Prime Fano threefolds of genus 12 with a $G_m$-action

We give an explicit construction of prime Fano threefolds of genus 12 with a $G_m$-action, describe their isomorphism classes and automorphism groups.


Introduction
We work over the field C of complex numbers. Recall that a prime Fano threefold is a smooth projective variety of dimension 3 with Pic(X) ZK X and −K X ample. The genus g(X) is defined by the equality (−K X ) 3 = 2g(X) − 2. We refer to [IP99] for the general theory of Fano varieties and their classification in dimension 3.
In this paper we discuss prime Fano threefolds of genus 12, which means (−K X ) 3 = 22. Specifically, we give an explicit description of those of them that admit a faithful action of a one-dimensional torus G m . Thus, this paper complements to [Pro90], [KPS16,], [DKK17], and [D08, D17,CS18]. To state the main result, we introduce some notation.
Theorem 1.3. There is a natural bijection between isomorphism classes of prime Fano threefolds of genus 12 with a faithful G m -action, and the set P 1 \ {0, 1, ∞} of smooth quadrics Q u passing through the curve Γ . Set u = u 0 /u 1 . If X m (u) denotes the threefold corresponding to the quadric Q u , then Aut(X m (u)) where the action of Z/2Z on G m in the semidirect product is given by the inversion.
The proof of the theorem takes the rest of the paper. We compute the automorphisms in Corollary 3.9, establish the bijection of isomorphism classes in Corollary 4.2, and identify the Mukai-Umemura threefold in Proposition 4.4.

A birational transformation
The proof of Theorem 1.3 is based on a Sarkisov link described in Theorem 2.2 below. To state it we need to remind the notion of quadratical normality for projective curves.
Recall that a curve in a projective space is called quadratically normal if restrictions of quadrics form a complete linear system on the curve. We will need the following evident observation.
Remark 2.1. A rational sextic curve Γ ⊂ P 4 is quadratically normal if and only if the linear system of quadrics passing through Γ is one-dimensional. Indeed, this follows immediately from the exact sequence since dim H 0 (P 4 , O P 4 (2)) = 15 and dim H 0 (Γ , O Γ (12)) = 13. In particular, a quadratically normal rational sextic curve in P 4 is not contained in a hyperplane.
The next result is the base of our construction.

Theorem 2.2. ([Tak89], [IP99])
Let X be a prime Fano threefold of genus 12 and let C be a smooth conic on X. Then there exists the following commutative diagram of birational maps • σ Q is the blow up of a smooth rational quadratically normal sextic curve Γ ⊂ Q, • σ X is the blow up of C, and χ is a flop.
Furthermore, let H X and H Q be the ample generators of the Picard groups Pic(X) and Pic(Q), respectively, and denote H X := σ * X H X and H Q := σ * Q H Q . Let E C := σ −1 X (C) and E Γ := σ −1 Q (Γ ) be the exceptional divisors of σ X and σ Q , respectively. Then (i) The map σ Q • χ : X Q ⊂ P 4 is given by the linear system |H X − 2E C | and contracts the unique divisor from |2H X − 5E C |.
(ii) The map σ X • χ −1 : Q X ⊂ P 13 is given by the linear system |5H Q − 2E Γ | and contracts the unique Proof. Let X be the blowup of X along C. The linear system | − K X | is base point free and defines a morphism that does not contract divisors by Lemma 2.4 below. Therefore, [Tak89] or [IP99,Theorem 4.4.11] applies and gives the diagram, and most of the details of the theorem. The only thing left is to show that the curve Γ is quadratically normal.
The argument of Remark 2.1 shows that the curve Γ is quadratically normal if and only if the linear system |2H Q − E Γ | is zero-dimensional. On the other hand, the map χ −1 identifies this linear system with |2(H X − 2E C ) − (2H X − 5E C )| = |E C |, which consists of a single divisor, hence the claim.
Lemma 2.4. Let X be the blowup of X along a smooth conic C. The linear system | − K X | is base point free and defines a morphism that does not contract divisors.
(2.5) see [IP99, Corollary 2.1.14(ii)], and so | − K X | embeds X into P 13 . Note that X ⊂ P 13 does not contain planes (because Pic(X) = ZH X ) and is an intersection of quadrics (see [Isk80, Proposition IV.1.3 and Theorem II.3.4]). Note that | − K X | = |H X − E C | is the strict transform of the linear system of hyperplane sections of X passing through the conic C. Their scheme-theoretic intersection is the intersection of the linear span of C with X. Since X is an intersection of quadrics and does not contain planes, this is equal to C. Therefore, the linear system | − K X | on X is base point free (moreover, it follows that the morphism defined by this linear system is birational, although we do not need this). In particular, −K X · γ ≥ 0 for any curve γ on X .
Assume γ is contracted by | − K X |, so that −K X · γ = 0. Then H X · γ 0, since γ is an effective curve, so since H X is nef we have H X · γ > 0. Therefore E C · γ > 0 and so which means that γ is contained in the base locus of the linear system |H X − 2E C |. The natural exact sequences and dim H 0 (C, O C (2)) = 3, dim H 0 (C, I C /I 2 C (2)) = 6 (the first equality is (2.5), the second is evident, and the third follows from (5.2)) imply that we have an inequality dim |H X − 2E C | ≥ 13 − 3 − 6 = 4.
Since moreover Pic(X) = ZH X , every hyperplane section of X is irreducible, and since the linear system |H X − 2E C | is of positive dimension, the only possible divisorial component of its base locus is E C . Thus, it remains to check that E C is not contracted by | − K X |. This, however, easily follows from the equality The construction of Theorem 2.2 can be reversed: Theorem 2.6. Let Q ⊂ P 4 be a smooth quadric and let Γ ⊂ Q be a smooth rational quadratically normal sextic curve. Then there is a smooth prime Fano threefold X of genus 12 and a smooth conic C ⊂ X related to (Q, Γ ) by the diagram (2.3).
Proof. Let Q be the blowup of Q along Γ . By Lemma 2.7 below the linear system | − Assume γ is a curve on Q contracted by the linear system |−K Q |, so that −K Q · γ = 0. Then H Q ·γ 0, since γ is an effective curve, so since H Q is nef we have H Q · γ > 0. Therefore E Γ · γ > 0 and so for all m ≥ 1.
Take m to be the maximal such that |2H Q − mE Γ | ∅ (by Remark 2.1 we have m ≥ 1). Let F ∈ |2H Q − mE Γ | be any member. Then γ ⊂ F. Since Γ does not lie in a hyperplane (Remark 2.1), such F is irreducible, and thus it remains to show that the morphism defined by the linear system | − K Q | does not contract F. This follows from Now, we can make a flop Q X and consider the Mori contraction X → X. Solving Diophantine equations, as in [IP99, §4.1] (see also [CM13]) one can show that X → X is the blowup of a smooth conic on a prime Fano threefold X of genus 12.
Lemma 2.7. Let Γ ⊂ P 4 be a smooth rational quadratically normal curve of degree 6. Then Γ is a schemetheoretic intersection of cubics.
Proof. By Remark 2.1 the curve Γ is not contained in a hyperplane. Hence [GLP83,Corollary] applies to Γ and shows that Γ is an intersection of cubics, unless it has a 4-secant line. It remains to show that a curve Γ with a 4-secant line is not quadratically normal.
Indeed, let L ⊂ P 4 be a 4-secant line and denote by D := L ∩ Γ the scheme-theoretic intersection (it is a zero-dimensional subscheme in Γ of length at least 4). Let I L ⊂ O P 4 and I D ⊂ O Γ be the ideal sheaves. The space H 0 (P 4 , I L (2)) of quadrics in P 4 containing L has codimension 3 in the space of all quadrics. On the other hand, its image in the space H 0 (Γ , O Γ (12)) is contained in the subspace H 0 (Γ , I D (12)), which has codimension at least 4. Therefore, the map H 0 (P 4 , O (2)) → H 0 (Γ , O Γ (12)) is not surjective.
Remark 2.8. The constructions of Theorem 2.2 and Theorem 2.6 are mutually inverse. Moreover, these constructions are functorial. In other words, let ϕ : X 1 → X 2 be an isomorphism of prime Fano threefolds of genus 12 and let C 1 ⊂ X 1 and C 2 = ϕ(C 1 ) be smooth conics on them. If (Q i , Γ i ) is the quadric with a sextic curve associated to the pair (X i , C i ) then the isomorphism ϕ extends in a unique way to an isomorphism of diagrams (2.3). In particular, it induces an isomorphism ψ : Conversely, let ψ : Q 1 → Q 2 be an isomorphism of smooth quadrics, and let Γ 1 ⊂ Q 1 and Γ 2 = ψ(Γ 1 ) be smooth rational quadratically normal sextic curves on them. If (X i , C i ) is the prime Fano threefold of genus 12 with a conic associated to the pair (Q i , Γ i ) then the isomorphism ψ extends in a unique way to an isomorphism of diagrams (2.3). In particular, it induces an isomorphism ϕ : X 1 → X 2 such that ϕ(C 1 ) = C 2 .
In particular, if the pair (Q, Γ ) corresponds to a pair (X, C), we have an isomorphism where Aut(X, C) ⊂ Aut(X) is the group of automorphisms of X that preserve the conic C and similarly Aut(Q, Γ ) ⊂ Aut(Q) is the group of automorphisms of Q that preserve the sextic Γ .
Remark 2.10. By using a relative version of MMP one can see that the constructions of Theorem 2.2 and Theorem 2.6 work in smooth families. Namely, if X → S is a smooth morphism whose fibers are prime Fano threefolds of genus 12 and C ⊂ X is a subscheme which is smooth over S and whose fibers are conics, there is a relative over S version of the diagram (2.3) with the same description of fibers. Similarly, if Q → S is a smooth morphism whose fibers are three-dimensional quadrics and Γ ⊂ Q is a subscheme which is smooth over S and whose fibers are rational quadratically normal sextic curves, there is a relative over S version of the diagram (2.3) and again with the same description of fibers.

Automorphisms
To apply the results of Section 2 for a description of prime Fano threefolds X of genus 12 with a G maction and of their automorphism groups, we need to show that any such X has a smooth conic that is Aut(X)-invariant. For this we will need a description of G m -invariant lines from [KPS16]. Recall the Mukai-Umemura threefold X MU , see [MU83].
Lemma 3.1. Let X be a prime Fano threefold of genus 12 with a faithful G m -action which is not isomorphic to the Mukai-Umemura threefold X MU . There are exactly two G m -invariant lines L 1 and L 2 on X, these lines are the only special lines on X, i.e., and they do not meet.
Proof. By [KPS16, Theorem 5.3.10] the variety X belongs to the family X m (u) that was described in [KPS16, Example 5.3.4]. Its Hilbert scheme of lines Σ(X) was described in [KPS16,Proposition 5.4.4] as the union of two smooth rational curves with two points of tangency as shown on a picture below: Let L 1 and L 2 be the lines on X corresponding to the singular points of Σ(X). Clearly, these lines are Now we can pass to conics. Recall that a conic on a projective variety X ⊂ P N is a subscheme C ⊂ X with Hilbert polynomial p C (t) = 1 + 2t. There are three types of conics -smooth conics, i.e., curves isomorphic to the second Veronese embedding of P 1 , reducible conics, i.e., unions of two lines meeting at a point, and non-reduced conics, i.e., non-reduced schemes C such that C red = L is a line and I L /I C O L (−1), see [KPS16, Lemma 2.1.1(ii)].
Proposition 3.2. Let X be a prime Fano threefold of genus 12 with a faithful G m -action which is not isomorphic to the Mukai-Umemura threefold X MU . Then X contains a smooth Aut(X)-invariant conic.
In Lemma 4.1(ii) we will show that an Aut(X)-invariant conic on such X is unique.
Proof. By [Pro90]  where is a line on S(X) = P 2 and the points c 1 and c 2 correspond to the non-reduced conics C 1 and C 2 .
In the first case the conic corresponding to the third point c 3 is Aut(X)-invariant. In the second case, if the points c 1 and c 2 lie on , the conic corresponding to the point c 0 is Aut(X)-invariant. Finally, if only one of the points c 1 and c 2 lies on , then this point is Aut(X)-invariant, hence the finite group Aut(X)/G m acting on has an invariant point, hence Aut(X)/G m is a cyclic group, hence it has yet another fixed point on , which then corresponds to an Aut(X)-invariant conic. Thus, in all these cases we have found an Aut(X)-invariant conic C on X that is distinct from the non-reduced conics C 1 and C 2 . The only thing left is to show that C is non-singular. Indeed, as we mentioned above X has no non-reduced conics distinct from C 1 and C 2 . On the other hand, if a G m -invariant conic C is a union of two distinct lines meeting at a point then each of these lines is G m -invariant, hence C = L 1 ∪L 2 by Lemma 3.1. But the G m -invariant lines L 1 and L 2 do not meet (again by Lemma 3.1), so this is also impossible.
By Remark 2.8, if X is a prime Fano threefold of genus 12 with a faithful G m -action and C ⊂ X is a smooth Aut(X)-invariant conic, the corresponding pair (Q, Γ ) has a faithful G m -action. We show below how Γ and Q look like.
Lemma 3.5. Assume G m acts faithfully on P 4 and Γ ⊂ P 4 is a G m -invariant smooth rational quadratically normal sextic curve. Then in suitable coordinates Γ is the image of the map (1.1).
Proof. By Remark 2.1 the curve Γ does not lie in a hyperplane. Therefore, we can assume that the action of G m on Γ is faithful and that the image of G m in Aut(Γ ) PGL 2 is the standard torus. Since Γ spans P 4 the embedding Γ → P 4 canonically factors as where the first arrow is the Veronese embedding of degree 6 and the second arrow is a linear projection with center a line. Since the composition is G m -equivariant, the line is G m -invariant. If y 0 , y 1 , y 2 , y 3 , y 4 , y 5 , y 6 are the standard weight coordinates on P 6 , the line is given by equations y i = 0 for i ∈ I, where I ⊂ {0, . . . , 6} is a subset of cardinality 5. Since the image of Γ is a smooth curve, this line does not intersect the tangent lines to Γ at 0 and ∞, hence the set I contains {0, 1, 5, 6}. If the fifth element of I is 2 or 4, the curve Γ has a 4-tangent line and so is not quadratically normal by the proof of Lemma 2.7. Hence the fifth element in I is 3.
Lemma 3.7. If Γ ⊂ P 4 is the curve defined by (1.1) and Q is a quadric from the pencil generated by the quadrics (1.2), then Aut(Q, Γ ) G m Z/2Z.
Proof. Since Γ spans P 4 , we have Aut(Q, Γ ) ⊂ Aut(Γ ) PGL 2 . Furthermore, it is easy to see that the points (1 : 0 : 0 : 0 : 0) and (0 : 0 : 0 : 0 : 1) are the only points on Γ that lie on the singular locus of one of the quadrics passing through Γ . Therefore, where G m is the torus that acts on Γ preserving the above two points (by rescaling one of the coordinates t 0 and t 1 ), and Z/2Z is generated by the involution ι ∈ Aut(Γ ), ι : (t 0 : t 1 ) −→ (t 1 : t 0 ) (3.8) that normalizes this torus. On the other hand, it is easy to see that both the torus and the involution preserve any quadric in the pencil passing through Γ , hence the claim.

Isomorphism classes
Now we switch to the proof of the part of Theorem 1.3 describing isomorphism classes of prime Fano threefolds X of genus 12 with a faithful G m -action. For this we will need a couple of observations about the action of Aut(X) on X. We denote by ι X : X → X the involution corresponding to (3.8) under the isomorphism (2.9). Recall that, as it was explained in the proof of Proposition 3.2, if X is not the Mukai-Umemura threefold X MU , there are precisely two special lines L 1 and L 2 on X.
Lemma 4.1. Let X be a prime Fano threefold of genus 12 with a faithful G m -action which is not isomorphic to the Mukai-Umemura threefold.
(i) The involution ι X swaps the special lines L 1 and L 2 on X; in particular X has no Aut(X)-invariant lines.
(ii) An Aut(X)-invariant conic on X is unique.
(iii) The non-reduced conics C 1 and C 2 supported on the G m -invariant lines L 1 and L 2 , and the smooth Aut(X)-invariant conic C are the only G m -invariant conics on X.
Proof. The first part of the lemma follows immediately from [KPS16,Proposition 5.4.6], where it was checked that Aut(X, L 1 ) G m . In a combination with Lemma 3.1 this proves that there are no Aut(X)-invariant lines on X.
To prove the uniqueness of the Aut(X)-invariant conic constructed in Proposition 3.2 we have to show that the action of the involution ι X on the G m -fixed locus S(X) G m in the Hilbert scheme of conics S(X) has a single fixed point. Recall that we have two possibilities (3.4).
In the first case the points c 1 and c 2 correspond to the non-reduced conics supported on L 1 and L 2 , hence they are swapped by the involution ι X . Thus the third point c 3 is the only point in S(X) G m fixed by ι X . Moreover, in this case there are exactly three G m -invariant conics on X. So, it remains to show that the second case of (3.4) does not occur.
In the second case the point c 0 is fixed by ι X , hence the points c 1 and c 2 corresponding to the nonreduced conics belong to the line . The involution ι X preserves , hence has two fixed points c and c on it, which are thus Aut(X)-invariant. So, we have a triple (c 0 , c , c ) of Aut(X)-invariant points on S(X), hence the image of Aut(X) in Aut(S(X)) PGL 3 is abelian. But the action of Aut(X) on S(X) is faithful by [KPS16,Lemma 4.3.4], so this contradicts the fact that the conjugation by ι acts on G m as inversion (see (3.8)).
Most of things we discussed so far were related to all X with a faithful G m -action except of the Mukai-Umemura threefold X MU . Now we note that the latter can also be covered by the same approach. We refer to [MU83] for a description of X MU and of its Hilbert schemes of lines.
Recall that Aut(X MU ) PGL 2 which acts on the Hilbert scheme of conics S(X MU ) P 2 as on the projectivization of an irreducible representation. In particular, there are two Aut(X MU )-orbits on S(X MU ). One of them is a conic S nr ⊂ S(X MU ) that parameterizes non-reduced conics in X MU (thus S nr also parameterizes lines on X MU , all of which are special). The complement S(X) \ S nr parameterizes smooth conics on X MU . In particular for every smooth conic C ⊂ X MU we have Aut(X MU , C) G m Z/2Z, the normalizer of a torus in PGL 2 . So, applying the construction of Theorem 2.2 to the pair (X MU , C) we obtain a pair (Q, Γ ), where Γ is the curve defined by (1.1) and Q is a smooth quadric from the pencil generated by (1.2). Moreover, since the group Aut(X MU ) acts transitively on smooth conics in X MU , Remark 2.8 shows that different choices of a smooth conic C produce the same pair (Q, Γ ). Thus, there is a particular quadric in the pencil corresponding to the Mukai-Umemura threefold. We will call it the Mukai-Umemura quadric and denote by Q MU . In Proposition 4.4 below we show Q MU = Q −1/4 . And meanwhile, we observe that the above argument proves the first part of Theorem 1.3.  To finish the proof of Theorem 1.3 it remains to identify the Mukai-Umemura quadric.
Proof. Recall the constructions of the Mukai-Umemura threefold X = X MU from [MU83]. Let M d denote the vector space of degree d homogeneous polynomials in two variables x and y with its natural GL 2 -action. Then X can be realized inside P(M 12 ⊕ M 0 ) P 13 as the closure of a PGL 2 -orbit: where φ 12 = xy(x 10 + 11x 5 y 5 + y 10 ).
We apply the construction of Theorem 2.2 to the pair (X, C). Consider the linear functions z i on P(M 12 ⊕ M 0 ) that take (f , c) ∈ M 12 ⊕ M 0 to the coefficient of f at x 12−i y i (then its G m -weight is i −6). Let alsoz be the linear function that takes (f , c) to c (its weight is 0).
We claim that the double projection ξ : X Q ⊂ P 4 of Theorem 2.2 is defined (in appropriate coordinates on P 4 ) by the map (4.6) Indeed, we know that the map should be G m -equivariant, and we know from Lemma 3.5 that the weights of G m on P 4 should be (−3, −2, 0, 2, 3). Clearly, the functions z 3 , z 4 , z 8 , and z 9 are the only linear functions of weights −3, −2, 2, and 3 respectively. On the other hand, there are two functions z 6 andz of weight zero, so we should take an appropriate linear combination of those. But this function should vanish at the point (φ 12 , 1) ∈ C, and so it remains to note that z 6 (φ 12 , 1) = 11, whilez(φ 12 , 1) = 1.
This computation completes the proof of Theorem 1.3.

Concluding remarks
To finish the paper we provide some extra details for the birational transformations of Theorems 2.2 and 2.6. Proof. Since the Hilbert scheme of conics on X is smooth (see (3.3)), the normal bundle of the smooth G m -invariant conic C (and, in fact, of any smooth conic on X) is If X X MU the conic C is the unique smooth G m -invariant conic and L 1 , L 2 are the only G m -invariant lines on X by Lemma 4.1. Therefore, the flopping locus of the map χ : X Q is the union of the strict transforms of the lines L 1 and L 2 . Note that the lines L 1 and L 2 do not meet by Lemma 3.1. If X X MU the explicit description of the Hilbert schemes of conics and lines shows that C is still the unique smooth G m -invariant conic and there are precisely two G m -invariant lines on X; for instance, if the curve C is given by (4.5), these lines are + a 2 y)x 11 } and L 2 = {(a 1 x + a 2 y)y 11 }.
Clearly, these lines do not meet. Note that in both cases L 1 ∩ L 2 = ∅ and the lines L 1 and L 2 are special, hence the normal bundles of their strict transforms in X are isomorphic to O ⊕ O (−2), hence the flop χ is given by Reid's pagoda.
In the following remark we describe the flopping locus and the exceptional divisor of the rational map ξ −1 : Q X.
Remark 5.3. The base locus of the pencil of quadrics Q 0 , Q ∞ is a surface F ⊂ P 4 of degree 4. This surface is the exceptional divisor of the map ξ −1 : Q X by Theorem 2.2 (ii). The singular locus of F consists of four ordinary double points and F contains exactly four lines 0,1 = P 0 , P 1 , 1,6 = P 1 , P 6 , 6,5 = P 6 , P 5 , 5,0 = P 5 , P 0 .
In particular, F is a toric del Pezzo surface with Pic(F) Z 2 and Cl(F) Z 2 ⊕ Z/2. One can also realize F as the anti-canonical image of the blowup F of P 1 × P 1 at four points (0, 0), (0, ∞), (∞, ∞), (∞, 0). The two 3-tangent lines 0,1 and 6,5 are contained in the flopping locus for χ −1 . Since by Proposition 5.1 the flopping locus consists of two irreducible components, the strict transforms in Q of these two lines form the whole flopping locus of χ −1 .
Finally, we give a description of one of the boundary points in the family P 1 \ {0, 1, ∞} of G m -invariant threefolds X. Note that the quadric Q 1 = {y 0 y 6 − y 1 y 5 = 0} is a cone over a smooth quadric surface and contains Γ . Its vertex P 3 = (0 : 0 : 1 : 0 : 0) is an ordinary double point on Q 1 that lies away from the surface F described in Remark 5.3. In particular, the point P 3 lies away from Γ and away from the flopping lines of χ −1 . Therefore, the construction of Theorem 2.6 can be applied to (Q 1 , Γ ) and produces a prime Fano threefold X m (1) of genus 12 with one ordinary double point and a faithful G m -action. On the other hand, Fano threefolds of genus 12 with a single ordinary double point were classified in [Pro16]. In the next proposition we identify X m (1) with a threefold of type (IV); a detailed description of such threefolds is given in [Pro16,§7].