aa r X i v : . [ m a t h . AG ] A ug THE CUBO-CUBIC TRANSFORMATION AND K3 SURFACES
FABIAN REEDE
Abstract.
In this note we observe that the Cremona transformation in Oguiso’s exam-ple of Cremona isomorphic but not projectively equivalent quartic K3 surfaces in P isthe classical cubo-cubic transformation of P . Introduction
Let D and D be two smooth hypersurfaces of degree d in P n which are isomorphic asabstract varieties. It is then a natural question if an isomorphism between them can beobtained by restricting an automorphism or a Cremona transformation of the ambient P n .We say D and D are projectively equivalent if there is an automorphism g : P n → P n which restricts to an isomorphism D → D . We say D and D are Cremona isomorphicif there is a Cremona transformation f : P n P n such that the restriction extends to anisomorphism D → D .Assume n ≥ n, d ) = (3 , D and D are projectively equivalent (so especially Cremona isomorphic) if they areisomorphic as abstract varieties.Oguiso showed in [Ogu17] that in the special case ( n, d ) = (3 ,
4) this result is no longertrue, by giving an example of two smooth quartic K3 surfaces in P which are Cremonaisomorphic but which are not projectively equivalent.The main observation of this note is that the Cremona transformation f : P P in Oguiso’s example is the classical cubo-cubic transformation of P , see [Noe71] or [SR49].In this note we work over the field C and the note consists of three sections. In the firstsection we recall two constructions of the cubo-cubic Cremona transformation. The secondsection contains a summary of Oguiso’s results. In the third section we combine the firsttwo sections and show how to understand Oguiso’s example in terms of the cubo-cubictransformation. 1. The cubo-cubic transformation
Let C be a general smooth irreducible curve of genus 3 and degree 6 in P . Thehomogeneous ideal of C is generated by four cubic polynomials f i ∈ C [ x ], which are the3 × × A ( x ) of linear forms, see e.g. [Ell75, Exemples 2],here x = [ x , x , x , x ] are the homogeneous coordinates on P . These polynomials definea rational map: ϕ : P P . It follows from [Dol12, Theorem 7.2.4] that this map is in fact birational with base scheme C and has multidegree (3 , ϕ of ϕ in P × P , that is we have the following diagram:Γ ϕ P P π π ϕ Mathematics Subject Classification.
Primary: 14E05; Secondary: 14E07, 14J28, 14M12.
Key words and phrases.
Cremona transformations, K3 surfaces, determinantal hypersurfaces. where π i : Γ ϕ → P is the i -th projection and ϕ = π ◦ π − .The graph Γ ϕ is the intersection of three general divisors of bidegree (1 ,
1) in P × P .These divisors are defined by the 3 rows of the matrix A ( x ). As the multidegree is (3 , ϕ is called cubo-cubic transformation.An equivalent definition of the cubo-cubic transformation is the following, see [Kat87]or [Dol12, Example 7.2.6]: let C be a general smooth irreducible curve of genus 3 anddegree 6 in P and denote the blow up of C in P by X . Then Pic( X ) = Z [ H ] ⊕ Z [ E ],where H is the pullback of a hyperplane in P to X via the blow up morphism σ : X → P and E is the exceptional divisor of σ . The linear system | H − E | defines a morphismΨ : X → P which can be shown to be of degree 1, hence Ψ is birational. The morphism Ψ contractsthe strict transform F of the trisecant surface Sec ( C ) with respect to σ to a curve C ′ ⊂ P with C ∼ = C ′ .We get a birational φ map from the following diagram: X P P σ Ψ φ We can see that φ = ϕ by noting that the graph Γ ϕ , i.e. the intersection of the threedivisors of bidegree (1 ,
1) in P × P , is the same as the blow-up of the curve C , see forexample [CCGK16, H.29, Lemma E.1] or [Dol12, 7.1.3]. Remark . The cubo-cubic Cremona transformation and most of its properties werealready known to Max Noether, see [Noe71, § Remark . The cubo-cubic transformation is special in the sense that it is the only non-trivial Cremona transformation of P that is resolved by just one blow up along a smoothcurve, see [Kat87, Proposition 2.1].2. Oguiso’s example
In [Ogu17, Theorem 1.5.] Oguiso constructs two quartic K3 surfaces S and S in P and a Cremona transformation τ : P P such that τ restricts to a birational map S S . This map must be an isomorphism as the canonical divisor of a K3 surfaceis nef. But these two surfaces are not projectively equivalent in P , that is there is no g ∈ Aut( P ) = PGL(4 , C ) with g ( S ) = S .The K3 surfaces are constructed as follows: pick three general divisors Q , Q and Q of bidegree (1 ,
1) in P × P , then V := Q ∩ Q ∩ Q is a smooth Fano threefold, birational to P in two different ways, given by restricting theprojections π i : P × P → P to V , i = 1 ,
2. Call these morphisms p and p . Oguisodefines the birational map τ : P P via: τ := p ◦ p − . This defines a non-trivial Cremona transformation of P , [Ogu17, Theorem 1.5.(2)].For a very general divisor Q of bidegree (1 , S := Q ∩ Q ∩ Q ∩ Q . Then S is a smooth K3 surface with NS( S ) = Z h ⊕ Z h with intersection matrix(( h i , h j )) i,j = (cid:18) (cid:19) . The restrictions of the p i for i = 1 ,
2, i.e. p i | S : S → p i ( S ), turn out to be isomorphismsgiven by the complete linear systems | h i | . We get two quartic K3 surfaces S i := p i ( S ) ⊂ P .The main result of Oguiso regarding these two surfaces is: HE CUBO-CUBIC TRANSFORMATION AND K3 SURFACES 3
Theorem 2.1. [Ogu17, Theorem 1.5.(3)]
The K3 surfaces S and S are Cremona iso-morphic but not projectively equivalent in P .Remark . The reason why S and S are not projectively equivalent is that there is no f ∈ Aut( S ) such that f ∗ h = h , see [Ogu17, Proposition 6.2., Lemma 6.4.].The whole construction is captured in the following diagram: SV P P S S | h | | h | p p τ ∼ = (cid:3) Remark . The K3 surfaces S and S are determinantal quartic K3 surfaces and werealready known to Cayley, see [Cay71] and [FGvGvL13].3. Cubo-cubic Cremona transformations and K3 surfaces
We want to bring the previous two sections together and understand Oguiso’s examplein terms of a cubo-cubic transformation. To do this, pick a general smooth irreduciblecurve C of genus 3 and degree 6 in P , with 3 × A ( x ) and write A ( x ) = x A + x A + x A + x A with A k = ( a kij ) i,j ∈ Mat(3 , , C ). This defines three 4 × B i = ( a kij ) j,k for i = 1 , , k is the index for the columns and j is the index for the rows.The three divisors Q , Q and Q of bidegree (1 ,
1) induced by A ( x ) (the equations ofthe graph of the cubo-cubic transformation induced by C ) are : Q i = X k,j =1 a kij x k y j = 0 ⊂ P × P ( i = 1 , , . We choose a very general fourth divisor Q of bidegree (1 ,
1) as described above: Q = X k,j =1 a k j x k y j = 0 ⊂ P × P . Thus S = Q ∩ Q ∩ Q ∩ Q ⊂ P × P is given by S = (cid:8) ( x , y ) ∈ P × P | M ( x ) · y t = 0 t (cid:9) for the 4 × M ( x ) = ( m ij ( x )) i,j with entries: m ij ( x ) = a ij x + a ij x + a ij x + a ij x . The observation(1) X k,j =1 a kij x k y j = X j =1 X k =1 a kij x k ! y j = X k =1 X j =1 a kij y j x k , implies the following identity: M ( x ) · y t = N ( y ) · x t with the 4 × N ( y ) = ( n ik ( y )) i,k given by n ik ( y ) = a ki y + a ki y + a ki y + a ki y . Oguiso proved in [Ogu17, Proposition 4.1.] that S = (cid:8) x ∈ P | det( M ( x )) = 0 (cid:9) and S = (cid:8) y ∈ P | det( N ( y )) = 0 (cid:9) . FABIAN REEDE
By construction the first three rows of M ( x ) are those of A ( x ). Using the Laplace expan-sion with respect to the last row shows that for every x ∈ C we have det( M ( x )) = 0 andhence C ⊂ S .Similarly the first three rows of N ( y ) define a general 3 × A ′ ( y ) whose 3 × C ′ with C ′ ⊂ S .The blow up X of C in P is given by Q ∩ Q ∩ Q , which by (1) is also the blow upof C ′ in P and C ∼ = C ′ by [Kat87, Proposition 1.3.].Denote by e S the strict transform of S in the blow up X of C in P and by e S thestrict transform of S in the blow up of C ′ . Using Oguiso’s results and the fact that C and C ′ are smooth we get S ∼ = e S = S = e S ∼ = S . Finally we look at the cubo-cubic Cremona transformation given by the curve C : X P P σ Ψ φ By the previous results the strict transform e S with respect to σ of the K3 surface S in X is the K3 surface S and it is also equal to the strict transform e S with respect to Ψof the K3 surface S .This implies that the birational map φ restricts to a birational map between S and S which extends to an isomorphism, because S and S are K3 surfaces, which have nefcanonical divisors. Hence S and S are Cremona isomorphic in P and the Cremonaisomorphism is induced by the cubo-cubic transformation given by the curve C . But byTheorem 2.1, they are not projectively equivalent in P . Remark . The construction of smooth determinantal quartic surfaces in P containinga curve of genus 3 and degree 6 can also be found in Beauville’s paper [Bea00, 6.7]. In factBeauville proves: a smooth quartic surface in P is determinantal if and only if containsa nonhyperelliptic curve of genus 3 embedded in P by a linear system of degree 6, see[Bea00, Corollary 6.6]. References [Bea00] Arnaud Beauville. Determinantal hypersurfaces.
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Institut f¨ur Algebraische Geometrie, Leibniz Universit¨at Hannover, Welfengarten 1,30167 Hannover, Germany
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