◦ ) s × P → X ([ g ] , b ) g ( ϕ ( b )) , whose restriction to G ◦ / ( G ◦ ) s gives κ . The image Z = Φ(Aut ◦ ( X ) / ( G ◦ ) s × P ) = Aut ◦ ( X ) · s strictly contains G · s , which is closed in X . Since Z is irreducible (because Aut ◦ ( X ) / ( G ◦ ) s × P is irreducible), we find dim Z > dim G · s = dim( G ◦ / ( G ◦ ) s × P ) .As Aut ◦ ( X ) /G ≃ C ∗ has dimension , we obtain dim(Aut ◦ ( X ) / ( G ◦ ) s × P ) = dim Z = dim( G · s ) + 1 . In particular,
Φ : Aut ◦ ( X ) / ( G ◦ ) s × P → Z is generically finite.We now prove that every fibre of Φ is finite. Suppose by contradiction thatsome irreducible curve Γ ⊆ Aut ◦ ( X ) / ( G ◦ ) s × P is sent onto a point q ∈ Z .For each g ∈ Aut ◦ ( X ) , the curve g · Γ is contracted onto g ( q ) , so the closure of { g · Γ | g ∈ Aut ◦ ( X ) } is a subvariety F ⊆ Aut ◦ ( X ) / ( G ◦ ) s × P that is sent by Φ ontoa subvariety F ′ = Φ( F ) of Z of dimension dim F ′ < dim F . Since Φ is genericallyfinite, dim F < dim(Aut ◦ ( X ) / ( G ◦ ) s × P ) . As F is invariant by Aut ◦ ( X ) , it has tobe equal to Aut ◦ ( X ) / ( G ◦ ) s ×{ b } for some b ∈ P . This implies that Γ = Γ ′ ×{ b } for some curve Γ ′ ⊆ Aut ◦ ( X ) / ( G ◦ ) s . Replacing Γ with g · Γ for some g ∈ Aut ◦ ( X ) , ONNECTED ALGEBRAIC GROUPS ACTING ON FANO FIBRATIONS OVER P we may assume that q = ϕ ( b ) . We then obtain Γ ′ · q = q . As s is an aut-minimalsection, Proposition 3.1.4 implies that Γ ′ · s = s . As ( G ◦ ) q = ( G ◦ ) s , the group G q / ( G ◦ ) s is finite, so Γ ′ is not contained in G , which means that α (Γ ′ ) = C ∗ . Foreach element g ∈ Aut ◦ ( X ) , there exists then h ∈ Γ ′ such that α ( g ) = α ( h ) , andthus g = g h for some g ∈ G . This gives g ( s ) = g h ( s ) = g ( s ) ⊆ G · s . This beingtrue for each g ∈ Aut ◦ ( X ) , it gives Z = Aut ◦ ( X ) · s = G · s , giving the desiredcontradiction.We denote by d the degree of the quasi-finite morphism Φ : Aut ◦ ( X ) / ( G ◦ ) s × P → Z , by Z the closure of Z in X and by R ( Z the subset R = { x ∈ Z | Φ − ( x ) contains less than d points } . As Z is Aut ◦ ( X ) -invariant and Φ is Aut ◦ ( X ) -equivariant, the set R is Aut ◦ ( X ) -invariant.We prove that R is closed in Z . By the Hironaka resolution of singularities,there is a projective variety W together with a morphism Φ : W → Z and an openimmersion ι : Aut ◦ ( X ) / ( G ◦ ) s × P → W such that Φ ◦ ι = Φ . Let Φ = σ ◦ µ where µ has connected fibres, and is therefore birational, and σ is finite. By the Zariskimain theorem [Sta19, Lemma 37.38], µ ◦ ι is an open immersion, therefore we canassume that Φ is finite. The set R is closed as it is the union of two closed sets: R = Z \ Z ∪ { x ∈ Z | Φ − ( x ) contains less than d points } . We now prove that π ( R ) = P . Suppose for contradiction that π ( R ) = P ,which implies that π ( R ) ⊆ { [0 : 1] , [1 : 0] } . Writing Z = Z ∩ π − ([1 : 1]) , thepreimage W = Φ − ( Z ) is projective (Lemma 2.3.1). It then suffices to showthat the morphism W → C ∗ obtained by composing α with the first projection Aut ◦ ( X ) / ( G ◦ ) s × P → Aut ◦ ( X ) / ( G ◦ ) s is surjective to obtain the desired contra-diction. For each µ ∈ C ∗ , we take g ∈ Aut ◦ ( X ) such that α ( g ) = µ and choose thepoint p ∈ s such that g ( ϕ ( p )) ∈ π − ([1 : 1]) .If R contains a section s ′′ , we have Aut ◦ ( X ) · s ′′ ⊆ R ( Aut ◦ ( X ) · s = Z andchoose a aut-minimal section s ′ contained in Aut ◦ ( X ) · s ′′ . Otherwise, we choosea point x ∈ R ∩ π − ([1 : 0]) and a section s ′′ ⊂ X through x and choose s ′′ such that − K X · s ′′ is minimal, among all sections through x (this is possible byLemma 3.1.1). If dim Aut ◦ ( X ) · s ′′ < dim Aut ◦ ( X ) · s , we choose a aut-minimalsection s ′ contained in Aut ◦ ( X ) · s ′′ . It remains to assume that dim Aut ◦ ( X ) · s ′′ ≥ dim Aut ◦ ( X ) · s ) and to derive a contradiction. As Aut ◦ ( X ) · x ⊆ R ∩ π − ([1 : 0]) ,we have dim Aut ◦ ( X ) · x < dim R < dim Z = dim Aut ◦ ( X ) · s . This, togetherwith dim Aut ◦ ( X ) · s ′′ ≥ dim Z implies the existence of an irreducible curve C ⊆ Aut ◦ ( X ) such that C · x = x and C · s ′′ = s ′′ , contradicting Proposition 3.1.4. (cid:3) The following proposition directly implies Theorem A.
Proposition 3.3.3.
Let π : X → P be a Mori fibre space such that a general fibre F satisfies ρ ( F ) ≥ and let s ⊂ X be an aut-minimal section such that Aut ◦ ( X ) · s is of minimal dimension ( i.e. dim(Aut ◦ ( X ) · s ) ≤ dim(Aut ◦ ( X ) · s ′ ) for each aut-minimal section s ′ ) . Then, the following holds: (1) S = Aut ◦ ( X ) · s = Aut ◦ ( X ) P · s = (Aut ◦ ( X ) P ) ◦ · s is a proper closed subsetof X ; (2) For each b ∈ P , the fibre π − ( b ) ∩ S of S → P is equal to (Aut ◦ ( X ) P ) ◦ · p ,where p ∈ s is such that π ( p ) = b . Proof.
We write G = Aut ◦ ( X ) P and denote by G ◦ the connected component ofthe identity. If a point p ∈ s were such that G ◦ · p were not closed, then for eachpoint q ∈ G ◦ · p \ G ◦ · p there would be an aut-minimal section s ′ ⊂ X that contains q (Lemma 3.3.1(2)), giving thus dim( G · s ′ ) < dim( G · s ) , impossible.The choice of s implies thus that G ◦ · p is closed for each p ∈ s , and thus that G · p is closed too. Lemma 3.3.1(3) then implies that G ◦ · s = (Aut ◦ ( X ) P ) ◦ · s is aclosed subset of X and that for each point b ∈ P , the fibre π − ( b ) ∩ S of S → P isequal to G ◦ · p = (Aut ◦ ( X ) P ) ◦ · p , where p ∈ s is the point such that π ( p ) = b . Thefact that G ◦ · s is a proper subset of X then follows from the fact that the generalfibre has F satisfies ρ ( F ) ≥ and from Lemma 3.2.1.Lemma 3.3.2 then implies that Aut ◦ ( X ) P · s = Aut ◦ ( X ) · s . (cid:3) The group
Aut ◦ ( X ) P The case of tori.
We start this section by proving Propositions B and C,stated in the introduction. We recall the statements for the sake of readability.
Proposition B.
Let π : X → P be a Mori fibre space such that a general fibre F satisfies ρ ( F ) ≥ . Assume that Aut ◦ ( X ) P = { g ∈ Aut ◦ ( X ) | πg = π } is either finite or a torus. Then Aut ◦ ( X ) is a torus of dimension r ∈ { dim(Aut ◦ ( X ) P ) , dim(Aut ◦ ( X ) P ) + 1 } . Moreover, if r ≥ , there is a smooth projective vari-ety C , a trivial P r -bundle Y → C and a birational map ψ : X Y such that ψ Aut ◦ ( X ) ψ − ( Aut ◦ ( Y ) .Proof. Let us write G = Aut ◦ ( X ) to simplify the notation. As the result is emptyif G is trivial, we may assume that dim G = r ≥ . The group G acts on P byBlanchard’s Lemma 2.1.2. This gives rise to an exact sequence → Aut ◦ ( X ) P → G → H → where H ⊆ Aut( P ) . As ρ ( F ) ≥ for a general fibre F , the number of singularfibres is at least so H is a torus of dimension or (Lemma 2.2.4). Hence, we find r ∈ { dim(Aut ◦ ( X ) P ) , dim(Aut ◦ ( X ) P ) + 1 } . If Aut ◦ ( X ) P is finite, then dim H =dim G = r , and [Bor91, IV.11.14, Corollary 1] provides the existence of a torus T ⊆ G of dimension r . Hence, G is a torus of dimension r . If Aut ◦ ( X ) P is a torusof positive dimension, it is contained in a maximal torus T of G , which containsa subtorus T ′ ⊆ T such that T ′ → H is an isogeny (again by [Bor91, IV.11.14,Corollary 1]). In particular, T /
Aut ◦ ( X ) P is isomorphic to H , so dim T = dim G and G = T is again a torus.We have proved that G is isomorphic to ( G m ) r = ( C ∗ ) r . There is a G -invariantopen subset of X isomorphic to ( G m ) r × U , where U is a smooth affine variety,such that G acts trivially on U and acts on ( G m ) r by multiplication (see [BFT19,Proposition 2.5.1] or [Pop16, Theorem 3]). Choosing a smooth projective variety C birational to U , and embedding ( G m ) r into P r , we obtain a birational map ψ : X P r × C such that ψGψ − ⊆ Aut ◦ ( P r × C ) . It remains to observe thatthis last inclusion is strict, as Aut( P r ) embeds into Aut ◦ ( P r × C ) . (cid:3) Orbits of small dimension.Proposition C.
Let π : X → B be a Mori fibre space such that Aut ◦ ( X ) B = { g ∈ Aut ◦ ( X ) | π ◦ g = π } ONNECTED ALGEBRAIC GROUPS ACTING ON FANO FIBRATIONS OVER P is a linear group of positive dimension and that no orbit of Aut ◦ ( X ) B is dense in afibre of π . Then, k = max { dim((Aut ◦ ( X ) B ) ◦ · x ) | x ∈ X } > and there is a Morifibre space Y → C with dim C ≥ dim X − k > dim B , and an Aut ◦ ( X ) -equivariantbirational map X Y .Proof. Let us fix a polarization H on X . By [Kol96, Theorem 3.21.3], for all positiveintegrs h, ℓ ≥ , there is a projective variety Chow h,ℓ ( X ) which parametrises theproper algebraic cycles of dimension h and degree ℓ with respect to H . For moredetails on Chow varieties and the construction of these, we refer to [Kol96] and[Ryd03]. For all h, ℓ ≥ , the group Aut ◦ ( X ) acts biregularly on Chow h,ℓ ( X ) .Let us write G = (Aut ◦ ( X ) B ) ◦ , so that k = max { dim( G · x ) | x ∈ X } . Byhypothesis, G is of positive dimension, so k > . The union of all orbits of dimension k is a dense open G -invariant subset U ⊆ X (Lemma 2.1.3).We denote by R the set of closures of k -dimensional orbits of the linear con-nected algebraic group G = (Aut ◦ ( X ) B ) ◦ . The set R is an uncountable subsetof S ℓ ≥ Chow k,ℓ ( X ) that is pointwise fixed by the action of the group G and leftinvariant by Aut ◦ ( X ) , since G is a normal subgroup of Aut ◦ ( X ) . There is thus aninteger ℓ ≥ such that R ∩ Chow k,ℓ ( X ) is uncountable.For each integer ℓ ≥ , we decompose the closure of R ∩ Chow k,ℓ ( X ) in Chow k,ℓ ( X ) into finitely many irreducible components R ℓ,j and consider the irreducible varieties Z ℓ,j = { ( x, [ t ]) ∈ X × R ℓ,j | x ∈ t } , which have dimension equal to dim R ℓ,j + k .The morphism [ ℓ,j Z ℓ,j → X given by the first projection is surjective, since U = X . On the other hand, thereare countably many pairs ( ℓ, j ) , therefore there exists a pair ( ℓ, j ) such that thefirst projection β : Z ℓ,j → X is surjective, which implies that R ℓ,j has dimension atleast dim X − k . We now prove that the morphism β is generically injective. Wedenote by p : Z ℓ,j → R ℓ,j the second projection. There is an open set V ⊆ R ℓ,j such that a point in V corresponds to a unique cycle in X . The morphism β is G -equivariant, therefore it sends fibres of p inside closures of orbits of the actionof G in X . We prove that β is injective on p − V ∩ β − U . For this, we take twopoints y, y ′ ∈ p − V ∩ β − U having the same image x = β ( y ) = β ( y ′ ) ∈ X . Wewrite y = ( x, [ t ]) and y ′ = ( x, [ t ′ ]) where t, t ′ ∈ R ℓ,j . Then, β ( p − t ) , β ( p − t ′ ) arecontained in the closure of the same orbit as they have the point x in common. Asthe orbit of x has dimension k , we have β ( p − t ) = β ( p − t ′ ) implying t = t ′ .Since the morphism β is generically injective, it is birational.As Aut ◦ ( X ) acts on S ′ = R ℓ,j and on Z ′ = Z ℓ,j , there is an action of Aut ◦ ( X ) onthe normalisations S and Z of S ′ and Z ′ . We obtain two Aut ◦ ( X ) -equivariant mor-phisms Z → X and Z → S ; the morphism Z → X is birational and a general fibre of Z → S is a unirational variety of dimension k , so dim S = dim Z − k = dim X − k > dim B , since we assumed k < dim X − dim B . Applying an Aut ◦ ( X ) -equivariant res-olution of singularities and an Aut ◦ ( X ) -equivariant resolution of indeterminacies,we can assume that S and Z are smooth. We run a K Z -MMP over S with scaling ofa relatively ample divisor. Since k > and because the fibres of Z → S are unira-tional, this MMP terminates by Lemma 2.2.3 with a Mori fibre space over S that wedenote by π ′ : Y → C → S . We have also dim C ≥ dim S = dim X − k > dim B . (cid:3) Symmetric smooth Fano threefolds
The list of symmetric smooth Fano threefolds.
In Table 5.1 we recallthe list [CFST16, Table 1] of all the smooth threefolds F with Picard rank at least that are fibres of klt Mori fibre spaces. These are varieties which are “symmetric”,in the sense that there is a finite group G ⊆ Aut( F ) such that Pic( F ) G has rank [Pro13, Theorem 1.2]. The numeration that we use is the one of [CFST16, Table 1].It is almost the same numeration as in [Pro13, Theorem 1.2]: Cases a and b correspond to (1.2.1) in [Pro13, Theorem 1.2] (which is also subdivided into twocases), and Cases , , . . . , correspond respectively to (1 . . , (1 . . , . . . , (1 . . in [Pro13, Theorem 1.2]. ρ ( F ) − K F Description of F a (6a) 2 12 A divisor of bidegree (2 , in P × P . b (6b) 2 12 A cover of a smooth divisor W of bidegree (1 , in P × P branched along a member of |− K W | . (12) 2 20 The blow-up of P along a curve of degree and genus which is an intersection of cubics. (28) 2 28 The blow-up of a smooth quadric Q ⊂ P along a smoothrational curve of degree 4 which spans P . (32) 2 48 A divisor of bidegree (1 , in P × P . (1) 3 12 A double cover of P × P × P branched along a memberof |− K P × P × P | . (13) 3 30 The blow-up of a smooth divisor of bidegree (1 , in P × P along a curve C of bidegree (2 , , such that C ֒ → W ֒ → P × P → P is an embedding for bothprojections P × P → P . (27) 3 48 P × P × P . (1) 4 24 A smooth divisor of multidegree (1 , , , in P × P × P × P . Table 1.
Smooth Fano varieties being general fibres of klt Mori fi-bre spaces. The first column indicates the numeration of [CFST16],which is also the numeration of [Pro13], and the second column in-dicates the numeration of [MM82].5.2.
Automorphisms of symmetric smooth Fano threefolds.
Because ofProposition B, the interesting threefolds F in the list are those such that Aut ◦ ( F ) is not a torus. We will show that Aut ◦ ( F ) is trivial in all cases except , , or (Proposition 5.2.2 below). The cases and consist of one isomorphism class,with automorphism group not being a torus; we moreover prove that in the fam-ilies and , there is only one isomorphism class where Aut ◦ ( F ) is not a torus(Section 5.3).To prove Proposition 5.2.2, we will need the following result whose proof isadapted from the proof of [Bea77, Proposition 1.2]. ONNECTED ALGEBRAIC GROUPS ACTING ON FANO FIBRATIONS OVER P Lemma 5.2.1.
Let S be a smooth threefold given either by ( i ) ([ x : x : x ] , ( u, v )) ∈ P × A (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:2) x x x (cid:3) · M ( u, v ) · x x x = 0 for some symmetric matrix M ∈ Mat × ( C [ u, v ]) or by ( ii ) (cid:26) ([ x : x ] , [ y : y ] , ( u, v )) ∈ P × P × A (cid:12)(cid:12)(cid:12)(cid:12)(cid:2) x x (cid:3) · M ( u, v ) · (cid:20) y y (cid:21) = 0 (cid:27) for some matrix M ∈ Mat × ( C [ u, v ]) . Let ∆ ⊆ A be the zero locus of det M andlet π : S → A be the projection on the last factor. The following hold: (1) ∆ is a reduced curve of A , smooth if S ⊆ P × P × A is as in ( ii ) and withonly ordinary double points if S ⊆ P × A is as in ( i ) ; (2) For each p ∈ A , the fibre π − ( p ) is isomorphic to a conic in P which is not reduced ( double line ) if p is a singular point of ∆; reduced and singular ( two distinct lines ) if p is a smooth point of ∆; smooth ( isomorphic to P ) if p does not belong to ∆ . Proof.
We take a point p ∈ A and consider the fibre π − ( p ) ⊂ X , and the matrix M ( p ) associated to p . If p ∆ , then M ( p ) is invertible, so π − ( p ) ⊂ X is a smoothconic in P (respectively a smooth curve of bidegree (1 , in P × P ). This givesthe third case of (2). We then assume that p ∈ ∆ , in which case M ( p ) is notinvertible, so π − ( p ) is not smooth. We prove that either p is a smooth point of ∆ ,or an ordinary double point of ∆ , and that these two cases give the first two casesof (2). For this, one can change coordinates on A and assume that p = (0 , . Wechoose r = 5 (respectively r = 3 ) and α , . . . , α r ∈ C [ u, v ] so that either M = α α α α α α α α α or (cid:18) α α α α (cid:19) and the equation of S is either ( i ) α x + α x x + α x + α x x + α x x + α x = 0 or ( ii ) α x y + α x y + α x y + α x y = 0 . We first assume that M ( p ) is the zero matrix (i.e. α i ( p ) = 0 for i = 0 , . . . , r ), andderive a contradiction. Each symmetric matrix R ∈ Mat × ( C ) (respectively eachmatrix R ∈ Mat × ( C ) ) defines a closed subset C R in P (respectively P × P ) by [ x x x ] · R · h x x x i = 0 (or [ x x ] · R · [ y y ] = 0 ). The corresponding subsets C ∂M∂u ( p ) and C ∂M∂v ( p ) for ∂M∂u ( p ) and ∂M∂v ( p ) have non-empty intersection; we can then changecoordinates in P (respectively P × P ) and assume that [1 : 0 : 0] (respectively ([1 : 0] , [1 : 0]) ) belongs to C ∂M∂v ( p ) ∩ C ∂M∂u ( p ) . This implies that ∂α ∂u ( p ) = ∂α ∂v ( p ) = 0 .Hence, the point ([1 : 0 : 0] , p ) (respectively ([1 : 0] , [1 : 0] , p ) ) is a singular pointof S , contradicting the smoothness assumption.We now assume that Ker M ( p ) has dimension . Changing coordinates in P (respectively P × P ), we may assume that M ( p ) is diagonal, with α ( p ) = 0 and α ( p ) = α ( p ) = 1 (respectively α ( p ) = 1 ). Hence, π − ( p ) is given by x + x = 0 (respectively x y = 0 ), which is isomorphic to the union of two lines in P . Thefact that S is smooth at ([1 : 0 : 0] , p ) (respectively ([1 : 0] , [1 : 0] , p ) ) impliesthat ( ∂α ∂u ( p ) , ∂α ∂v ( p )) = (0 , . The form of the diagonal matrix M ( p ) implies that det M − α has multiplicity along p , which implies that det M has multiplicity at p , so p is a smooth point of ∆ .The remaining case is when Ker M ( p ) has dimension . Since M ( p ) = 0 , thiscase only occurs for P × A . We may change coordinates and assume that M ( p ) is diagonal with α ( p ) = 1 and α i ( p ) = 0 for i = 1 , . . . , . Hence π − ( p ) is definedby x = 0 , which is a double line.It remains to prove that p is an ordinary double point of ∆ . For each i ∈{ , , } we denote by a i = u ∂α i ∂u ( p ) + v ∂α i ∂v ( p ) the linear part of α i , so that α i − a i has multiplicity at least at p . As S is smooth along π − ( p ) , the polynomial a x + a x x + a x has multiplicity at any point of π − ( p ) (note that α x x + α x x + α x has multiplicity along any point of π − ( p ) as x and α , α vanishon it). The smoothness of the point ([1 : 0 : 0] , p ) is equivalent to to the condition a = 0 . After a linear change of coordinates of A , we may assume that a = u .We then replace x by x + ξx for some ξ ∈ C . Under this change of coordinates a becomes a + 2 a ξ , and we may assume that a = ǫv for some ǫ ∈ C . We thenwrite a = λu + µv for some λ, µ ∈ C . For each θ ∈ C , the point ([ θ : 1 : 0] , p ) issmooth, so we get = θ a + θa + a = ( θ + λ ) u + ( θǫ + µ ) v. Choosing θ such that θ = − λ , we get θǫ + µ = 0 and − θǫ + µ = 0 , so =( θǫ + µ )( − θǫ + µ ) = λǫ + µ .The polynomial det( M ) − α α + α , has multiplicity at least at the origin, so p is a singular point of ∆ and it remains to see that a a − a is not a square. Thisis because a a − a = u ( λu + µv ) − ǫ v , whose discriminant is λǫ + µ . (cid:3) Proposition 5.2.2.
Let F be a smooth Fano threefold being in the list of Table . or equivalently of [CFST16, Table 1] ) . If Aut ◦ ( F ) is not trivial, then F belongsto the families , , or respectively (28) , (13) , (32) , (27)) in the notation of [CFST16] ( respectively of [MM82] ) .Proof. We study the list case-by-case and prove that
Aut( F ) is finite in cases a , b , , and . We use the notation of [CFST16], i.e. the first column. a : In case a , F is a hypersurface of bidegree (2 , in P × P , that we canview as F = n ([ x : x : x ] , [ y : y : y ]) ∈ P × P (cid:12)(cid:12)(cid:12)(cid:2) y y y (cid:3) · M ( x ) · h y y y i = 0 o where M is a symmetric × -matrix whose coefficients are homogeneous poly-nomials of degree in C [ x , x , x ] . We consider the first projection π : F → P ,whose fibres are conics. As F is smooth, the curve ∆ ⊂ P given by the polynomial det( M ) , which parametrises the singular fibres, is reduced and has only ordinarydouble points (it follows from Lemma 5.2.1 applied to affine charts of P ).By Blanchard’s Lemma 2.1.2, the group Aut ◦ ( F ) acts on P via a connectedsubgroup H ⊆ Aut( P ) ≃ PGL ( C ) . We will prove that H is trivial, which im-plies that Aut ◦ ( F ) is trivial, as we can make the same argument with the otherprojection. The group H preserves the reduced curve ∆ ⊂ P of degree . Supposefirst that an irreducible component C of ∆ is not a line. The action of H on C gives an injective group homomorphism H ֒ → Aut( C ) . If C is not rational, then H is trivial, as Aut( C ) does not contain any connected linear algebraic group ofpositive dimension, so we may assume that C is rational. Every singular pointof C is an ordinary node, so if there are at least two singular points, Aut ◦ ( C ) is ONNECTED ALGEBRAIC GROUPS ACTING ON FANO FIBRATIONS OVER P trivial. Hence we may assume that deg( C ) ∈ { , } . Then C ′ = ∆ \ C intersects C in deg( C ) · deg( C ′ ) ≥ points, all fixed by H , which implies again that H istrivial. The remaining case is when ∆ is a union of lines, no of which have acommon point. One may change coordinates so that four of the lines are given by xyz ( x + y + z ) = 0 , which implies that H is trivial, as it has to leave each of thefour lines invariant. b : In case b , F is a cover of a smooth divisor W of bidegree (1 , in P × P branched along a member D of |− K W | . We consider the composition p i : F → W → P of the covering with the projection on the i -th factor, and observethat fibres are conics. By Blanchard’s Lemma 2.1.2, Aut ◦ ( F ) acts on P , makingthe morphism equivariant. Hence, the group Aut ◦ ( F ) acts on W , via a connectedalgebraic subgroup H ⊆ Aut ◦ ( W ) that preserves D . As W is smooth, the divisor D is also smooth, and satisfies K D = 0 . The Kodaira dimension of D being non-negative, Aut ◦ ( D ) does not contain any connected linear algebraic group of positivedimension, so the action of H on D is trivial, and thus H is trivial, as the set ofthe fixed points of a non-trivial automorphism of P is a finite union of lines andisolated points. In case , the blow-up F → P is Aut ◦ ( F ) -equivariant (Lemma 2.1.2). Onemay thus see Aut ◦ ( F ) as the subgroup of Aut( P ) = PGL ( C ) preserving the blown-up curve Γ ⊂ P of degree and genus . As this curve is not contained in a plane,the action on Γ gives an injective group homomorphism Aut ◦ ( F ) ֒ → Aut ◦ (Γ) . Thisshows that Aut ◦ ( F ) is trivial as Aut(Γ) is finite.
In case , F is a double cover of P × P × P whose branch locus is a divi-sor D of tridegree (2 , , . We consider the morphism F → P × P obtainedby projecting onto the first two coordinates. As the fibres are connected, byBlanchard’s Lemma 2.1.2 the group Aut ◦ ( F ) acts on P × P . Doing the samewith the other projections on two factors, one obtains a group homomorphism Aut ◦ ( F ) → Aut ◦ ( P × P × P ) ≃ PGL ( C ) whose image is a connected group H that preserves the divisor D . As F is smooth, the divisor D is also smooth.Moreover, K D ∼ so the Kodaira dimension of F is non-negative, which impliesthat Aut ◦ ( F ) does not contain any connected linear algebraic group of positivedimension. The action of H on F is then trivial, which implies that H is trivial. In case , F is a smooth divisor of multidegree (1 , , , in P × P × P × P =( P ) . We can view F as the set (cid:26) ([ x : x ] , [ y : y ] , [ u : u ] , [ v : v ]) ∈ ( P ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:2) x x (cid:3) · M ( u, v ) · (cid:20) y y (cid:21) = 0 (cid:27) where M is a symmetric × -matrix whose coefficients are homogeneous polynomi-als of bidegree (1 , in C [ u , u ][ v , v ] . We consider the projection π : F → P × P onto the last two factors. By Lemma 5.2.1 applied to affine charts of P × P , as F is smooth, the curve ∆ ⊂ P defined by the polynomial det( M ) , parametrising thesingular fibres, is smooth. By Blanchard’s Lemma 2.1.2, the group Aut ◦ ( F ) actson P × P via a connected subgroup H ⊆ Aut ◦ ( P × P ) ≃ PGL ( C ) × PGL ( C ) .We will prove that H is trivial, which implies that Aut ◦ ( F ) is trivial, as we canmake the same argument with the projection onto the first two factors. The group H preserves the smooth curve ∆ ⊂ P × P which is of bidegree (2 , and is thusof genus . As Aut(∆) does not contain any connected linear algebraic group ofpositive dimension, the group H acts trivially on ∆ , and is thus the trivial group, as the set of fixed points of every non-trivial element of Aut ◦ ( P × P ) consists ofunions of isolated points and fibres of the projections. (cid:3) Explicit descriptions of the groups of automorphisms of positive di-mension.
According to Proposition 5.2.2, the only smooth Fano threefolds F inthe list of Table . for which Aut ◦ ( F ) is not trivial belong to the families , , or . We now describe Aut ◦ ( F ) in these cases, and prove that in each family thereare finitely many isomorphism classes for which Aut ◦ ( F ) is not a torus.In Case , F = P × P × P , so Aut ◦ ( F ) = PGL ( C ) × PGL ( C ) × PGL ( C ) (thisis classical and follows from Blanchard’s Lemma 2.1.2 applied to the projections F → P ). We then consider the cases , and in Lemmata 5.3.1, 5.3.2 and 5.3.4respectively. Lemma 5.3.1.
Let F be the blow-up of a smooth quadric Q ⊂ P along a smoothrational curve C of degree which spans P ( Case of Table . . (1) Aut ◦ ( F ) is either trivial, or isomorphic to G a , G m or PGL ( C ) , all casesbeing possible. (2) If Aut ◦ ( F ) ≃ PGL ( C ) , then up to a change of coordinates, Q is given by x x − x x + 3 x = 0 and C is the image of the Veronese embedding of degree of P . (3) If Aut ◦ ( F ) ≃ G a , then up to a change of coordinates, Q is given by x x − x x + 3 x + x x − x = 0 and C is the image of the Veronese embedding of degree of P . Moreover, there is a unique point p ∈ F fixed by Aut ◦ ( F ) .Proof. By Blanchard’s Lemma 2.1.2, the morphism η : F → Q is Aut ◦ ( F ) -equivariant,so Aut ◦ ( F ) is conjugate via η to the connected component H ◦ of the group H = { g ∈ Aut( Q ) | g ( C ) = C } = { g ∈ Aut( P ) | g ( C ) = C, g ( Q ) = Q } containing the identity. Changing coordinates on P , we may assume that C is theimage of the Veronese embedding τ : P → P , [ u : v ] [ u : u v : u v : uv : v ] . In particular, H is contained in ˆ H = { g ∈ Aut( P ) | g ( C ) = C } , which is isomorphicto Aut( C ) ≃ Aut( P ) ≃ PGL ( C ) . We then choose the following basis of the vectorspace of polynomials of degree vanishing on C : f = x x − x x + 3 x , f = x x − x f = x x − x , f = x x − x x ,f = x x − x x , f = x x − x . One then verifies that f is invariant by ˆ H ≃ PGL ( C ) . ( i ) Suppose first that some torus G m is contained in H . Conjugating by anelement of ˆ H , the torus acts on the image of the Veronese embedding via τ as [ u : v ] [ u : ξv ] , ξ ∈ C ∗ and thus acts on P as [ x : · · · : x ] [ x : ξx : ξ x : ξ x : ξ x ] . Replacing x i with ξ i x i in f , . . . , f yields ξ f , ξ f , ξ f , ξ f , ξ f , ξ f .Hence, the equation of Q is either given by f + κf = 0 for some κ ∈ C , or isgiven by f i for some i ∈ { , . . . , } . This latter case is impossible as Q is smooth,so we are in the former case. If κ = 0 , then H ≃ PGL ( C ) , as f is invariant by ˆ H . We may thus assume that κ = 0 and prove that H = G m ⋊ h σ i , where σ is theinvolution σ : [ x : x : · · · : x : x ] [ x : x : · · · : x : x ] , which preserves Q .As PGL ( C ) acts -transitively on P , the group G m ⋊ h σ i , which is the group of ONNECTED ALGEBRAIC GROUPS ACTING ON FANO FIBRATIONS OVER P elements preserving { [1 : 0] , [0 : 1] } , is maximal in ˆ H . Hence, it remains to seethat H = ˆ H when κ = 0 . We simply consider the automorphism of P given by [ u : v ] [ u : v + u ] , which induces the automorphism ν : [ x : · · · : x ] [ x : x + x : x + 2 x + x : x + 3 x + 3 x + x : x + 4 x + 6 x + 4 x + x ] on P ,and observe that ν does not preserve Q . ( ii ) Suppose now that no torus G m is contained in H . This implies that either H is finite (in which case Aut ◦ ( F ) is trivial) or H ◦ ≃ G a . Conjugating by an elementof ˆ H , we may assume that H corresponds to [ u : v ] [ u : v + ξu ] , ξ ∈ C , and thuscontains the element ν given above. We then compute that f ◦ ν, . . . , f ◦ ν are equalto f , f , f + f , f + f + f , f + 2 f + 3 f + 6 f + f , f + f + 2 f + 6 f + 2 f + f respectively. This implies that the equation of Q is a linear combination of f and f . As { f = 0 } is singular and no torus is contained in H , the equation is ofthe form f + λf for some λ ∈ C ∗ . Conjugating by [ u : v ] [ u : ξv ] for some ξ ∈ C ∗ we obtain the equation f + f . We now prove that H ◦ is indeed isomorphicto G a for this equation. Indeed, otherwise H ◦ would be bigger and thus wouldcontain the torus [ u : v ] [ u : ξv ] , ξ ∈ C ∗ , which is impossible because f + f is not invariant under this torus. It remains to show that there is only one fixedpoint for Aut ◦ ( F ) . As q = [0 : · · · : 0 : 1] is the only point of P fixed by ν , allfixed points of Aut ◦ ( F ) are contained in the preimage of q in F , which is a curve e isomorphic to P . It remains then to see that the action of Aut ◦ ( F ) on e is nottrivial. Note that the tangent hyperplane of Q at q is given by x = 0 . The tangentline of C at q is given by x = x = x = 0 . The quadrics f , f , f then generatethe conormal bundle of C at q and f , f , f are singular at q . Since f ◦ ν = f , f ◦ ν = f + 2 f + 3 f + 6 f + f and f ◦ ν = f + f + 2 f + 6 f + 2 f + f , theaction of Aut ◦ ( F ) on e is not trivial. (cid:3) A smooth hypersurface of bidegree (1 , in P × P is isomorphic to P ( T P ) , viaany of the two projections. In the next lemma we recall the proof of this classicalfact for the reader’s convenience. Lemma 5.3.2.
Let F ⊂ P × P be a smooth hypersurface of bidegree (1 ,
1) (
Case of Table . . Then, the following hold: (1) Changing coordinates on P × P , the threefold F is given by F = (cid:26) ([ x : x : x ] , [ y : y : y ]) ∈ P × P (cid:12)(cid:12)(cid:12)(cid:12) P i =0 x i y i = 0 (cid:27) . (2) The group
PGL ( C ) acts faithfully on F via PGL ( C ) × F −→ F (cid:16) A, (cid:16)h x x x i , h y y y i(cid:17)(cid:17) (cid:16) A · h x x x i , t A − · h y y y i(cid:17) . Moreover, this actions provides an isomorphism
PGL ( C ) ≃ Aut ◦ ( F ) .Proof. (1): The variety F is given by n ([ x : x : x ] , [ y : y : y ]) ∈ P × P (cid:12)(cid:12)(cid:12) [ x x x ] · M · h y y y i = 0 o for some matrix M ∈ Mat × ( C ) . After a change of coordinates on P × P of theform ( A, B ) with A, B ∈ GL ( C ) , we can assume that M is diagonal with all entries equal to either or . Indeed, in the new coordinates, F is given by the matrix t A − M B − . As F is smooth, this implies that M is the identity matrix, so F = n ([ x : x : x ] , [ y : y : y ]) ∈ P × P (cid:12)(cid:12)(cid:12)P i =0 x i y i = 0 o . (2): The group PGL ( C ) acts faithfully on F via PGL ( C ) × F −→ F (cid:16) A, (cid:16)h x x x i , h y y y i(cid:17)(cid:17) (cid:16) A · h x x x i , t A − · h y y y i(cid:17) . In particular, the group
Aut ◦ ( F ) contains PGL ( C ) . It remains to see that eachelement of Aut ◦ ( F ) is of this form. To see this, we first use Blanchard’s Lemma 2.1.2for the two projections F → P , which are P -bundles, and obtain that each elementof Aut ◦ ( F ) is of the form ( x, y ) ( Ax, By ) for some A, B ∈ PGL ( C ) . Applyingthe element ( x, y ) ( t Bx, B − y ) , we may assume that B is the identity. It remainsto see that A is the identity too. Denoting by π : F → P the second projection,the automorphism ( x, y ) ( Ax, y ) leaves invariant every fibre of π . In particular,it preserves π − ([1 : 0 : 0]) , π − ([0 : 1 : 0]) and π − ([0 : 0 : 1]) , so A is diagonal.It moreover preserves π − ([1 : 1 : 1]) , whose image under the first projection is theline x + x + x = 0 , and thus A is the identity. (cid:3) Case of Table . is presented in [Pro13] as an intersection of three hypersur-faces of tridegree (0 , , , (1 , , and (1 , , in P × P × P . In [Pro13, Case1.2.6, page 422], it is explained that varieties of that form are isomorphic to smoothvarieties as in Case . Lemma 5.3.3 below gives an explicit way to see the converse.Lemma 5.3.4 then describes the group of automorphisms. Lemma 5.3.3.
Let T ⊂ P × P be a smooth hypersurface of bidegree (1 , . Let C ⊂ T be a smooth curve of bidegree (2 , , such that the projection to any of thetwo P gives an embedding C ֒ → P . Denoting by η : F → T the blow-up of C andby E ⊂ F the exceptional divisor, the following hold: (1) The threefold F is a smooth Fano threefold of Picard rank . (2) The divisor D = − K F − E is ample. (3) The linear system | D + K F | gives a morphism κ : F → P . Moreover, themorphism κ × η gives a closed embedding F → P × P × P , that sends F onto athe intersection of three hypersurfaces of tridegree (0 , , , (1 , , and (1 , , .Proof. After changing coordinates in P × P , we may assume that C is the imageof the morphism τ : P ≃ −→ C , [ u : v ] ([ u : uv : v ] , [ u : uv : v ]) .We have K T = − H , where H ⊂ T is the intersection of T with a hypersurfaceof P × P of bidegree (1 , . Since K F = η ∗ ( K T ) + E , we obtain D + K F = − K F − E = − η ∗ ( K T ) − E = η ∗ ( H ) − E . Therefore the linear system | D + K F | is the linear system of strict transforms of hypersurfaces of bidegree (1 , through C .The vector space of polynomials of bidegree (1 , vanishing along C is of dimension , generated by x y − x y , x y − x y , x y − x y , x y − x y . The equation of T is a linear combination of the four above polynomials. We observethat it is linearly independent of the first three. Indeed, for any ( a , a , a ) ∈ C \ { } , the hypersurface of P × P given by a ( x y − x y ) + a ( x y − x y ) + a ( x y − x y ) is singular at ([ a : a : a ] , [ a : a : a ]) ∈ P × P , as the derivativewith respect to x i or y i is zero for all i . This implies that the intersection of T with ONNECTED ALGEBRAIC GROUPS ACTING ON FANO FIBRATIONS OVER P the diagonal of P × P is equal to C , that the linear system of hypersurfaces of T of degree (1 , through C is of dimension , and that the rational map κ : F P induced by | D + K F | is equal to κ = θ ◦ η , where θ : P × P P is given by θ (cid:0) ([ x : x : x ] , [ y : y : y ]) (cid:1) = [ x y − x y : x y − x y : x y − x y ] . We consider the variety W = { ( x, y, z ) ∈ P × P × P | x z + x z + x z = 0 , y z + y z + y z = 0 } and observe that ǫ : W → P × P , ( x, y, z ) ( x, y ) is the blow-up of the diagonal ∆ ⊂ P × P . This can for instance be seen on the local charts U ij ⊂ P × P given by x i y j = 0 with i, j ∈ { , , } . The inverse of the blow-up is ( x, y ) ( x, y, θ ( x, y )) . Hence, ǫ : ǫ − ( T ) → T is the blow-up of T ∩ ∆ = C , with inverse ( x, y ) ( x, y, κ ( x, y )) . It follows that the map κ × η : F P × P × P is anisomorphism onto its image F ≃ −→ ǫ − ( T ) . This proves (3), which in turn impliesthat D + K F = η ∗ ( H ) − E is semiample but not big.To see that − K F = 2 η ∗ ( H ) − E is ample, one observes that for each curve Γ ⊂ F not contracted by η , one has Γ · ( − K F ) ≥ Γ · η ∗ ( H ) > since η ∗ ( H ) − E is nefand that Γ · ( − K F ) ≥ Γ · ( η ∗ ( H ) − E ) > for curves Γ contracted by η . Moreover, ( − K F ) = 30 > , so we conclude by the Nakai-Moishezon criterion.We can also see that F is in [MM82, n ◦
13 of Table 3]. This gives (1) and thenimplies that D is ample, since D + K F is semiample. (cid:3) Lemma 5.3.4.
Let F be the blow-up of a smooth divisor T ⊂ P × P of bidegree (1 , along a curve C of bidegree (2 , , such that C ֒ → T ֒ → P × P → P is anembedding for both projections P × P → P , assume that F is smooth ( Case ofTable . . Then, the following hold: (1) Aut ◦ ( F ) is either isomorphic to G m , or to G a or to PGL ( C ) , all cases beingpossible. (2) If Aut ◦ ( F ) ≃ PGL ( C ) , then F is isomorphic to F = ([ x : x : x ] , [ y : y : y ] , [ z : z : z ]) ∈ ( P ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) P i =0 x i y i = 0 P i =0 x i z i = 0 P i =0 y i z i = 0 . The projection F → P × P on two different factors yields a birational morphism F → T , where T = n ([ x : x : x ] , [ y : y : y ]) ∈ P × P (cid:12)(cid:12)(cid:12)P i =0 x i y i = 0 o , whichis the blow-up of Γ = n ([ x : x : x ] , [ x : x : x ]) ∈ P × P (cid:12)(cid:12)(cid:12)P i =0 x i = 0 o .For each matrix M ∈ PGL ( C ) such that t M · M = id , we have an automorphismof F given by F −→ F (cid:16)h x x x i , h y y y i , h z z z i(cid:17) (cid:16) M · h x x x i , M · h y y y i , M · h z z z i(cid:17) . and every automorphism of F is of this form. This gives an isomorphism Aut ◦ ( F ) ≃ PO ( C ) ≃ PGL ( C ) . (3) If Aut ◦ ( F ) ≃ G a , there is only one isomorphism class for F . This latter isisomorphic to the intersection of three hypersurfaces of P × P × P of tridegree (0 , , , (1 , , and (1 , , such that the group Aut ◦ ( F ) acts on each of the threecopies of P by fixing exactly one point. Proof.
By definition, we have a birational morphism η : F → T , where T ⊂ P × P is a smooth divisor of bidegree (1 , . This morphism is the blow-up of a curve C ofbidegree (2 , . By Blanchard’s Lemma 2.1.2, the morphism η : F → T is Aut ◦ ( F ) -equivariant, so Aut ◦ ( F ) is conjugate via η to the group H = { g ∈ Aut ◦ ( T ) | g ( C ) = C } . Changing coordinates, by Lemma 5.3.2(1) we may assume that T = n ( x, y ) ∈ P × P (cid:12)(cid:12)(cid:12)P i =0 x i y i = 0 o . Moreover, the embeddings of C into P givenby each of the two projections π , π : P × P → P induce isomorphisms of C withtwo conics C ⊂ P and C ⊂ P . We may apply an automorphism of T of the form ( t A − , A ) (cf. Lemma 5.3.2(2)) and assume that C is defined by P i =0 x i = 0 .By Lemma 5.3.2, PGL ( C ) ≃ Aut ◦ ( T ) acts on T by ( x, y ) ( Ax, t A − y ) . Thesubgroup of PGL ( C ) that preserves the conic C ⊂ P is the projective orthogonalgroup PO ( C ) = { M ∈ PGL ( C ) | M = t M } ≃ PGL ( C ) . Hence, the group H iscontained in the subgroup ˆ H ⊂ Aut ◦ ( T ) given by ˆ H = { ( x, y ) ( M x, M y ) | M ∈ PO ( C ) } ≃ PO ( C ) ≃ PGL ( C ) . The closed curve Γ = { ([ x : x : x ] , [ x : x : x ]) ∈ P × P | P i =0 x i = 0 } ⊂ T given by the diagonal embedding of C into T is invariant by ˆ H , so one obtains H = ˆ H if C = Γ . In any case, C is contained inthe surface ( π ) − ( C ) ⊂ T , which is isomorphic to P × P , via τ : P × P ≃ −→ ( π ) − ( C )([ a : b ] , [ c : d ]) ([ a − b : i ( a + b ) : 2 ab ] , [ ac − bd : i ( ac + bd ) : ad + bc ]) . Moreover, the isomorphism τ sends the diagonal of P × P onto Γ . As ˆ H ≃ PGL ( C ) acts on ( π ) − ( C ) ≃ P × P via a faithful action on the first coordinate(corresponding to the action of PO ( C ) on C ) and preserves the diagonal, theaction on P × P is the diagonal action (for a suitable isomorphism ˆ H ≃ PGL ( C ) ).In particular, Γ is the unique curve of ( π ) − ( C ) that is invariant by ˆ H .The curve C is the image by τ of a curve C ′ ⊂ P × P of bidegree (1 , . If C isnot equal to Γ , it intersects Γ in two or one point. In the first case, we may apply anelement of ˆ H and assume that the two points are the image by τ of ([0 : 1] , [0 : 1]) and ([1 : 0] , [1 : 0]) , which implies that C ′ is given by ad + ξbc for some ξ ∈ C \ {− } .Hence, H is isomorphic to C ∗ , acting as ([ a : b ] , [ c : d ]) ([ λa : b ] , [ λc : d ]) . Inthe second case, we may assume that the point is ([0 : 1] , [0 : 1]) . Hence, C ′ is given by ad − bc + ξac for some ξ ∈ C ∗ . Applying an element of the form ([ a : b ] , [ c : d ]) ([ λa : b ] , [ λc : d ]) , we may assume that ξ = 1 . The group is thenisomorphic to G a , via ([ a : b ] , [ c : d ]) ([ a : b + µa ] , [ c : d + µc ]) . This achieves theproof of (1).(2): If Aut ◦ ( F ) ≃ PGL ( C ) , then in the above description, C is given by Γ . Wewrite as above F = (cid:26) ( x, y, z ) ∈ ( P ) (cid:12)(cid:12)(cid:12)(cid:12)X i =0 x i y i = X i =0 x i z i = X i =0 y i z i = 0 (cid:27) , and consider the rational map τ : T F given by ( x, y ) ( x, y, [ x y − x y : x y − x y : x y − x y ]) , which is PO ( C ) -equivariant, with an action on F given by ( x, y, z ) ( M x, M y, M z ) (follows from the fact that τ corresponds to thecross-product). Lemma 5.3.3 implies that τ ◦ η : F F is an isomorphism andthus that the projection F → T is the blow-up of C . ONNECTED ALGEBRAIC GROUPS ACTING ON FANO FIBRATIONS OVER P (3): If Aut ◦ ( F ) ≃ G a , then the curve C is the image by τ of the curve of P × P given by ad − bc + ac = 0 . Hence, C = { τ ([ a : b ] , [ a : b − a ]) | [ a : b ] ∈ P } = { ([ a − b : i ( a + b ) : 2 ab ] , [ a + ab − b : i ( a + b − ab ) : 2 ab − a ]) | [ a : b ] ∈ P } . The action of
Aut ◦ ( F ) ≃ G a on any of the two P preserves the conic and fixes aunique point of the conic, so has a unique point fixes on P . This point is equal to τ ([0 : 1] , [0 : 1]) . Using the embedding of F into P × P × P of Lemma 5.3.3, onegets an action of G a on the third factor too, with a unique fixed point, indeed, theprojection on the last two coordinates gives again a birational morphism which isthe blow-up of a curve of bidegree (2 , (see [Pro13, Case 1.2.6, page 422]) and wecan use the same argument as above. (cid:3) Corollary 5.3.5.
There are exactly two smooth Fano threefolds F which satisfy ρ ( F ) ≥ , Aut ◦ ( F ) ≃ PGL ( C ) and which can occur as general fibres of a Morifibre space. These two threefolds are the following: ( A ) The blow-up of the quadric Q ⊂ P given by x x − x x + 3 x = 0 alongthe image of the Veronese embedding of degree of P . ( B ) The threefold ( ( x, y, z ) ∈ ( P ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) X i =0 x i y i = X i =0 x i z i = X i =0 y i z i = 0 ) . Proof.
Let F be a Fano threefold with ρ ( F ) ≥ and Aut ◦ ( F ) ≃ PGL ( C ) , whichoccur as a general fibre of a Mori fibre space. By [CFST16, Theorem 1.4], thethreefold needs to be in the list of Table . or equivalently of [CFST16, Table 1] ) .Since Aut ◦ ( F ) is not trivial, Proposition 5.2.2 implies that F belongs to the families , , or .In Case , Lemma 5.3.1 proves that F is isomorphic to the threefold ( A ) above.In Case , it is impossible to have Aut ◦ ( F ) ≃ PGL ( C ) (Lemma 5.3.2).In Case , Lemma 5.3.4 proves that F is isomorphic to the threefold ( B ) above.In Case , we have F ≃ ( P ) , contradicting Aut ◦ ( F ) ≃ PGL ( C ) . (cid:3) Symmetric birational maps from ( P ) or P ( T P ) Symmetric birational maps from P × P × P . In order to constructbirational maps from a Mori fibre space X → B with general fibre a smooth Fanothreefold F , we need to understand rational maps from F to other varieties, whichare symmetric enough. Those will be typically induced by sublinear systems of − mK F for some positive m ∈ Q . In this section, we study the case of F = P × P × P . Lemma 6.1.1.
Let C be a curve of tridegree (1 , , in F = P × P × P , let η : ˆ F → F be the blow-up of F along C , with exceptional divisor E . (1) The threefold ˆ F is a smooth Fano threefold of Picard rank . (2) The divisor D = − K ˆ F − E is ample. (3) The linear system | D + K ˆ F | gives a birational morphism ˆ F → P , which isthe contraction of the strict transforms of divisors of F of tridegree (0 , , , (1 , , , (1 , , through C , or equivalently the blow-up of three skew lines of P . Proof.
We have K F = − H , where H ⊂ F is a hypersurface of tridegree (1 , , .Since K ˆ F = η ∗ ( K F ) + E , we obtain D + K ˆ F = − K ˆ F − E = − η ∗ ( K F ) − E = η ∗ ( H ) − E . Changing coordinates, we may assume that C = { ([ u : v ] , [ u : v ] , [ u : v ]) | [ u : v ] ∈ P } . The divisors H , H , H ⊆ F of tridegree (0 , , , (1 , , , (1 , , through C are then given by H = { x y − x y = 0 } , H = { y z − y z = 0 } , H = { x z − x z = 0 } . The rational map τ : F P induced by |− K ˆ F − E | is then given by hyper-surfaces of tridegree (1 , , through C and thus given by ([ x : x ] , [ y : y ] , [ z : z ]) [ y ( x z − x z ) : y ( x z − x z ) : z ( x y − x y ) : z ( x y − x y )] . Its inverse τ − : P P × P × P is given by [ w : x : y : z ] ([ w − y : x − z ] , [ w : x ] , [ y : z ]) . We observe that τ − contracts the smooth quadric surface S = { wz − xy = 0 } ⊂ P onto the curve C , and that τ contracts respectively H , H , H onto the three skewlines ℓ , ℓ , ℓ ⊂ S ⊂ P given by ℓ = { y = z = 0 } , ℓ = { w − y = x − z = 0 } , ℓ = { w = x = 0 } . Denote by κ : X → P the blow-up of ℓ , ℓ , ℓ . For i ∈{ , , } , we denote by π i : P × P × P → P the i -th projection, and observethat π i ◦ τ − : P P is the linear projection away from the line ℓ i . Hence, π i ◦ τ − ◦ κ : X → P is a morphism. This being true for the three projections, thebirational map τ − ◦ κ : X → P × P × P is a morphism. This birational morphismbetween two smooth threefolds contracts the strict transform of S , isomorphic to P × P , onto the curve C ≃ P , and is thus the blow-up of C . This achieves theproof of (3).To prove (1), one can compute the cone of effective curves and prove that it ispolyhedral, like in Lemma 6.1.3 and check that − K ˆ F is ample. Equivalently, onecan see that ˆ F appears in the classifcation of Fano threefolds (see [MM82, n ◦ D + K ˆ F is big and nef,and (1) implies that − K ˆ F is ample, so D ample. (cid:3) Remark . In Lemma 6.1.1, note that τ : P × P × P P is PGL ( C ) -invariant, where (cid:18) a bc d (cid:19) ∈ PGL ( C ) acts on P × P × P and P as ([ x : x ] , [ y : y ] , [ z : z ]) ([ ax + bx : cx + dx ] , [ ay + by : cy + dy ] , [ az + bz : cz + dz ]) and [ w : x : y : z ] [ aw + bx : cw + dx : ay + bz : cy + dz ] . Lemma 6.1.3.
Let p be a point of F = P × P × P and let ℓ , ℓ , ℓ ⊂ F be the threecurves of tridegree (1 , , , (0 , , , (0 , , passing through p . Let η : F → F bethe blow-up of F at p and let η : F → F be the blow-up at the strict transforms ˜ ℓ , ˜ ℓ , ˜ ℓ of ℓ , ℓ , ℓ . Denoting by E i ⊂ F i the exceptional divisor of η i and writingagain E ⊂ F for the strict transform of E ⊂ F , the following hold: (1) The divisor D = − K F − E − E is big and nef. (2) The linear system | D + K F | gives a birational morphism τ : F → P , whichis the contraction of the exceptional divisors of η and the strict transforms of thedivisors of F of tridegree (1 , , , (0 , , , (0 , , through p . It is also the blow-upof three non-collinear points of P followed by the blow-up of the strict transformsof the three lines through two of them. ONNECTED ALGEBRAIC GROUPS ACTING ON FANO FIBRATIONS OVER P (3) The birational map τ = τ ◦ η − : F P is obtained by the flop of thecurves ˜ ℓ , ˜ ℓ , ˜ ℓ followed by the contraction of the strict transforms of the divisorsof F of tridegree (1 , , , (0 , , , (0 , , through p . (4) The divisor A = − ( η ) ∗ K F − E is an ample divisor of F and the unionof the curves of F having intersection with A is E ∪ ˜ ℓ ∪ ˜ ℓ ∪ ˜ ℓ .Proof. We denote by H , H , H ⊂ F the divisors of tridegree (1 , , , (0 , , , (0 , , respectively through p . This gives, for all i, j, k with { , , } = { i, j, k } ,that ℓ k = H i ∩ H j , H i · ℓ j = 0 and H i · ℓ i = 1 . The cone of curves of F is thengenerated by ℓ , ℓ , ℓ , and one has − K F = 2 H + 2 H + 2 H , so − K F · ℓ i = 2 foreach i ∈ { , , } .We denote by e ⊂ E ⊂ F a line in E ≃ P , by ˜ ℓ i and H i the strict transformsof ℓ i and H i on F , giving ( η ) ∗ ( H i ) = H i + E , for i = 1 , , . For all i, j ∈ { , , } with i = j , one finds the following intersection numbers: E H i H j e − ℓ i − This implies that the cone of curves of F is polyhedral, generated by ˜ ℓ , ˜ ℓ , ˜ ℓ , e .Indeed, each irreducible curve C of F is either contained in E ≃ P and thusequivalent to a positive multiple of e , or is the strict transform of a curve of F ,so equal to P a i ˜ ℓ i + be with b ∈ Z and a , a , a ≥ , a + a + a ≥ . If C isnot equal to ˜ ℓ , ˜ ℓ or ˜ ℓ , then it is not contained in H i and H j for two distinct i, j ∈ { , , } . Choosing k with { i, j, k } = { , , } we obtain ≤ H i · C = b − a j − a k and ≤ H j · C = b − a i − a k , which implies that b ≥ . We moreover obtain K F = ( η ) ∗ ( K F ) + 2 E = − P i =1 H i − E . We now use this to prove (4). Firstly, the divisor A = − ( η ) ∗ K F − E =2 P i =1 H i + 5 E is ample as A · ˜ ℓ = A · ˜ ℓ = A · ˜ ℓ = A · e = 1 . As every line in E ≃ P is equivalent to e , its intersection with A is . The union of curves havingintersection with A thus contains E ∪ ˜ ℓ ∪ ˜ ℓ ∪ ˜ ℓ . Conversely, an irreduciblecurve C ⊂ F not contained in E is numerically equivalent to P a i ˜ ℓ i + be with a , a , a , b ≥ and a + a + a ≥ . Moreover, if it is not equal to ˜ ℓ , ˜ ℓ or ˜ ℓ ,then b ≥ , as we observed before, so C · A = a + a + a + b ≥ . This achievesthe proof of (4). ℓ ℓ ℓ H H H • p ˜ ℓ ˜ ℓ ˜ ℓ e e e E H H H s s s s s s e e e f f f f f f E H H H E E E Figure 1.
The divisors and curves on F , F and F respectively.The morphism η : F → F being the blow-up of ˜ ℓ , ˜ ℓ , ˜ ℓ , we denote by E i theirreducible component of E lying over ˜ ℓ i , for i = 1 , , . The divisor E i is isomorphic to P × P as it is isomorphic to the exceptional divisor of the blow-up of F along ℓ i . We write s i ⊂ E i for a section of E i → ˜ ℓ i of self-intersection and f i ⊂ E i a fibre of η ; in particular f i · E i = − . We then denote again by E , H , H , H the strict transforms of the same surfaces on F and obtain η ∗ ( E ) = E and η ∗ ( H ) = H + E + E .Since s · η ∗ ( H ) = ( η ) ∗ ( s ) · H = ˜ ℓ · H = − and s · H = s · E = 0 , wefind s · E = − . Similarly, one obtains s i · E i = − for each i ∈ { , , } .For all distinct i, j ∈ { , , } , we also denote by e ij ⊂ F the strict transform ofthe line of E ≃ P that intersects ˜ ℓ i and ˜ ℓ j (see Figure 1). This gives E · e ij = η ∗ ( E ) · e ij = E · ( η ) ∗ ( e ij ) = E · e = − . We similarly obtain E · f i = E · f j = 0 .For all i, j, k with { i, j, k } = { , , } , one finds the following intersection num-bers: H i H k E E i E k e ij − − s i − f i − The fact that e ij · H k = − can be computed as follows: e · H = ( η ) ∗ e · H = e · η ∗ ( H ) = e · ( H + E + E ) = e · H + 2 .We now prove that the cone of effective curves of F is polyhedral and generatedby s , s , s , f , f , f , e , e , e , by proving that every irreducible curve C is anon-negative linear combination of these. If C is contained in one of the surfaces E , E i , H i , i ∈ { , , } this is true as our curves include all extremal rays of thesedel Pezzo surfaces. We may thus assume that C is not contained in E or in anyof the E i or H i . The curve C is the strict transform of η ( C ) ≡ P i =1 a i ˜ ℓ i + be for some a , a , a , b ≥ . This gives C ≡ P i =1 a i s i + be + P i =1 c i f i for some c , c , c ∈ Z . As ≤ H · C = c − b we find that c ≥ . Similarly, ≤ H i · C = c i for i = 2 , , proving the statement on the Mori cone of F .We have E = P i =1 E i and D = − K F − E − E . Writing η = η ◦ η , weget K F = η ∗ ( K F ) + 2 E + E = − P i =1 H i − E − E .D = − η ∗ ( K F ) − E − E = 3 P i =1 H i + 5 E + 4 E This implies that D · e ij = 0 , D · s i = 1 and D · f i = 2 . for all distinct i, j ∈ { , , } .The divisor D is thus nef. As − K F is big (it is very ample), the divisor − η ∗ ( K F ) = P i =1 H i + 3 E + 2 E is big. Since D + η ∗ ( K F ) = 2 P i =1 H i + 2 E + 2 E is effective, this implies that D is big. This achieves the proof of (1).To prove (2), we change coordinates, assume that p = ([0 : 1] , [0 : 1] , [0 : 1]) andtake coordinates ([ x : x ] , [ y : y ] , [ z : z ]) on F . This implies that H , H , H are respectively given by x = 0 , y = 0 and z = 0 . We consider the linear system | D + K F | on F . Since D + K F = X i =1 H i + E + E = − η ∗ ( K F ) − E − E , ONNECTED ALGEBRAIC GROUPS ACTING ON FANO FIBRATIONS OVER P it corresponds to the strict transform of hyperplane sections of F = P × P × P oftridegree (1 , , having multiplicity at p and passing through the three curves ℓ , ℓ , ℓ . This last condition is in fact implied by the first. The linear systemcorresponds then to the toric birational map τ : P × P × P P given by τ : ([ x : x ] , [ y : y ] , [ z : z ]) [ x y z : x y z : x y z : x y z ] , whose inverse is given by τ − : [ w : x : y : z ] ([ w : x ] , [ w : y ] , [ w : z ]) , and which restricts to an isomorphism F \ ( H ∪ H ∪ H ) ≃ −→ P \ H w , where H w ⊂ P is the hyperplane given by w = 0 . Hence, the linear system | D + K F | yields the toric birational map τ = τ ◦ η : F P .We prove now that τ is a morphism, i.e. that it is defined at every point q ∈ F .As τ is defined outside of ℓ ∪ ℓ ∪ ℓ , we may assume that η ( q ) ∈ ℓ ∪ ℓ ∪ ℓ . Usingthe action of Sym on x, y, z , we may assume that η ( q ) ∈ ℓ . If η ( q ) = p , then η ( q ) belongs to the image of the open embedding A ֒ → F, ( r, s, t ) ([1 : r ] , [ s : 1] , [ t :1]) . The morphism η corresponds on this chart to the blow-up of s = t = 0 , thatis { (( r, s, t ) , [ u : v ]) ∈ A × P | sv = tu } . Hence, τ is locally given by (( r, s, t ) , [ u : v ]) [ st : rst : t : s ] = [ sv : rsv : v : u ] = [ tu : rtu : v : u ] and is then well-definedat every point. The remaining case is where η ( q ) = p , so η ( q ) belongs to the surface E ⊂ F isomorphic to P . We study τ = τ ◦ η : F P in a neighbourhood of E . For this, we take the open embedding A ֒ → F, ( r, s, t ) ([ r : 1] , [ s : 1] , [ t : 1]) ,and obtain that η is the blow-up of the origin of A in this chart, corresponding to { (( r, s, t ) , [ u : v : w ]) ∈ A × P | su = rv, sw = tv, rw = tu } . The rational map η is locally given by (( r, s, t ) , [ u : v : w ]) [ rst : st : rt : rs ] = [ rvw : vw : uw : uv ] .The divisor E corresponds to (0 , , × P , so η is defined at every point of E except the three toric points. These are exactly the points where ( η ) − isnot an isomorphism. Using the symmetry, we may assume that η ( q ) = E ∩ ˜ ℓ ,corresponding to ((0 , , , [0 : 0 : 1]) ∈ A × P . We choose the open embedding A ֒ → A × P , ( a, b, c ) (( ac, bc, c ) , [ a : b : 1]) , and see η as the blow-up of theline a = b = 0 . This latter is given by { (( a, b, c ) , [ α : β ]) ∈ A × P | aβ = bα } and τ is given by (( a, b, c ) , [ α : β ]) [ abc : b : a : ab ] = [ aβc : β : α : aβ ] , and is thusdefined at every point.We have now proven that τ = | D + K F | : F → P is a birational morphism.Since τ ( H \ ( ℓ ∪ ℓ )) = [0 : 1 : 0 : 0] , the morphism τ contracts the surface H ⊂ F onto [0 : 1 : 0 : 0] . Similarly, the surfaces H , H ⊂ F are contracted onto [0 : 0 : 1 : 0] and [0 : 0 : 0 : 1] . The above description of τ implies also that E iscontracted to the curve w = x = 0 , so E and E are contracted onto w = y = 0 and w = z = 0 . One can then either check in coordinates or use the universal propertyof blowing-ups, to see that τ is exactly the blow-up of [0 : 1 : 0 : 0] , [0 : 0 : 1 : 0] and [0 : 0 : 0 : 1] , followed by the blow-up of the strict transform of the three linesthrough these points. This achieves the proof of (2).It remains to prove (3). We have already shown that τ is obtained by blowing-up the curves ˜ ℓ , ˜ ℓ , ˜ ℓ , then contracting their divisors E , E , E ≃ P × P “in theother direction” and then contracting the strict transforms of the divisors of F oftridegree (1 , , , (0 , , , (0 , , through p onto [0 : 1 : 0 : 0] , [0 : 0 : 1 : 0] and [0 : 0 : 0 : 1] . As ˜ ℓ , ˜ ℓ , ˜ ℓ ⊂ F are extremal and have intersection with thecanonical, the blow-up of them followed by the contracting of the divisors “in theother direction” simply consists of three Atiyah flops. (cid:3) Symmetric birational maps from P ( T P ) . We now describe symmetric bi-rational maps from a smooth hypersurface of P × P of bidegree (1 ,
1) (
Case ofTable . , which is isomorphic to P ( T P ) (Lemma 5.3.2).The following lemma is similar to Lemma 6.1.1. Lemma 6.2.1.
Let C be a curve of bidegree (1 , in P × P contained in a smoothhypersurface F of bidegree (1 , . Let η : ˆ F → F be the blow-up of F at C , withexceptional divisor E . (1) The threefold ˆ F is a smooth Fano threefold of Picard rank . (2) The divisor D = − K ˆ F − E is ample. (3) The linear system | D + K ˆ F | gives a birational morphism ˆ F → Q , where Q isa smooth quadric in P , which is the contraction of the strict transforms of divisorsof F of bidegree (0 , , (1 , through C , or equivalently the blow-up of two skewlines of Q .Proof. We have K F = − H , where H ⊂ F is the intersection of F with a hy-persurface of P × P of bidegree (1 , . Since K ˆ F = η ∗ ( K F ) + E , we obtain D + K ˆ F = − K ˆ F − E = − η ∗ ( K F ) − E = η ∗ ( H ) − E . Changing coordinates, wemay assume that F = (cid:26) ([ x : x : x ] , [ y : y : y ]) ∈ P × P (cid:12)(cid:12)(cid:12)(cid:12) P i =0 x i y i = 0 (cid:27) . (Lemma 5.3.2). We may then apply an element of PGL ( C ) as in Lemma 5.3.2(2)and assume that the projection of C onto the first coordinate is given by x = 0 ,so C is given by C = { ([0 : u : v ] , [ αu + βv : − v : u ]) | [ u : v ] ∈ P } for some α, β ∈ C . Applying an automorphism of the form ([ x : x : x ] , [ y : y : y ]) ([ x : x − βx : x + αx ] , [ y + βy − αy : y : y ]) , we may assume that α = β = 0 .The divisors H , H ⊆ F of bidegree (0 , , (1 , through C are then givenrespectively by H = { y = 0 } , H = { x = 0 } . The linear system |− K ˆ F − E | is the linear system of strict transforms ofhypersurfaces of bidegree (1 , through C and thus the rational map τ : F Q ⊂ P induced by it is given by ([ x : x : x ] , [ y : y : y ]) [ x y : x y : x y : x y : x y ] . Its image is Q = { [ z : · · · : z ] ∈ P | z + z z + z z = 0 } . The inverse τ − : Q F is given by [ z : · · · : z ] ([ z : z : z ] , [ z : z : z ]) . We observe that τ − contracts the smooth quadric surface S = { z = 0 , z z + z z = 0 } ⊂ Q ⊂ P onto the curve C , and that τ contracts respectively H , H onto the two skew lines ℓ , ℓ ⊂ S given by ℓ = { z = z = z = 0 } and ℓ = { z = z = z = 0 } . Denote by κ : X → Q the blow-up of ℓ , ℓ . For i ∈ { , } , we denoteby π i : P × P × P the i -th projection, and observe that π i ◦ τ − : P P is thelinear projection away from the line ℓ i . Hence, π i ◦ τ − ◦ κ : X → P is a morphism.This being true for the two projections, the birational map τ − ◦ κ : X → F is amorphism. This birational morphism between two smooth threefolds contracts the ONNECTED ALGEBRAIC GROUPS ACTING ON FANO FIBRATIONS OVER P strict transform of S , isomorphic to P × P onto the curve C ≃ P , and is thus theblow-up of C . This achieves the proof of (3).To prove (1), one can compute the cone of effective curves, prove that it ispolyhedral like in Lemma 6.1.3 and check that − K ˆ F is ample. Equivalently, ˆ F isa smooth Fano threefold appearing in the Mori-Mukai classification (see [MM82,n ◦
20 of Table 3]). The proof of (2) can be done as follows: we first observe that (3)implies that D + K ˆ F is big and nef, and (1) implies that − K ˆ F is ample, so D isample. (cid:3) The following result, and its proof, are very similar to Lemma 6.1.3. The maindifference is the morphism induced by | D + K F | : it is birational with image P inthe case of ( P ) and it is not birational in the case of F ⊂ P × P , as it is therestriction of P × P P × P where each factor is a projection from a point. Lemma 6.2.2.
Let p = ( p , p ) be a point of P × P , contained in a smoothhypersurface F of bidegree (1 , . Let ℓ , ℓ ⊂ F be the two curves of bidegree (1 , , (0 , that pass through p . Let η : F → F be the blow-up of F at p andlet η : F → F be the blow-up at the strict transforms ˜ ℓ , ˜ ℓ of ℓ , ℓ . Denotingby E i ⊂ F i the exceptional divisor of η i and writing again E ⊂ F for the stricttransform of E ⊂ F , the following hold: (1) The divisor D = − K F − E − E is big and nef. (2) The linear system | D + K F | gives a morphism τ : F → P × P ⊂ P , withgeneral fibres isomorphic to P , corresponding to projections P P away from p i on the two factors of P × P . (3) The divisor A = − ( η ) ∗ K F − E is an ample divisor of F and the unionof the curves of F having intersection with A is E ∪ ˜ ℓ ∪ ˜ ℓ ∪ ˜ ℓ .Proof. Changing coordinates, we may assume that F = n ([ x : x : x ] , [ y : y : y ]) ∈ P × P (cid:12)(cid:12)(cid:12)P i =0 x i y i = 0 o . (Lemma 5.3.2). We may then apply an element of PGL ( C ) as in Lemma 5.3.2(2)and assume that p = [1 : 0 : 0] and p = [0 : 1 : 0] . This gives ℓ = { ([ u : 0 : v ] , [0 : 1 : 0]) , [ u : v ] ∈ P } ,ℓ = { ([1 : 0 : 0] , [0 : u : v ]) , [ a : b ] ∈ P } . We denote by H , H ⊂ F the divisors given by x = 0 , y = 0 respectively. Weobserve that the involution σ ∈ Aut( F ) given by ([ x : x : x ] , [ y : y : y ]) ([ y : y : y ] , [ x : x : x ]) exchanges ℓ and ℓ , exchanges H and H and fixes p .The divisors H and H generate the cone of effective divisors of F and satisfy H ∩ H = ℓ ∪ ℓ . For { i, j } = { , } , the surface H i is isomorphic to F , theprojection on the i -th factor gives a P -bundle on a line of P , the curve ℓ j ⊂ H being a fibre and the projection on the j -th factor gives the contraction F → P ofthe ( − -curve ℓ i ⊂ H i . We also have H i · ℓ j = 0 and H i · ℓ i = 1 when { i, j } = { , } .The cone of curves of F is then generated by ℓ , ℓ , and one has − K F = 2 H + 2 H ,so − K F · ℓ i = 2 for each i ∈ { , } .We denote by e ⊂ E ⊂ F a line in E ≃ P , by ˜ ℓ i and H i the strict transformsin F of ℓ i and H i , giving ( η ) ∗ ( H i ) = H i + E , for i = 1 , . For { i, j } = { , } , one finds the following intersection numbers: e ˜ ℓ i ˜ ℓ j E − H i − This implies that the cone of curves on F is polyhedral, generated by ˜ ℓ , ˜ ℓ , e .Indeed, each irreducible curve C of F is either contained in E ≃ P and thusequivalent to a positive multiple of e , or is the strict transform of a curve of F , soequal to P a i ˜ ℓ i + be with b ∈ Z and a , a ≥ . There is moreover i ∈ { , } suchthat a i ≥ . If C is not equal to ˜ ℓ i , then it is not contained in H i , for some rightchoice of H i in its pencil. We obtain ≤ H i · C = b − a i , which implies that b ≥ .Moreover, we obtain K F = ( η ) ∗ ( K F ) + 2 E = − H − H − E .H H • pℓ ℓ H H E e e e ˜ ℓ ˜ ℓ Figure 2.
The divisors and curves on F and F respectively.We now use this to prove (3). Firstly, the divisor A = − ( η ) ∗ K F − E =2 H + 2 H + 3 E is ample as A · ˜ ℓ = A · ˜ ℓ = A · e = 1 . As every line in E ≃ P is equivalent to e , its intersection with A is . The union of curves havingintersection with A thus contains E ∪ ˜ ℓ ∪ ˜ ℓ ∪ ˜ ℓ . Conversely, an irreduciblecurve C ⊂ F not contained in E is numerically equivalent to P a i ˜ ℓ i + be with a , a , b ≥ and a + a ≥ . Moreover, if it is not equal to ˜ ℓ or ˜ ℓ , then b ≥ , aswe observed before, so C · A = a + a + b ≥ . This achieves the proof of (3).The morphism η : F → F being the blow-up of ˜ ℓ , ˜ ℓ , we denote by E i theirreducible component of E lying over ˜ ℓ i , for i = 1 , . We then denote again by E , H , H the strict transforms of the same surfaces on F . The morphism E i → ˜ ℓ i is a P -bundle, and H ∩ E i , H ∩ E i are two sections. We now prove that E i isisomorphic to P × P and that H j ∩ E i has self-intersection or , if j = i or j = i respectively. To see this, we can look at the blow-up of ℓ i ⊂ F . Using thesymmetry, we assume i = 1 ; this allows to work on the open subset U ⊂ F where y = 1 , isomorphic to P × A , via ([ u : v ] , ( a, b )) ([ u : − au − bv : v ] , [ a : 1 : b ]) .The curve ℓ is given in this chart by a = b = 0 , so the exceptional divisor isisomorphic to P × P . The surface H and H are given by au + bv = 0 and a = 0 and thus their strict transform intersect the exceptional divisor along sections ofself-intersection and respectively.For { i, j } = { , } , we write s i = E i ∩ H j , which is a section of E i → ˜ ℓ i ofself-intersection and f i ⊂ E i a fibre; in particular f i · E i = − . As δ i = H i ∩ E i has bidegree (1 , in E i ≃ P × P , we find H i · f i = H i · s i = 1 . Moreover, H j · δ i = H j · f i = 1 , which implies that H j · s i = 0 , since δ i ≡ f i + s i . ONNECTED ALGEBRAIC GROUPS ACTING ON FANO FIBRATIONS OVER P H H E ˜ e ˜ e e f f s s s s f f E E Figure 3.
The divisors and curves on F respectively.We now denote by e ⊂ F the strict transform of the unique line of E ≃ P thatintersects ˜ ℓ and ˜ ℓ . It then satisfies e · E = e · E = 1 and since ( η ) ∗ ( e ) = e .To compute s i ·E i and e · H i , we use η ∗ ( H i ) = H i + E + E . We have s i · η ∗ ( H i ) =( η ) ∗ ( s i ) · H i = ˜ ℓ i · H i = 0 , which gives s i · E i = − since s i · E j = 0 and s i · H i = 1 . Similarly, e · η ∗ ( H i ) = ( η ) ∗ ( e ) · H i = e · H i = 1 , which gives e · H i = − . Wealso compute e · E = e · ( η ) ∗ ( E ) = e · E = − .For { i, j } = { , } , one finds the following intersection numbers: s i s j f i f j e H i − E − E i − − We now prove that the cone of effective curves of F is polyhedral and generated by s , s , f , f , e , by proving that every irreducible curve C is a non-negative linearcombination of these. If C is contained in one of the surfaces E , H , H , E or E this is true as our curves include all extremal rays of these smooth toric surfaces.We may thus assume that C is not contained in E , E , E or H or H . The curve C is the strict transform of η ( C ) ≡ a ˜ ℓ + a ˜ ℓ + be for some a , a , b ≥ . Thisgives C ≡ a s + a s + be + c f + c f for some c , c ∈ Z . For { i, j } = { , } we have ≤ ( E i + H i ) · C = c j .We have E = E + E and D = − K F − E − E . Writing η = η ◦ η , we get η ∗ ( K F ) = − H − H − E − E and obtain K F = η ∗ ( K F ) + 2 E + E = − H − H − E − E ,D = − η ∗ ( K F ) − E − E = 3 H + 3 H + 2 E + 4 E .D + K F = − η ∗ ( K F ) − E − E = H + H + E This implies that D · e = 0 , D · s i = 1 and D · f i = 2 for all i ∈ { , } .The divisor D is thus nef. As − K F is big (it is very ample), the divisor − η ∗ ( K F ) = H + H + 2 E + 2 E is big. Since D + η ∗ ( K F ) = 2 H + 2 H + 2 E is effective, this implies that D is big. This achieves the proof of (1).To prove (2), we consider the linear system | D + K F | on F . Its elements arethe strict transforms of hyperplane sections of F ⊂ P × P of bidegree (1 , havingmultiplicity at p and passing through the two curves ℓ , ℓ . This last conditionis in fact implied by the first. The linear system then induces the rational map τ : P × P P given by τ : ([ x : x : x ] , [ y : y : y ]) [ x y : x y : x y : x y ] , whose image is contained in the smooth quadric Q = { [ x : · · · : x ] ∈ P | x x = x x } ≃ P × P . The rational map to P × P is then given by τ ′ : ([ x : x : x ] , [ y : y : y ]) ([ x : x ] , [ y : y ]) and thus corresponds to the projections P P away from p i on the two factors i = 1 , of P × P .It remains to see that τ = τ ′ ◦ η : F → P × P is a morphism, i.e. that it isdefined at every point q ∈ F . As τ is defined outside of ℓ ∪ ℓ , we may assume that η ( q ) ∈ ℓ ∪ ℓ . After composing with the automorphism σ , we may assume that η ( q ) ∈ ℓ . If η ( q ) = p , then η ( q ) belongs to the image of the open embedding A ֒ → F, ( r, s, t ) ([1 : r : s ] , [ − s − rt : t : 1]) . The morphism η corresponds on this chartto the blow-up of r = s = 0 , that is { (( r, s, t ) , [ u : v ]) ∈ A × P | rv = su } . Hence, τ is locally given by (( r, s, t ) , [ u : v ]) ([ u : v ] , [ t : 1]) and is then well-defined at everypoint. The remaining case is where η ( q ) = p , so η ( q ) belongs to the surface E ⊂ F isomorphic to P . We study τ = τ ′ ◦ η : F P in a neighbourhood of E . Forthis, we take the open embedding A ֒ → F, ( r, s, t ) ([1 : r : s ] , [ − r − st : 1 : t ]) ,and obtain that η is the blow-up of the origin of A in this chart, correspondingto { (( r, s, t ) , [ u : v : w ]) ∈ A × P | su = rv, sw = tv, rw = tu } . The rational map η is locally given by (( r, s, t ) , [ u : v : w ]) ([ u : v ] , [ − u − tv : w ]) . The divisor E corresponds to (0 , , × P , so η is defined at every point of E except thetwo points [0 : 0 : 1] and [0 : 1 : 0] . These are exactly the points where ( η ) − isnot an isomorphism. Using the symmetry, we may assume that η ( q ) = E ∩ ˜ ℓ ,corresponding to ((0 , , , [0 : 0 : 1]) ∈ A × P . We choose the open embedding A ֒ → A × P , ( a, b, c ) (( ac, bc, c ) , [ a : b : 1]) , and see η as the blow-up of theline a = b = 0 . This latter is given by { (( a, b, c ) , [ α : β ]) ∈ A × P | aβ = bα } and τ is given by (( a, b, c ) , [ α : β ]) ([ α : β ] , [ − a − bc : 1]) and is thus defined at everypoint. This achieves the proof of (2). (cid:3) Mori fibre spaces with general fibres isomorphic to ( P ) or P ( T P ) . In this section we prove Theorem D. Propositions 7.2.1 and 7.3.1 deal with Morifibre spaces with general fibres isomorphic to P × P × P and P ( T P ) respectively.In Lemma 7.1.1 we prove that the horizontal subvarieties of such Mori fibre spacesmark balanced subvarieties on the general fibres.7.1. Balanced curves and divisors on a general fibre.Lemma 7.1.1.
We write F = P × P × P and denote by F ⊂ P × P a smoothdivisor of bidegree (1 , . Let i ∈ { , } , let π : X → B be a Mori fibre space with thegeneral fibre isomorphic to F i and let Y ( X be a proper irreducible closed subsetwith π ( Y ) = B . Then, the intersection of Y with a general fibre of π correspondsin F i to one of the following: (1) A divisor linearly equivalent to aK F i for some a ∈ Q . (2) A curve C ⊂ P × P × P of tridegree ( a, a, a ) ( i = 1) or a curve C ⊂ P × P of bidegree ( a, a ) ( i = 2) for some integer a ≥ . Moreover, the degree of π i : C → π i ( C ) is the same for all i , where π i is the projection onto each factor of P × P × P ( resp. of P × P ) . (3) Finitely many points.
ONNECTED ALGEBRAIC GROUPS ACTING ON FANO FIBRATIONS OVER P Proof.
Assume that the intersection of Y with the general fibre of π has dimensionat least . Even if case (1) directly follows from the definition of Mori fibre space,we will address case (1) and (2) at the same time, as the proof is the same.As general fibres are isomorphic to the smooth Fano threefold F i , the genericfibre of π is a smooth Fano threefold F defined over the field K = C ( B ) . We denoteby L = K an algebraic closure of K , and by F L the geometric generic fibre. Wenow prove that F L is isomorphic to ( F i ) L . By [Via13, Lemma 2.1], there is a fieldisomorphism C ≃ −→ L which induces an isomorphism ψ : F i ≃ −→ F L . In particular,the Picard rank of F L is − i ∈ { , } and K F L = K F i = − . If i = 1 , this impliesthat F L is either isomorphic to ( P ) or P × F [MM82, Table 3], the secondcase being impossible because of the existence of the isomorphism ψ . If i = 2 , thisimplies that F L is isomorphic to a hypersurface of bidegree (1 , in P × P [MM82,Table 2].Let G = Gal( L/K ) be the Galois group. Then, we have an isomorphism NS( F ) ≃ NS( F L ) G [Kol96, Chapter II, Proposition 4.3]. Since π is a Mori fibration, thePicard rank of F is equal to , so rk NS( F ) = rk NS( F L ) G = 1 . As F L is isomorphicto ( P L ) or to a hypersurface of ( P L ) of bidegree (1 , , the Galois group has topermute the factors in a transitive way. This implies that the cone of curves of F L is of rank , and that every curve on F L corresponds in ( P L ) or ( P L ) to acurve of multidegree ( a, a, a ) or ( a, a ) for some integer a ≥ ; and the same holdsfor hypersurfaces.We now consider the proper irreducible closed subset Y ( X . If the intersectionof Y with a general fibre F is not finite, it is either a curve or a divisor. The genericfibre of Y → B is a curve or a divisor Y ⊂ F , corresponding in F L to a balancedcurve (of multidegree ( a, a, a ) or ( a, a ) for some integer a ≥ , with projections oneach factor of the same degree, as stated in (2)) or a balanced divisor (equivalentto a multiple of the canonical divisor). Restricting to a general fibre, we obtain thecases (1) and (2). (cid:3) Mori fibre spaces with general fibres isomorphic to ( P ) .Proposition 7.2.1. Let π : X → P be a Mori fibre space whose general fibres areisomorphic to ( P ) . Then, there is an Aut ◦ ( X ) -equivariant commutative diagram X Y P π ϕ π Y where ϕ is birational and π Y : Y → P is a Mori fibre space whose general fibresare isomorphic to P .Proof. By Theorem A there is a section s ⊂ X of π such that the following holds:the set S = Aut ◦ ( X ) · s = Aut ◦ ( X ) P · s = (Aut ◦ ( X ) P ) ◦ · s is a proper closed subset of X such that for each b ∈ P , the fibre S b = π − ( b ) ∩ S of π | S : S → P is equal to π − ( b ) ∩ S = (Aut ◦ ( X ) P ) ◦ · p, where p ∈ s is the point such that π ( p ) = b . We first observe that dim( S ) = 3 leads to a contradiction. For a general b ∈ P ,the fibre S b is then a surface in π − ( b ) ≃ P × P × P which has tridegree ( a, a, a ) forsome integer a ≥ (Lemma 7.1.1). As S b is the orbit of a point by (Aut ◦ ( X ) P ) ◦ ,which acts on P × P × P via a subgroup of PGL ( C ) × PGL ( C ) × PGL ( C ) , itis rational and therefore its canonical divisor is not pseudoeffective, whence a = 1 .Hence the morphisms induced by projections S t → P × P are birational, butnot isomorphisms (this can be shown for instance by computing K S t ). The actionof (Aut ◦ ( X ) P ) ◦ on P × P × P yields an action on P × P , by Blanchard’sLemma 2.1.2, which cannot be transitive, contradicting the fact that S t is an orbit.We now study the case where dim( S ) = 2 . For a general b ∈ P , the fibre S b isthen a curve in π − ( b ) ≃ P × P × P which has tridegree ( a, a, a ) for some integer a ≥ (Lemma 7.1.1). We then observe that a = 1 . Indeed, if a ≥ , the projectiononto each factor would be a finite ramified cover, so the action given by Blanchard’sLemma cannot be transitive, contradicting the fact that S b is an orbit.We now denote by ˆ X → X the blow-up of S , with exceptional divisor E . Let U ⊂ P be the open set over which b X → P is smooth. We consider the divisor − K ˆ X − E . By Lemma 6.1.1 we have − K ˆ X − E = K ˆ X + D where D is relativelybig and nef over U . By Theorem 2.1.4, the divisor − K ˆ X − E induces a morphismover U which, again by Lemma 6.1.1, on each fibre contracts the strict transformsof the three divisors of tridegree (1 , , , (1 , , , (0 , , through S b . This givesan Aut ◦ ( X ) -equivariant birational morphism ˜ X U → Y U , where Y U → U has fibresisomorphic to P (again by Lemma 6.1.1). By Lemma 2.2.1 we get a Mori fibrespace Y → P that is Aut ◦ ( X ) -birational to X over P and whose general fibresare isomorphic to P .It remains to study the case where dim( S ) = 1 , which implies that S = s is asection, pointwise fixed by (Aut ◦ ( X ) P ) ◦ and invariant by Aut ◦ ( X ) .We now denote by η : X → X the blow-up of S with exceptional divisor E .Let H be an ample divisor of the form − η ∗ K X − E + η ∗ π ∗ α where α is sufficientlyample on P . We consider the projective variety Chow , ( X ) which parametrisesthe proper algebraic cycles of dimension and degree with respect to H . If ˆ F is ageneral fibre of π ◦ η : X → P , by Lemma 6.1.3(4) the only -cycles of degree withrespect to H contained in ˆ F are the strict transforms of the three curves through F ∩ S of tridegree (1 , , , (0 , , , (0 , , and the lines in E . Moreover, if α issufficiently ample, the only 1-cycles of degree with respect to H are contained infibres of π ◦ η . We set U = { ( x, [ t ]) ∈ X × Chow , ( X ) | x ∈ t } .Therefore the image of the first projection U → X is a subvariety of X of theform Z ∪ Z ′ ∪ E where Z is horizontal, and Z ′ is vertical.The subvariety Z is such that its intersection with the general fibre of π ◦ η isthe union of the strict transforms of the three curves through F ∩ S of tridegree (1 , , , (0 , , , (0 , , . We set η : X → X the blow-up of X along Z . We set E the exceptional divisor and write again E the strict transform of the exceptionaldivisor E of η . Let U ⊂ P be the open set over which X → P is smooth. Thedivisor − K X − E − E is relatively semiample over U by Theorem 2.1.4 as itis the sum of K X and of − K X − E − E which by Lemma 6.1.3 is relativelybig and nef over U . It induces then a morphism ( X ) U → Y U → U where thegeneral fibre of Y U → U is P . By Lemma 2.2.1 we get a Mori fibre space Y → P that is Aut ◦ ( X ) -birational to X over P and whose general fibre is P (using againLemma 6.1.3). (cid:3) ONNECTED ALGEBRAIC GROUPS ACTING ON FANO FIBRATIONS OVER P Mori fibre spaces with general fibres isomorphic to P ( T P ) .Proposition 7.3.1. Let π : X → P be a Mori fibre space whose general fibres areisomorphic to a smooth hypersurface of P × P of bidegree (1 , . Then, there isan Aut ◦ ( X ) -equivariant commutative diagram X YB P π ϕ π Y π B where ϕ is birational and π Y : Y → B is a Mori fibre space whose general fibres areeither isomorphic to P or a smooth quadric Q ⊂ P ( in this latter case, B → P is an isomorphism ) .Proof. Let F = { ([ x : x : x ] , [ y : y : y ]) ∈ P × P | P x i y i = 0 } . The generalfibres of π are isomorphic to F (Lemma 5.3.2).We apply Theorem A and obtain a section s ⊂ X of π such that the followingholds: the set S = Aut ◦ ( X ) · s = Aut ◦ ( X ) P · s = (Aut ◦ ( X ) P ) ◦ · s is a proper closed subset of X , and such that for each b ∈ P , the fibre S b = π − ( b ) ∩ S of π | S : S → P is equal to π − ( b ) ∩ S = (Aut ◦ ( X ) P ) ◦ · p, where p ∈ s is the point such that π ( p ) = b .We first observe that dim( S ) = 3 leads to a contradiction. For a general b ∈ P ,the fibre S b is then a surface in π − ( b ) ≃ F which is the intersection of F with ahypersurface of P × P of bidegree ( a, a ) for some integer a ≥ (Lemma 7.1.1).As S b is the orbit of a point by (Aut ◦ ( X ) P ) ◦ , its canonical divisor is not pseu-doeffective, so a = 1 . Hence the morphism induced by any projection S t → P isbirational, but not an isomorphism (this can be shown for instance by computing K S t ). The action of (Aut ◦ ( X ) P ) ◦ on P yields an action on P , by Blanchard’sLemma 2.1.2, which cannot be transitive, contradicting the fact that S t is an orbit.We now study the case where dim( S ) = 2 . For a general b ∈ P , the fibre S b is then a curve C b in π − ( b ) ≃ F ⊂ P × P which has bidegree ( a, a ) for someinteger a ≥ (Lemma 7.1.1). As C b is an orbit, it is smooth, and it is rationalsince (Aut ◦ ( X ) P ) ◦ acts on P × P via a subgroup of PGL ( C ) × PGL ( C ) , byBlanchard’s Lemma 2.1.2. The degree of π i : C b → π i ( C b ) is the same for i = 1 , ,where π i : P × P is the projection on each factor (Lemma 7.1.1). So π i ( C b ) isa curve for each i . Moreover, C b → π i ( C b ) is an isomorphism, as otherwise theramification points would be fixed (using again Blanchard’s Lemma, we have anaction on each P ). There are thus two possibilities: either a = 1 , or a = 2 and theprojections to each factor are embeddings.We now denote by ˆ X → X the blow-up of S , with exceptional divisor E . Let U ⊂ P be the open set over which b X → P is smooth. We consider the divisor − K ˆ X − E . By Lemma 6.2.1 if a = 1 or Lemma 5.3.3 if a = 2 we have − K ˆ X − E = K ˆ X + D where D is relatively big and nef (in fact relatively ample) over U . ByTheorem 2.1.4, the divisor − K ˆ X − E induces a morphism over U . If a = 1 , the morphism is birational, and contracts on each fibre the strict trans-forms of the two divisors of bidegree (0 , , (1 , through S b (Lemma 6.2.1). Thisgives an Aut ◦ ( X ) -equivariant birational morphism ˜ X U → Y U , where Y U → U is a morphism with fibres isomorphic to a smooth quadric Q ⊂ P (again byLemma 6.2.1). By Lemma 2.2.1, we get a Mori fibre space Y → P that is Aut ◦ ( X ) -birational to X over P and whose general fibres are isomorphic to Q . This con-cludes the proof in this case, and the variety B in the statement is isomorphic to P .If a = 2 , the morphism induced by − K ˆ X − E is not birational. On each fibre,it gives a morphism to P with general fibres isomorphic to P (Lemma 5.3.3). Wethen apply Lemmas 2.2.2 and 2.2.3 to get an Aut ◦ ( X ) -equivariant birational map ϕ : X Y , where Y → B is a Mori fibre space with general fibres isomorphic to P .We moreover obtain a morphism B → P which makes the diagram commutative,as in the statement.It remains to study the case where dim( S ) = 1 , which implies that S = s isa section, pointwise fixed by (Aut ◦ ( X ) P ) ◦ and invariant by Aut ◦ ( X ) . The prooffollows the same lines as the proof of Proposition 7.2.1.We denote by η : X → X the blow-up of S with exceptional divisor E . Let H be an ample divisor of the form − η ∗ K X − E + η ∗ π ∗ α where α is sufficientlyample on P . We consider the projective variety Chow , ( X ) which parametrisesthe proper algebraic cycles of dimension and degree with respect to H . If ˆ F is ageneral fibre of π ◦ η : X → P , by Lemma 6.2.2(3) the only -cycles of degree withrespect to H contained in ˆ F are the strict transforms of the two curves through F ∩ S of bidegree (1 , , (0 , and the lines in E . Moreover, if α is sufficientlyample, the only -cycles of degree with respect to H are contained in fibres of π ◦ η . We set U = { ( x, [ t ]) ∈ X × Chow , ( X ) | x ∈ t } .Therefore the image of the first projection U → X is a subvariety of X of theform Z ∪ Z ′ ∪ E where Z is horizontal and Z ′ is vertical.The subvariety Z is such that its intersection with the general fibre of π ◦ η is the union of the strict transforms of the two curves through F ∩ S of bidegree (1 , , (0 , . We set η : X → X the blow-up of X along Z . We set E theexceptional divisor and denote again by E the strict transform of the exceptionaldivisor of η . Let U ⊂ P be the open set over which X → P is smooth. Thedivisor − K X − E − E is relatively semiample over U by Theorem 2.1.4 as it isthe sum of K X and of − K X − E − E which by Lemma 6.2.2 is relatively bigand nef over U . Therefore, the divisor − K X − E − E induces a morphism,which, on each fibre of ( X ) U → U , is a morphism to P × P with general fibreisomorphic to P (again by Lemma 6.2.2). We then apply Lemmas 2.2.2 and 2.2.3to get an Aut ◦ ( X ) -equivariant birational map ϕ : X Y , where Y → B is a Morifibre space with general fibres isomorphic to P . We moreover obtain a morphism B → P which makes the diagram commutative, as in the statement. (cid:3) We can now achieve this text by proving Theorem D.
Proof of Theorem D.
Let π : X → P be a Q -factorial terminal Mori fibre spacesuch that a general fibre F is a smooth threefold of Picard rank ≥ , and such that Aut ◦ ( X ) is not trivial. The general fibres of π belong to one of the families listedin Table . (follows from [CFST16, Theorem 1.4]). ONNECTED ALGEBRAIC GROUPS ACTING ON FANO FIBRATIONS OVER P Suppose first that
Aut ◦ ( X ) P = { g ∈ Aut ◦ ( X ) | πg = π } is finite. In this case, Proposition B implies that Aut ◦ ( X ) is a torus of dimension and provides an Aut ◦ ( X ) -equivariant birational map X P × Z where Z is aterminal threefold. We are then in Case (1) of Theorem D, with Y = P × Z and B = Z .We may now assume that Aut ◦ ( X ) P is of positive dimension. Let us write k = max { dim((Aut ◦ ( X ) P ) ◦ · x ) | x ∈ X } > for the maximal dimension of anorbit of (Aut ◦ ( X ) P ) ◦ (equivalently of Aut ◦ ( X ) P ) on X .If k = 1 , then Proposition C gives an Aut ◦ ( X ) -equivariant birational map X Y , where Y → B is a Mori fibre space, with dim B = 3 . In this case, the generalfibres are isomorphic to P , so we are in Case (1) of Theorem D.We now assume that k ≥ . By Lemma 2.1.3, a general fibre F of π : X → P then satisfies dim Aut ◦ ( F ) ≥ . By Proposition 5.2.2, the general fibres belong tothe families , , or of Table . . We do a case-by-case analysis.If the general fibres belong to the family , they are all such that Aut ◦ ( F ) ≃ PGL ( C ) and are isomorphic to the blow-up of the quadric Q ⊂ P of equation x x − x x + 3 x = 0 along the image of the Veronese embedding of degree of P (Lemma 5.3.1). We are then in Case ( i ) of Theorem D.If the general fibres belong to the family , they are all such that Aut ◦ ( F ) ≃ PGL ( C ) and are isomorphic to ( ( x, y, z ) ∈ ( P ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) X i =0 x i y i = X i =0 x i z i = X i =0 y i z i = 0 ) . (Lemma 5.3.4). We are then in Case ( ii ) of Theorem D.If the general fibres belong to the families and , Proposition 7.3.1 and Proposi-tion 7.2.1 provide an Aut ◦ ( X ) -equivariant birational map X Y where Y → B isa Mori fibre space whose general fibres are either isomorphic to P , P or a smoothquadric Q ⊂ P . We are in Case (1) of Theorem D. (cid:3) References [Bea77] Arnaud Beauville. Variétés de Prym et jacobiennes intermédiaires.
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