Rigidity of projective symmetric manifolds of Picard number 1 associated to composition algebras

To each complex composition algebra $\mathbb{A}$, there associates a projective symmetric manifold $X(\mathbb{A})$ of Picard number one, which is just a smooth hyperplane section of the following varieties ${\rm Lag}(3,6), {\rm Gr}(3,6), \mathbb{S}_6, E_7/P_7.$ In this paper, it is proven that these varieties are rigid, namely for any smooth family of projective manifolds over a connected base, if one fiber is isomorphic to $X(\mathbb{A})$, then every fiber is isomorphic to $X(\mathbb{A})$.


Introduction
Throughout this paper, we work over the complex number field. A smooth projective variety X is said to be rigid if for any smooth projective family over a connected base with one fiber isomorphic to X, all fibers are isomorphic to X. It is a difficult and subtle problem to prove the rigidity. Even for rational homogeneous varieties G/P of Picard number 1, the rigidity does not always hold. To wit, let B 3 /P 2 be the variety of lines on a 5-dimensional smooth hyperquadric Q 5 . An explicit family specializing B 3 /P 2 to a smooth projective G 2 -variety is constructed by Pasquier and Perrin in [PP10]. In [HL23], it is shown that this is the only smooth non-isomorphic specialization of B 3 /P 2 . It turns out that B 3 /P 2 is the only exception among all G/P of Picard number 1, as shown by the following. The key ingredient for the proof is the VMRT theory developed by Hwang and Mok. In the simplest case of a projective manifold X covered by lines (which is the case for our paper), the VMRT C x ⊂ PT x X at a general point x is just the Hilbert scheme of lines through x. This projective subvariety C x ⊂ PT x X encodes a lot of global geometry of X, and in some cases, we can even recognize X from its VMRT at general points.
As G/P is locally rigid, we only need to prove that G/P is rigid under specialization; namely, for a smooth projective family X → ∆ such that X t ≃ G/P for all t 0, we have X 0 ≃ G/P . The proof essentially consists of two steps: the first is to show the VMRT of X 0 is isomorphic to that of G/P , and the second is to use the recognization of G/P from its VMRT.
In many cases, the invariance of the VMRT can be proved, while the recognization problem is in general much more difficult. In [Par16], it is observed that for odd Lagrangian Grassmannians (which are not homogeneous), one can directly show that H 1 (X 0 , T X 0 ) = 0 by using VMRT theory. Hence X 0 is locally rigid and isomorphic to nearby fibers, which proves the rigidity for odd Lagrangian Grassmannians.
The goal of this paper is to prove the rigidity for projective symmetric varieties associated to composition algebras. Recall that there are exactly four complex composition algebras: A = C, C ⊕ C, H C , O C . To such an A, we can associate algebraic groups SL 3 (A) and SO 3 (A) with an involution θ such that SL 3 (A) θ = SO 3 (A). The quotient SL 3 (A)/ SO 3 (A) is a symmetric homogeneous space, which admits a unique smooth equivariant completion of Picard number 1, denoted by X(A). It turns out that X(A) is a smooth hyperplane section of one of the following varieties (cf. [Ruz10]): Lag(3, 6), Gr(3, 6), S 6 , E 7 /P 7 , where Lag(3, 6) is the Lagrangian Grassmannian associated to C 6 and S 6 is the 15-dimensional spinor variety. The main result of this paper is the following.

Theorem 1.2. For any complex composition algebra A, the variety X(A) is rigid.
We first remark that for A = C, X(A) is a Mukai variety, so its smooth deformation is again a Mukai variety, hence again a hyperplane section of Lag(3, 6) by the classification of Mukai varieties. This shows that X(A) is rigid in this case. We will assume A C in the following.
The rigidity problem of X(A) was studied by Kim and Park in [KP19]. When A = H C or O C , they prove the invariance of the VMRT and, moreover, observe that dim H 1 (X 0 , T X 0 ) ≤ 1. If H 1 (X 0 , T X 0 ) = 0, then X 0 is locally rigid, and thus it is isomorphic to nearby fibers X(A). When dim H 1 (X 0 , T X 0 ) = 1, X 0 is an equivariant compactification of the vector group G n a with n = dim X(A). With the help of [Wiś91], this result can be easily extended to the case A = C ⊕ C.
To prove Theorem 1.2, we will exclude the case of equivariant compactifications. Let X → ∆ be a specialization of X(A); i.e. X t ≃ X(A) for all t 0 and X 0 is an equivariant compactification of G n a . The Lie algebra of the automorphism group of the central fiber is given by aut(X 0 ) ≃ C n ⋊ (so 3 (A) ⊕ C). We will consider a family of tori H t ⊂ Aut 0 (X t ) induced from a maximal torus of SO 3 (A) and then take a connected component Y of the torus-fixed locus X H . As the rank of sl 3 (A) is 2 more than that of so 3 (A), there is an extra 2-dimensional torus acting on Y t for t 0. It turns out that Y → ∆ is a family of smooth projective surfaces with general fiber Y t isomorphic to the blowup of P 2 along three coordinate points. The central fiber Y 0 is an equivariant compactification of G 2 a . By delicate computations, we will show that Y 0 is isomorphic to the blowup of P 2 along three colinear points. On the other hand, the involution θ on SL 3 (A) induces an involution Θ on the family Y /∆, which preserves the boundaries of Y t for all t. It turns out that the involution Θ 0 : Y 0 → Y 0 sends extremal rays of the Mori cone NE(Y 0 ) to non-extremal rays, which gives a contradiction.

Acknowledgements
We are very grateful to the referee for their careful reading and helpful suggestions.

Composition algebras and associated Lie groups
Let A R be one of the four real normed division algebras, R, C, H, O, which admits an involution x →x, called conjugation. It is well known that the fixed points under this conjugation are exactly the base field R. Note that H, O are non-commutative and ab =bā for all a, b ∈ A R .
Let A = A R ⊗ R C be the complexification of A R , which is one of the following: C, C ⊕ C, H C , O C . The algebra structure on A is given by (a ⊗ c, a ′ ⊗ c ′ ) → aa ′ ⊗ cc ′ for multiplication and a ⊗ c =ā ⊗c for conjugation. Note that the conjugation fixes exactly elements in C. It turns out that A is a composition algebra and any finite-dimensional composition algebra over C is isomorphic to one of these A (see for example [VGO90, Chapter 5, Section 1]).
We consider the following vector space of A-Hermitian matrices of order 3 with coefficients in A: It turns out that J 3 (A) has the structure of a Jordan algebra with multiplication given by A • B = 1 2 (AB + BA), where AB is the usual matrix multiplication. The comatrix of A ∈ J 3 (A) is defined as com(A) = A 2 − tr(A)A + 1 2 ((tr(A)) 2 − tr(A 2 )) Id .
, we have the following explicit formulae: It then follows that Let us have a closer look at the C-valued polynomial det on J 3 (A). For A, B, C ∈ J 3 (A), define Then it follows that The following table gives the corresponding groups: Consider the two matrices Note that M 2 12 = M 2 23 = Id. Define Similarly, we can define σ 23 by using M 23 .

The involution on Lie algebras
There exists an involution θ : SL 3 (A) → SL 3 (A) coming from the symmetry of the Dynkin diagram of SL 3 (A), which satisfies SL 3 (A) θ = SO 3 (A). The quotient SL 3 (A)/ SO 3 (A) is a symmetric homogeneous space. The involution θ induces an involution (still denoted by θ) on sl 3 (A) whose fixed locus is so 3 (A). We have the following description of these Lie algebras: The following result is well known, but we include a proof here for the reader's convenience.  (ii) The map is an injective homomorphism of groups, and the associated homomorphism of Lie algebras is In particular, via ν and µ we can regard T 0 as a 2-dimensional torus of SL 3 (A) with Lie algebra h 0 ⊂ sl 3 (A). (iii) Take any σ ∈ S 3 ⊂ SO 3 (A). Then the inner automorphism and the adjoint representation of SL 3 (A) give rise to Inn σ (T 0 ) = T 0 and Ad σ (h 0 ) = h 0 . More precisely, Proof.
Hence T 0 is an abelian group. It follows that the group structure is the same as that of the 2-dimensional torus.
(ii) Assume A ∈ ker(ν); then B = ABA for any B ∈ J 3 (A), which implies that A 2 = Id and BA = (ABA)A = AB. This implies A = ±Id. Since A ∈ T 0 , we get A = Id. The assertion on Lie algebras follows immediately. ( It follows that Proof. By Lemma 2.4, h 0 is the Lie algebra of a torus in SL 3 (A).

Lemma 2.6. Let h 1 be a Cartan subalgebra of so 3 (A), and let
. Since k and h 1 are Cartan subalgebras of sl 3 (A) and so 3 (A), respectively, dim(k/h 1 ) = rank(sl 3 (A)) − rank(so 3 (A)). Then k/h 1 has the dimension as stated. It remains to prove that h = k. The space J 3 (A) 0 is an irreducible module of SO 3 (A). More precisely, we have The Hence k = h, completing the proof. □ Proposition 2.7.

(i) Given a maximal torus T 1 of SO 3 (A), the identity component T of its centralizer is a maximal torus of
Proof. Claim (i) follows immediately from Lemma 2.6. By Lemma 2.5, there is a maximal torus T ′ of is a maximal torus of SO 3 (A), and T 0 ≃ T ′ /(T ′ ∩ SO 3 (A)). Both T 1 and T ′ 1 are maximal tori of SO 3 (A), so there exists a g ∈ SO 3 (A) such that T ′ 1 = gT 1 g −1 . By Lemma 2.6 (resp. by the choice of T ), the maximal torus T ′ (resp. T g := gT g −1 ) is the identity component of the centralizer of T ′ 1 (resp. gT 1 g −1 ). As T ′ 1 = gT 1 g −1 , we have T ′ = T g , which concludes the proof. □

The symmetric variety X(A)
We consider the following rational map: We denote by G ω (A 3 , A 6 ) the closure of the image of Φ, which turns out to be a rational homogeneous space corresponding to the third row in Freudenthal's magic square of varieties (cf. [LM01]).
By [LM01,Proposition 4.1], the action of SL 3 (A) on P(J 3 (A)) has a unique closed orbit, denoted by AP 2 , which is just one of the four Severi varieties. Note that AP 2 is also the variety of lines on G ω (A 3 , A 6 ) through a fixed point.
The following table collects information about all these varieties: Lag(3, 6) Gr(3, 6) S 6 E 7 /P 7 AP 2 ν 2 (P 2 ) P 2 × P 2 Gr(2, 6) E 6 /P 1 Let X(A) be the closure of the image under Φ of the cubic hypersurface t 3 = det(A) in P(C ⊕ J 3 (A)), which is a hyperplane section of G ω (A 3 , A 6 ). We call X(A) the symmetric manifold associated to the composition algebra A.
We can now summarize some properties of X(A) as follows.
, is isomorphic to the blowup of P 2 along its three coordinate points.
By the proof of Lemma 2.9, the composition of T 0 -equivariant maps T 0 → T 0 · o → M 1 is diag(λ 1 , λ 2 , λ 3 ) → [λ 2 1 : λ 2 2 : λ 2 3 ]. The action of S 3 on T 0 extends to M 1 by permuting coordinates. This action of S 3 lifts to Y (A) since the blowup center is S 3 -stable. Furthermore, the lifting coincides with the restriction to Y (A) of the S 3 -action on X(A) because these two actions coincide on the open torus orbit of Y (A). Note that σ sends P i to P σ (i) and sends the line joining P i and P j to the line joining P σ (i) and P σ (j) . The conclusion follows. □ Now we study the involution θ on X(A).
Proof. Since Y (A) is a toric variety, its Picard group is generated by prime boundary divisors D 1 , D 2 , D 3 , E 1 , E 2 and E 3 . The line joining P 1 and P 2 and the line joining P 1 and P 3 are linearly equivalent in M 1 . Pulling back to Y (A), one gets • Pic(Y (A)) σ 12 is a free abelian group of rank 3 with basis {D 1 + E 2 , E 1 + E 2 , E 3 }; • Pic(Y (A)) σ 13 is a free abelian group of rank 3 with basis {D 1 + E 3 , E 1 + E 3 , E 2 }; • Pic(Y (A)) σ 23 is a free abelian group of rank 3 with basis {D 1 , E 1 , E 2 + E 3 }; • Pic(Y (A)) σ 123 = Pic(Y (A)) σ 321 = Pic(Y (A)) S 3 is a free abelian group of rank 2 with basis {D 1 + E 2 + E 3 , E 1 + E 2 + E 3 }.
Proof. As Y (A) is the blowup of P 2 along three points, Pic(Y (A)) is freely generated by E 1 , E 2 , E 3 and the pullback of a line in P 2 , namely D 1 + E 2 + E 3 , which gives the first claim.
Then c = a + b and It follows that Pic(Y (A)) σ 12 is of rank 3 and D 1 + E 2 , E 1 + E 2 , E 3 is a Z-basis.
The proofs for other claims are similar. □

Invariance of varieties of minimal rational tangents
For a uniruled projective manifold X, let RatCurves n (X) denote the normalization of the space of rational curves on X (see [Kol96, Proposition II.2.11]). Every irreducible component K of RatCurves n (X) is a (normal) quasi-projective variety equipped with a quasi-finite morphism to the Chow variety of X; the image consists of the Chow points of irreducible, generically reduced rational curves. There is a universal family U with projections υ : U → K, µ : U → X, and υ is a P 1 -bundle (for these results, see [Kol96, Proposition II.2.11 and Theorem II.2.15]).
For any x ∈ X, let U x := µ −1 (x) and K x := υ(U x ). We call K a family of minimal rational curves if K x is non-empty and projective for a general point x. There is a rational map ι x : K x PT x X (the projective space of lines in the tangent space at x) that sends any curve which is smooth at x to its tangent direction. The closure of the image of ι x is denoted by C x and called the variety of minimal rational tangents (VMRT) at the point x. By [HM04, Theorem 1] and [Keb02, Theorem 3.4], composing ι x with the normalization map K n x → K x yields the normalization of C x . Also, K n x is a union of components of the variety RatCurves n (x, X) defined in [Kol96, II.(2.11.2)] and hence is smooth for x ∈ X general by [Kol96, Corollary II.3.11.5]. In this case, U x ≃ K n x is smooth, and the rational map ι x induces a birational morphism U x ≃ K n x → C x , which is still denoted by ι x by abuse of notation. Since U x is both the normalization of K x and that of C x , we call U x the normalized Chow space or the normalized VMRT.
By Proposition 2.8, the variety X(A) is covered by lines, and its VMRT at a general point is just the variety of lines through that point, denoted by C(A) ⊂ P(V A ) with V A being the tangent space of X(A) at a general point, which is respectively ν 2 (Q 1 ), PT * P 2 , Gr ω (2, 6), F 4 /P 1 with the natural embedding. For a family of smooth projective varieties π : X → ∆ with X t ≃ X(A) for all t 0, we take a general section τ : ∆ → X such that τ(t) is a general point in X t for all t ∈ ∆. By considering the VMRT of X t at τ(t), we get a family of embedded projective subvarieties with general fibers isomorphic to C(A) ⊂ P(V A ).
We first prove the invariance of the VMRT, which means that the central fiber has the same VMRT as the general fiber. Note that the cases when A = H C or O C are proved in [KP19]. For the reader's convenience, we include the proof for all three cases here. Proof. Firstly we show that the normalized Chow space U x 0 of X 0 at a general point x 0 is isomorphic to C(A). Take a general section t ∈ ∆ → x t ∈ X t of π passing through the general point x 0 in X 0 . Shrinking ∆ if necessary, we can assume that x t is general in X t for each t 0. The normalized Chow spaces U x t along this section give a family of smooth projective varieties such that U x t ≃ C(A) for t 0. If A C ⊕ C, then U x 0 ≃ C(A) by Theorem 1.1. Now assume A = C ⊕ C. Consider the normalization map It follows that this equality also holds for t = 0, which implies that U x 0 is a Fano threefold with index 2 and we can apply [Wiś91,Theorem] to deduce that U x 0 ≃ P(T * P 2 ) (note that in [Wiś91], the projectivisation is taken in the sense of Grothendieck).
Recall that for a smooth projective subvariety Z ⊂ PV , the variety of tangential lines of Z is the subvariety T Z ⊂ Gr(2, V ) ⊂ P(∧ 2 V ) consisting of the tangential lines of Z. By [FL20, Lemma 2.12], the tangential variety of P(T * P 2 ) is non-degenerate. By [Hwa01, Proposition 2.6], the tangential varieties of Gr ω (2, 6) and F 4 /P 1 are both non-degenerate. Now we can use the same argument as that of [FL20, Proposition 3.9] to conclude the proof. □ Recall that a vector group of dimension g is the additive group G To prove Theorem 1.2, it suffices to exclude case (ii) in Proposition 3.2. In the following, we will assume case (ii) to deduce a contradiction.

Reduction to a family of surfaces
Let V = π * T X /∆ , which is a vector bundle over ∆ such that V t ≃ aut(X t ) ≃ sl 3 (A) for t 0. Let W ⊂ V be the subbundle such that W t ≃ so 3 (A) for t 0, which is the Lie algebra of the stabilizer of τ(t) ∈ X t . It follows that W 0 ≃ so 3 (A). For each t ∈ ∆, the fiber V t is a completely reducible so 3 (A)-module, which is isomorphic to so 3 (A) ⊕ J 3 (A) 0 . By a dimension check, V 0 ≃ C n ⋊ so 3 (A) ⊂ aut(X 0 ). Our construction and argument here for V and W is an analogue of the proof of [FL20, Lemma 4.11].
We fix a family of Cartan subalgebra H ⊂ W ; i.e. H t is a Cartan subalgebra of W t for all t. Consider H ⊂ V defined byH It follows thatH t is a Cartan subalgebra of V t for all t 0, by Lemma 2.6. Note that rk(H) = rk(H) + 2 (as

A C).
For t ∈ ∆, let H t = exp(H t ) ⊂ Aut 0 (X t ) τ(t) , which is a family of tori. SetH t = exp(H t ) ⊂ Aut 0 (X t ). (i) The map Y → ∆ is a smooth family of projective surfaces. (ii) For each t ∈ ∆, Y t is the closure ofH t · τ(t) in X t . (iii) When t 0, Y t is isomorphic to Y (A), the blowup of P 2 along three coordinate points.
Proof. By Białynicki-Birula's theorem on torus actions [Bia73], the map Y → ∆ is a smooth family of projective varieties. For each t ∈ ∆, the representation of W t ≃ so 3 (A) on T τ(t) X t coincides with that on J 3 (A) 0 , and the subspace T τ(t) Y t is contained in the H t -eigenspace of weight zero, which is of dimension 2.
Then Y t is the closure ofH t · τ(t) in X t , and it is a projective surface. This proves (i) and (ii). By Proposition 2.7, when t 0, the projective surface Y t is isomorphic to the closure of T 0 · o in X(A). Then (iii) follows from Lemma 2.9. By the structure of V 0 ,H 0 is the semi-direct product of the torus H 0 and a vector group G 2 a . Then (iv) follows from (ii). □ It follows that the Y t are quasi-homogeneous for all t ∈ ∆. Denote by ∂Y t the boundary, i.e. the complement of the open orbit. Let ∂Y be the closure of ∪ t 0 ∂Y t , and let (∂Y ) t be the fiber of ∂Y over t ∈ ∆.
Lemma 3.4. We have (∂Y ) t = ∂Y t as sets for each t ∈ ∆.
Proof. When t 0, it is immediate from the construction that (∂Y ) t = ∂Y t . The subvariety (∂Y ) t ⊂ Y t is stable under the vector fields inH t for t 0. By continuity, this is also the case for t = 0. Consequently, (∂Y ) 0 has no intersection with the open orbitH t · τ(0) on Y 0 , implying (∂Y ) 0 ⊂ ∂Y 0 as sets.
Since (∂Y ) t = ∂Y t is the anticanonical divisor on Y t when t 0, −K Y 0 is given by the divisor (∂Y ) 0 (as a scheme-theoretic divisor, so each irreducible component has a multiplicity). As Y 0 is an equivariant compactification of a vector group, the support of its G 2 a -stable anticanonical divisor is the whole boundary by [HT99, Theorem 2.7]. It follows that (∂Y ) 0 = ∂Y 0 as sets. □ In the following, we will construct an involution that acts well on Y /∆. Proof. For each t 0, W t so 3 (A) is contained in the isotropic subalgebra at τ(t) ∈ X t , and V t /W t is an irreducible representation of so 3 (A) isomorphic to the representation T τ(t) X t of the isotropic subalgebra. For each t ∈ ∆, the evaluation of vector fields at τ(t) ∈ X t gives rise to an injective homomorphism of W t -modules V t /W t → T τ(t) X t , whence V t /W t is isomorphic to the irreducible so 3 (A)-module T o X(A) by Proposition 3.1 and by the fact that dim V t /W t = dim X t . Since so 3 (A) is a simple Lie algebra, for each t ∈ ∆, the module V t is isomorphic to so 3 (A)⊕T o X(A). Since the two direct summands so 3 (A) and T o X(A) are irreducible modules that are not isomorphic to each other, this decomposition is unique. Then we obtain a direct sum decomposition of the holomorphic family V /∆ of so 3 (A)-modules V = W ⊕ M. When t 0, is already a decomposition into irreducible so 3 (A)-modules. By the uniqueness of the decomposition, This gives an automorphism ξ of the vector bundle V /∆ of order 2. When t 0, ξ t is nothing but the involution θ ∈ Aut(so 3 (A)).
Proof. (i) Take any t ∈ ∆. Let G t be the connected algebraic subgroup of Aut 0 (X t ) with Lie algebra V t ⊂ aut(X t ), and denote by H t ⊂ G t the isotropic subgroup at τ(t) ∈ X t . Then Then ξ induces a biholomorphic map Θ : , and thus dΘ t preserves the VMRT of X o t . By continuity, dΘ 0 preserves the VMRT of X o 0 . By the extension theorem of Cartan-Fubini type [HM01, Main Theorem], we can extend Θ 0 to a biholomorphic map X 0 → X 0 . When t 0, we have identifications X o t = SL 3 (A)/ SO 3 (A) and Θ t = θ. Then (ii) and (iii) follow.
(iv) Note that H ⊂ W andH ⊂ W are ξ-stable. Then the open orbit of Y t , t ∈ ∆, is Θ t -stable. Hence the closure Y t and the boundary ∂Y t is Θ t -stable. □

The central fiber as a blowup of P 2
Recall that general fibers of the smooth family Y /∆ of rational projective surfaces are of Picard number 4, as is the special fiber Y 0 . As Y 0 is an equivariant compactification of a vector group, its boundary is of pure codimension 1 and spans Pic(Y 0 ) freely. In particular, ∂Y 0 has four irreducible components, say F 0 , F 1 , F 2 and F 3 ; thus Pic By Lemma 3.4, the divisors D i,0 and E j,0 of Y 0 lie in the boundary. We will find out what D i,0 and E j,0 are in the following.
There is an SO 3 (A)-action on the family X /∆ that is isotropic along the section τ(∆), and the associated Lie algebra is W t so 3 (A) for each t ∈ ∆.
Lemma 3.7. For each t ∈ ∆, the subvariety Y t ⊂ X t as well as the boundary ∂Y t are stable under the action of S 3 ⊂ SO 3 (A). Furthermore, σ · D i = D σ (i) and σ · E j = E σ (j) for σ ∈ S 3 and 1 ≤ i, j ≤ 3.
Proof. For any t ∈ ∆, the restriction map Pic(Y /∆) → Pic(Y t ) is an isomorphism which is compatible with the S 3 -action. Then Pic(Y 0 ) Γ ≃ Pic(Y t ) Γ for any subgroup Γ of S 3 and any t 0. In particular, Pic(Y 0 ) σ 12 ≃ Pic(Y t ) σ 12 is of rank 3 by Proposition 2.13. Let S := {F 0 , F 1 , F 2 , F 3 } be the set of irreducible components of ∂Y 0 . By [FL20,Lemma 4.16], the rank of Pic(Y 0 ) σ 12 is given by the number of σ 12 -orbits on S. Hence S consists of three orbits under the σ 12 -action; we may assume they are {F 1 , F 2 }, {F 3 } and {F 0 }.
Applying the action of σ 23 on Y /∆, by a similar argument there exists a subset T ⊂ S such that T consists of two elements permuted by σ 23 and each element of the set S \ T is σ 23 -stable.
Proof. Since D i,t and E i,t are contained in ∂Y t when t 0, the same holds when t = 0. Then there exist non-negative integers d 0 , . Applying the formula to σ = σ 23 , we have D 1,0 = d 0 F 0 + d 1 F 1 + d 2 F 3 + d 3 F 2 , implying d 2 = d 3 . The conclusions for E 1,0 and E σ (1),0 can be obtained similarly. □ Now we compute the anticanonical divisor of the central fiber Y 0 .
By [HT99,Theorem 2.7], the support of −K Y 0 is the whole boundary, which implies d 0 + e 0 ≥ 1 and d 1 + e 2 = d 2 + e 1 ≥ 1. □ To prove that Y 0 is the blowup of P 2 along three colinear points, we start with the following.
Proposition 3.11 (cf. [HT99, Section 5]). Every G 2 a -surface admits a G 2 a -equivariant morphism onto P 2 or a Hirzebruch surface F n . The boundary of P 2 consists of a unique line. The boundary of F n consists of two lines; one is a fiber, and the other is a minimal section.
Let l 1 and l 2 be two lines of F n such that F n \ C 2 = l 1 ∪ l 2 , where l 1 is the section of F n → P 1 and l 2 is a fiber of F n → P 1 . Then the anticanonical divisor of F n is given by −K F n = 2l 2 + (n + 2)l 1 .
(n + 2)l 1 2l 2 F n Since n ≥ 2, the blowup of F n along any point on l 1 ∪ l 2 would yield a surface S and an exceptional divisor E such that −K S = al 1 + bl 2 + cE, where a, b, c are distinct positive integers.
Any further blowup of S will produce a surfaceS whose anticanonical divisor −KS has at least three distinct coefficients, which is different from the form −K Y 0 = aF 0 + b(F 1 + F 2 + F 3 ). Hence by Proposition 3.11, Y 0 is the G 2 a -equivariant blowup of P 2 , F 0 or F 1 . □ Proposition 3.13. There is a G 2 a -equivariant birational morphism Φ : Y 0 → P 2 such that l 0 = Φ(F 0 ) is the boundary ∂P 2 = P 2 \ C 2 and p i = Φ(F i ), 1 ≤ i ≤ 3, are three distinct points on l 0 . Moreover, −K Y 0 = 3F 0 + 2(F 1 + F 2 + F 3 ), and we have Proof. In the following, we will apply the similar idea of comparing coefficients of −K Y 0 = aF 0 +b(F 1 +F 2 +F 3 ) and G 2 a -equivariant blowups of P 2 , F 0 = P 1 × P 1 and F 1 . Case 1: blow up from P 2 . There is only one possibility of blowing up P 2 to get Y 0 , that is, blow up three distinct points on l 0 = P 2 \ C 2 . Case 2: blow up from P 1 × P 1 . There is only one possibility of blowing up F 0 = P 1 × P 1 to get Y 0 , as follows, where l 1 ∪ l 2 = F 0 \ C 2 , l 3 is the exceptional divisor of the first blowup of the point p 1 ∈ l 1 ∩ l 2 , and l 4 is the exceptional divisor of the second blowup of a point p 2 ∈ l 3 \ (l 1 ∪ l 2 ). Case 3: blow up from F 1 . There is only one possibility of blowing up F 1 to get Y 0 , as follows, where l 1 ∪ l 2 = F 1 \ C 2 such that −K F 1 = 3l 1 + 2l 2 , l 3 is the exceptional divisor of the first blowup of a point p 1 ∈ l 1 \ l 2 , and l 4 is the exceptional divisor of the second blowup of a point p 2 ∈ l 1 \ (l 2 ∪ l 3 ). All three cases above yield the same Y 0 , which is the blowup of P 2 along three colinear points. Hence, −K Y 0 = 3F 0 + 2(F 1 + F 2 + F 3 ). By Corollary 3.10, we have d 0 + e 0 = d 1 + e 2 = d 2 + e 1 = 1 and d 2 + e 2 = 0 Indeed, by comparing the coefficients of −K Y 0 = 3F 0 + 2(F 1 + F 2 + F 3 ) = 3(d 0 + e 0 )F 0 + (d 1 + 2d 2 + e 1 + 2e 2 )(F 1 + F 2 + F 3 ) we get d 0 + e 0 = 1 and d 1 + 2d 2 + e 1 + 2e 2 = (d 1 + e 2 ) + (d 2 + e 1 ) + (d 2 + e 2 ) = 2. By Corollary 3.10, d i , e j , 0 ≤ i, j ≤ 2, are all non-negative and d 1 + e 2 = d 2 + e 1 ≥ 1. So we have d 0 + e 0 = d 1 + e 2 = d 2 + e 1 = 1 and d 2 + e 2 = 0. □ Remark 3.14. By choosing a family of three points in general position on P 2 degenerating to three colinear points, we can construct a smooth projective family Z/∆ such that Z t ≃ Y (A) for each t 0 while Z 0 is a blowup of P 2 along three colinear points. But in our situation, we have an extra involution Θ which prevents this situation. Now we are ready to prove Theorem 1.2.
Proof of Theorem 1.2. The case of A = C follows from the classification of Mukai varieties. Now assume A C. Take X → ∆ to be a specialization of X(A). Assume that X 0 is not isomorphic to X(A). By Proposition 3.2, X 0 is an equivariant compactification of G n a . By Proposition 3.3, we have a smooth family of surfaces Y → ∆ with the central fiber Y 0 being a G 2 a -surface. By Proposition 3.13, we may assume D i,0 = F 0 + F i , 1 ≤ i ≤ 3, and E j,0 = F j , 1 ≤ j ≤ 3 (the proof for the other case is similar). By Lemma 2.11 and Proposition 3.6, the involution Θ satisfies Θ(D i ) = E i and Θ(E i ) = D i . It follows that Θ 0 (F 0 + F i ) = F i and Θ 0 (F i ) = F 0 + F i . Consider the Mori cone NE(Y 0 ), which is the numerical effective cone of curves of Y 0 . Since each F i has negative self intersection, each F i spans an extremal ray of NE(Y 0 ). Then F 0 + F i is an interior point of a 2-dimensional extremal face of NE(Y 0 ). Since Θ 0 induces an isomorphism of the Mori cone NE(Y 0 ), it cannot send the extremal ray of F i to the non-extremal ray of F 0 + F i . This contradiction shows that X 0 ≃ X(A). □