The cubo-cubic transformation and K3 surfaces

TL;DR

This paper identifies the Cremona transformation in Oguiso's example of quartic K3 surfaces as the classical cubo-cubic transformation, highlighting a specific geometric correspondence.

Contribution

It reveals that the Cremona transformation in Oguiso's example is actually the classical cubo-cubic transformation, connecting specific K3 surfaces with classical birational geometry.

Findings

The Cremona transformation in Oguiso's example is the classical cubo-cubic transformation.

It demonstrates a link between K3 surfaces and classical birational transformations.

The note clarifies the geometric nature of the transformation involved.

Abstract

In this note we observe that the Cremona transformation in Oguiso's example of Cremona isomorphic but not projectively equivalent quartic K3 surfaces in three-dimensional projective space is the classical cubo-cubic transformation.

Equations

$$\phi :{\mathbb{P}}^3$$

$${\Gamma }_{\phi }$$

$$\Psi :X\to {\mathbb{P}}^3$$

$$X$$

$$V:=Q_1\cap Q_2\cap Q_3$$

$$\tau :=p_2\circ p_1^{-1}.$$

$$S:=Q_1\cap Q_2\cap Q_3\cap Q_4.$$

$$((h_i,h_j){)}_{i,j}=\left(\begin{array}{cc}4& 6\\ 6& 4\end{array}\right).$$

$$S$$

$$A(\text{x})=x_1{A}_1+x_2{A}_2+x_3{A}_3+x_4{A}_4$$

$$Q_i=\left(\sum _{k,j=1}^4{a}_{ij}^k{x}_k{y}_j=0\right)\subset {\mathbb{P}}^3\times {\mathbb{P}}^3(i=1,2,3).$$

$$Q_4=\left(\sum _{k,j=1}^4{a}_{4j}^k{x}_k{y}_j=0\right)\subset {\mathbb{P}}^3\times {\mathbb{P}}^3.$$

$$S=\left\{(\text{x},\text{y})\in {\mathbb{P}}^3\times {\mathbb{P}}^3|M(\text{x})\cdot {\text{y}}^t=0^t\right\}$$

$$m_{ij}(\text{x})=a_{ij}^1{x}_1+a_{ij}^2{x}_2+a_{ij}^3{x}_3+a_{ij}^4{x}_4.$$

$$\sum _{k,j=1}^4{a}_{ij}^k{x}_k{y}_j=\sum _{j=1}^4\left(\sum _{k=1}^4{a}_{ij}^k{x}_k\right)y_j=\sum _{k=1}^4\left(\sum _{j=1}^4{a}_{ij}^k{y}_j\right)x_k,$$

$$M(\text{x})\cdot {\text{y}}^t=N(\text{y})\cdot {\text{x}}^t$$

$$n_{ik}(\text{y})=a_{i1}^k{y}_1+a_{i2}^k{y}_2+a_{i3}^k{y}_3+a_{i4}^k{y}_4.$$

$$S_1=\left\{\text{x}\in {\mathbb{P}}^3|\text{det}(M(\text{x}))=0\right\}\text{and}{S}_2=\left\{\text{y}\in {\mathbb{P}}^3|\text{det}(N(\text{y}))=0\right\}.$$

$$S_1\cong {\tilde{S}}_1=S={\tilde{S}}_2\cong S_2.$$

$$X$$

Full text

Introduction

Let $D_{1}$ and $D_{2}$ be two smooth hypersurfaces of degree $d$ in $\mathbb{P}^{n}$ which are isomorphic as abstract varieties. It is then a natural question if an isomorphism between them can be obtained by restricting an automorphism or a Cremona transformation of the ambient $\mathbb{P}^{n}$.

We say $D_{1}$ and $D_{2}$ are projectively equivalent if there is an automorphism $g:\mathbb{P}^{n}\rightarrow\mathbb{P}^{n}$ which restricts to an isomorphism $D_{1}\rightarrow D_{2}$. We say $D_{1}$ and $D_{2}$ are Cremona isomorphic if there is a Cremona transformation $f:\mathbb{P}^{n}\dashrightarrow\mathbb{P}^{n}$ such that the restriction extends to an isomorphism $D_{1}\rightarrow D_{2}$.

Assume $n\geq 3$ and $(n,d)\neq(3,4)$, then by a result of Matsumura and Monsky, [MM64], $D_{1}$ and $D_{2}$ are projectively equivalent (so especially Cremona isomorphic) if they are isomorphic as abstract varieties.

Oguiso showed in [Ogu17] that in the special case $(n,d)=(3,4)$ this result is no longer true, by giving an example of two smooth quartic K3 surfaces in $\mathbb{P}^{3}$ which are Cremona isomorphic but which are not projectively equivalent.

The main observation of this note is that the Cremona transformation $f:\mathbb{P}^{3}\dashrightarrow\mathbb{P}^{3}$ in Oguiso’s example is the classical cubo-cubic transformation of $\mathbb{P}^{3}$, see [Noe71] or [SR49].

In this note we work over the field $\mathbb{C}$ and the note consists of three sections. In the first section we recall two constructions of the cubo-cubic Cremona transformation. The second section contains a summary of Oguiso’s results. In the third section we combine the first two sections and show how to understand Oguiso’s example in terms of the cubo-cubic transformation.

1. The cubo-cubic transformation

Let $C$ be a general smooth irreducible curve of genus 3 and degree 6 in $\mathbb{P}^{3}$. The homogeneous ideal of $C$ is generated by four cubic polynomials $f_{i}\in\mathbb{C}[\textit{{x}}]$, which are the $3\times 3$ minors of a general $3\times 4$ matrix $A(\textit{{x}})$ of linear forms, see e.g. [Ell75, Exemples 2], here $\textit{{x}}=\left[x_{1},x_{2},x_{3},x_{4}\right]$ are the homogeneous coordinates on $\mathbb{P}^{3}$. These polynomials define a rational map:

[TABLE]

It follows from [Dol12, Theorem 7.2.4] that this map is in fact birational with base scheme $C$ and has multidegree $(3,3)$. Furthermore the map it is resolved by the graph $\Gamma_{\varphi}$ of $\varphi$ in $\mathbb{P}^{3}\times\mathbb{P}^{3}$, that is we have the following diagram:

[TABLE]

where $\pi_{i}:\Gamma_{\varphi}\rightarrow\mathbb{P}^{3}$ is the $i$-th projection and $\varphi=\pi_{2}\circ\pi_{1}^{-1}$.

The graph $\Gamma_{\varphi}$ is the intersection of three general divisors of bidegree $(1,1)$ in $\mathbb{P}^{3}\times\mathbb{P}^{3}$. These divisors are defined by the 3 rows of the matrix $A(\textit{{x}})$. As the multidegree is $(3,3)$ the birational map $\varphi$ 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 and degree 6 in $\mathbb{P}^{3}$ and denote the blow up of $C$ in $\mathbb{P}^{3}$ by $X$. Then $\text{Pic}(X)=\mathbb{Z}\left[H\right]\oplus\mathbb{Z}\left[E\right]$, where $H$ is the pullback of a hyperplane in $\mathbb{P}^{3}$ to $X$ via the blow up morphism $\sigma:X\rightarrow\mathbb{P}^{3}$ and $E$ is the exceptional divisor of $\sigma$. The linear system $|3H-E|$ defines a morphism

[TABLE]

which can be shown to be of degree 1, hence $\Psi$ is birational. The morphism $\Psi$ contracts the strict transform $F$ of the trisecant surface $Sec_{3}(C)$ with respect to $\sigma$ to a curve $C^{\prime}\subset\mathbb{P}^{3}$ with $C\cong C^{\prime}$.

We get a birational $\phi$ map from the following diagram:

[TABLE]

We can see that $\phi=\varphi$ by noting that the graph $\Gamma_{\varphi}$, i.e. the intersection of the three divisors of bidegree $(1,1)$ in $\mathbb{P}^{3}\times\mathbb{P}^{3}$, is the same as the blow-up of the curve $C$, see for example [CCGK16, H.29, Lemma E.1] or [Dol12, 7.1.3].

Remark 1.1*.*

The cubo-cubic Cremona transformation and most of its properties were already known to Max Noether, see [Noe71, §2].

Remark 1.2*.*

The cubo-cubic transformation is special in the sense that it is the only non-trivial Cremona transformation of $\mathbb{P}^{3}$ that is resolved by just one blow up along a smooth curve, see [Kat87, Proposition 2.1].

2. Oguiso’s example

In [Ogu17, Theorem 1.5.] Oguiso constructs two quartic K3 surfaces $S_{1}$ and $S_{2}$ in $\mathbb{P}^{3}$ and a Cremona transformation $\tau:\mathbb{P}^{3}\dashrightarrow\mathbb{P}^{3}$ such that $\tau$ restricts to a birational map $S_{1}\dashrightarrow S_{2}$. This map must be an isomorphism as the canonical divisor of a K3 surface is nef. But these two surfaces are not projectively equivalent in $\mathbb{P}^{3}$, that is there is no $g\in\text{Aut}(\mathbb{P}^{3})=\text{PGL}(4,\mathbb{C})$ with $g(S_{1})=S_{2}$.

The K3 surfaces are constructed as follows: pick three general divisors $Q_{1}$, $Q_{2}$ and $Q_{3}$ of bidegree $(1,1)$ in $\mathbb{P}^{3}\times\mathbb{P}^{3}$, then

[TABLE]

is a smooth Fano threefold, birational to $\mathbb{P}^{3}$ in two different ways, given by restricting the projections $\pi_{i}:\mathbb{P}^{3}\times\mathbb{P}^{3}\rightarrow\mathbb{P}^{3}$ to $V$, $i=1,2$. Call these morphisms $p_{1}$ and $p_{2}$. Oguiso defines the birational map $\tau:\mathbb{P}^{3}\dashrightarrow\mathbb{P}^{3}$ via:

[TABLE]

This defines a non-trivial Cremona transformation of $\mathbb{P}^{3}$, [Ogu17, Theorem 1.5.(2)].

For a very general divisor $Q_{4}$ of bidegree $(1,1)$, avoiding the Noether-Lefschetz locus, define:

[TABLE]

Then $S$ is a smooth K3 surface with $\text{NS}(S)=\mathbb{Z}h_{1}\oplus\mathbb{Z}h_{2}$ with intersection matrix

[TABLE]

The restrictions of the $p_{i}$ for $i=1,2$, i.e. ${p_{i}}_{|S}:S\rightarrow p_{i}(S)$, turn out to be isomorphisms given by the complete linear systems $|h_{i}|$. We get two quartic K3 surfaces $S_{i}:=p_{i}(S)\subset\mathbb{P}^{3}$.

The main result of Oguiso regarding these two surfaces is:

Theorem 2.1**.**

[Ogu17, Theorem 1.5.(3)]** The K3 surfaces $S_{1}$ and $S_{2}$ are Cremona isomorphic but not projectively equivalent in $\mathbb{P}^{3}$.

Remark 2.2*.*

The reason why $S_{1}$ and $S_{2}$ are not projectively equivalent is that there is no $f\in\text{Aut}(S)$ such that $f^{*}h_{1}=h_{2}$, see [Ogu17, Proposition 6.2., Lemma 6.4.].

The whole construction is captured in the following diagram:

[TABLE]

Remark 2.3*.*

The K3 surfaces $S_{1}$ and $S_{2}$ are determinantal quartic K3 surfaces and were already 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 example in terms of a cubo-cubic transformation. To do this, pick a general smooth irreducible curve $C$ of genus 3 and degree 6 in $\mathbb{P}^{3}$, with $3\times 4$ matrix $A(\textit{{x}})$ and write

[TABLE]

with $A_{k}=(a_{ij}^{k})_{i,j}\in\text{Mat}(3,4,\mathbb{C})$. This defines three $4\times 4$ matrices $B_{i}=(a_{ij}^{k})_{j,k}$ for $i=1,2,3$ where $k$ is the index for the columns and $j$ is the index for the rows.

The three divisors $Q_{1}$, $Q_{2}$ and $Q_{3}$ of bidegree $(1,1)$ induced by $A(\textit{{x}})$ (the equations of the graph of the cubo-cubic transformation induced by $C$) are :

[TABLE]

We choose a very general fourth divisor $Q_{4}$ of bidegree $(1,1)$ as described above:

[TABLE]

Thus $S=Q_{1}\cap Q_{2}\cap Q_{3}\cap Q_{4}\subset\mathbb{P}^{3}\times\mathbb{P}^{3}$ is given by

[TABLE]

for the $4\times 4$ matrix $M(\textit{{x}})=\left(m_{ij}(\textit{{x}})\right)_{i,j}$ with entries:

[TABLE]

The observation

[TABLE]

implies the following identity:

[TABLE]

with the $4\times 4$ matrix $N(\textit{{y}})=\left(n_{ik}(\textit{{y}})\right)_{i,k}$ given by

[TABLE]

Oguiso proved in [Ogu17, Proposition 4.1.] that

[TABLE]

By construction the first three rows of $M(\textit{{x}})$ are those of $A(\textit{{x}})$. Using the Laplace expansion with respect to the last row shows that for every $\textit{{x}}\in C$ we have $\text{det}(M(\textit{{x}}))=0$ and hence $C\subset S_{1}$.

Similarly the first three rows of $N(\textit{{y}})$ define a general $3\times 4$ matrix $A^{\prime}(\textit{{y}})$ whose $3\times 3$ minors give rise to a genus 3 and degree 6 curve $C^{\prime}$ with $C^{\prime}\subset S_{2}$.

The blow up $X$ of $C$ in $\mathbb{P}^{3}$ is given by $Q_{1}\cap Q_{2}\cap Q_{3}$, which by (1) is also the blow up of $C^{\prime}$ in $\mathbb{P}^{3}$ and $C\cong C^{\prime}$ by [Kat87, Proposition 1.3.].

Denote by $\widetilde{S}_{1}$ the strict transform of $S_{1}$ in the blow up $X$ of $C$ in $\mathbb{P}^{3}$ and by $\widetilde{S}_{2}$ the strict transform of $S_{2}$ in the blow up of $C^{\prime}$. Using Oguiso’s results and the fact that $C$ and $C^{\prime}$ are smooth we get

[TABLE]

Finally we look at the cubo-cubic Cremona transformation given by the curve $C$:

[TABLE]

By the previous results the strict transform $\widetilde{S}_{1}$ with respect to $\sigma$ of the K3 surface $S_{1}$ in $X$ is the K3 surface $S$ and it is also equal to the strict transform $\widetilde{S}_{2}$ with respect to $\Psi$ of the K3 surface $S_{2}$.

This implies that the birational map $\phi$ restricts to a birational map between $S_{1}$ and $S_{2}$ which extends to an isomorphism, because $S_{1}$ and $S_{2}$ are K3 surfaces, which have nef canonical divisors. Hence $S_{1}$ and $S_{2}$ are Cremona isomorphic in $\mathbb{P}^{3}$ and the Cremona isomorphism is induced by the cubo-cubic transformation given by the curve $C$. But by Theorem 2.1, they are not projectively equivalent in $\mathbb{P}^{3}$.

Remark 3.1*.*

The construction of smooth determinantal quartic surfaces in $\mathbb{P}^{3}$ containing a curve of genus $3$ and degree $6$ can also be found in Beauville’s paper [Bea00, 6.7]. In fact Beauville proves: a smooth quartic surface in $\mathbb{P}^{3}$ is determinantal if and only if contains a nonhyperelliptic curve of genus 3 embedded in $\mathbb{P}^{3}$ by a linear system of degree 6, see [Bea00, Corollary 6.6].