The cubo-cubic transformation and K3 surfaces

Fabian Reede

The cubo-cubic transformation and K3 surfaces

T0 review · 0 major / 4 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read This note identifies the Cremona map in a known pair of quartic K3 surfaces with the classical cubo-cubic transformation of projective 3-space.

desk verdict A clean, useful identification of Oguiso's Cremona map as the classical cubo-cubic transformation; worth a serious referee. read the letter →

arxiv 1908.05548 v1 pith:XHLBWEBK submitted 2019-08-15 math.AG

Fabian Reede This is my paper · ORCID

classification math.AG MSC 14E0514E0714J2814M12

keywords CremonatransformationsK3surfacesdeterminantalhypersurfacescubo-cubictransformationquarticbirationalgeometrygenus-3degree-6curveprojective3-space

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper claims that the Cremona transformation appearing in a well-known example of two quartic K3 surfaces in projective 3-space is not a new or exotic map: it is the classical cubo-cubic transformation, already studied in the nineteenth century. The cubo-cubic transformation is the birational self-map of $\mathbb{P}^3$ obtained by blowing up a smooth genus-3 degree-6 curve and then contracting the proper transform of its trisecant surface. The paper shows that the threefold obtained from three general bidegree-$(1,1)$ divisors — the graph of the cubo-cubic map — is exactly the threefold used to construct the two K3 surfaces, and that the two quartic surfaces are the two determinantal hypersurfaces $\det(M(x))=0$ and $\det(N(y))=0$. A sympathetic reader should care because this collapses a seemingly special counterexample to projective equivalence into a classical construction, and it explains why the two surfaces are determinantal quartics.

What carries the argument

The load-bearing object is the graph of the cubo-cubic transformation: the intersection $Q_1\cap Q_2\cap Q_3$ of three divisors of bidegree $(1,1)$ in $\mathbb{P}^3\times\mathbb{P}^3$, which is also the blow-up of $\mathbb{P}^3$ along a genus-3 degree-6 curve $C$. The map is defined by the $3\times3$ minors of a $3\times4$ matrix $A(x)$ of linear forms, and the bilinear identity $M(x)y^t=N(y)x^t$ makes the two projections of the graph symmetric. This symmetry is what lets the single surface $S$ serve simultaneously as the strict transform of $S_1$ under one projection and the strict transform of $S_2$ under the other.

What would settle it

Compute the strict transform inside the blow-up $X=\operatorname{Bl}_C\mathbb{P}^3$ of the quartic $S_1=\{\det(M(x))=0\}$ and compare it with the preimage of $S_2=\{\det(N(y))=0\}$ under the second projection. If the two surfaces in $X$ differ at even one smooth point, the paper's identification fails; the paper predicts they coincide as the same K3 surface $S$.

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Extended reading notes

Core claim

The central claim is that the birational map $\tau$ from the earlier example is the cubo-cubic transformation $\varphi$ associated to a general smooth genus-3 degree-6 curve $C\subset\mathbb{P}^3$. Precisely: if the graph of $\varphi$ is $Q_1\cap Q_2\cap Q_3\subset\mathbb{P}^3\times\mathbb{P}^3$, and a very general fourth bidegree-$(1,1)$ divisor $Q_4$ is added, then $S=Q_1\cap Q_2\cap Q_3\cap Q_4$ is a K3 surface whose two projections give the two quartic surfaces $S_1=\{\det M(x)=0\}$ and $S_2=\{\det N(y)=0\}$. Because the identity $M(x)y^t=N(y)x^t$ holds, the strict transform of $S_1$ in the blow-up of $C$ is the same as the strict transform of $S_2$ in the blow-up of the curve $C'$; hence the cubo-cubic map restricts to a birational map $S_1\dashrightarrow S_2$ that extends to an isomorphism. This identifies the previously constructed Cremona isomorphism with the classical cubo-cubic transformation.

Load-bearing premise

The argument's load-bearing premise is that, inside the blow-up of $\mathbb{P}^3$ along the curve $C$, the strict transform of the first quartic surface is exactly the same surface $S$ as the strict transform of the second quartic surface; if those two strict transforms differed, the cubo-cubic map would not restrict to an isomorphism between the two quartics.

Editorial extensions

If this is right

The two quartic K3 surfaces in the example are determinantal quartic surfaces, cut out by the $4\times4$ determinants $\det(M(x))=0$ and $\det(N(y))=0$.
The base curve $C$ of genus 3 and degree 6 lies on $S_1$, and its counterpart $C'$ lies on $S_2$; the cubo-cubic map contracts the trisecant surface of $C$ onto $C'$.
The isomorphism between $S_1$ and $S_2$ is realized by a Cremona transformation of the ambient $\mathbb{P}^3$, so the example sits inside the classical cubo-cubic family rather than outside it.
Because the cubo-cubic transformation is the only non-trivial Cremona map of $\mathbb{P}^3$ resolved by a single blow-up along a smooth curve, the example is special in a precise birational sense.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

If the identification is correct, the phenomenon 'Cremona isomorphic but not projectively equivalent' for quartic K3 surfaces is governed by the geometry of genus-3 degree-6 curves: different choices of $C$ should produce a family of such pairs, all linked by cubo-cubic maps.
The matrix identity suggests a symmetric construction: any $4\times4$ matrix of linear forms whose first three rows define a smooth genus-3 degree-6 curve gives an $S_1$, and the transposed construction gives $S_2$; testing whether a generic such matrix yields non-projectively equivalent surfaces would extend the example.
This connects the example to the classical theory of determinantal quartic surfaces: since a smooth quartic is determinantal exactly when it contains a nonhyperelliptic genus-3 curve of degree 6, the cubo-cubic construction may be the geometric source of that determinantal structure.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Dear colleague,

Read Reede's note on the cubo-cubic transformation and Oguiso's K3 surfaces. The one thing to know: this is a short observation, but it is correct and it actually explains something. Oguiso constructed a Cremona transformation tau between two quartic K3 surfaces that are not projectively equivalent. Reede shows tau is the classical cubo-cubic transformation: the three bidegree (1,1) equations cutting out Oguiso's Fano threefold V are exactly the rows of a 3x4 matrix A(x) whose 3x3 minors define a genus 3 degree 6 curve C, and V is the graph of the cubo-cubic map associated to C. The symmetric identity M(x) y^t = N(y) x^t does the work, and the strict transforms of the two K3 surfaces in the common blow-up of C are identified with the complete intersection S. That is a genuinely new connection and a nice way to see Oguiso's example not as an exotic construction but as the classical map applied to determinantal quartic surfaces.

The paper does this without new machinery. It uses standard results from Katz, Dolgachev, and Oguiso in a straightforward way, and the citation pattern is fine—no self-cites, and the external results are appropriately flagged. The mathematical core is a direct computation, not an appeal to authority.

Where are the soft spots? The glue point is the equality of the strict transforms: S1-tilde = S = S2-tilde. The paper states it in a few lines, and the justifications are spread between the matrix identity and Oguiso's smoothness. A referee might ask for a slightly more explicit argument here, but it is not a gap. The stress-test's reasoning—det M(x) = 0 away from C, the first three rows determine y, so the strict transforms agree on a dense open set, then the divisor class argument in the smooth threefold X—checks out. The paper could include that in one sentence, but its absence is a minor presentational issue, not a flaw.

The non-projective equivalence of S1 and S2 is imported from Oguiso, so the note does not reprove the negative side of the story. That is the right division of labor, but readers should know the exoticness of the example still lives in Oguiso's paper. The note explains the positive side: which Cremona map works.

Who is this for? People who work with Cremona transformations, determinantal quartic surfaces, or Oguiso's example. It is a useful bridge between classical algebraic geometry and a modern counterexample. The paper deserves a serious referee and likely acceptance with minor comments.

Referee Report

0 major / 4 minor

Summary. The paper observes that the Cremona transformation τ in Oguiso's example of two smooth quartic K3 surfaces in P3 that are Cremona isomorphic but not projectively equivalent is, in fact, the classical cubo-cubic transformation. After recalling two standard presentations of the cubo-cubic transformation (via the graph of a general genus-3 degree-6 curve in P3, and via the blow-up of P3 along that curve), the author shows that Oguiso's threefold V = Q1 ∩ Q2 ∩ Q3 is precisely the graph of the cubo-cubic transformation defined by a matrix A(x), and that the K3 surface S = Q1 ∩ Q2 ∩ Q3 ∩ Q4 is simultaneously the strict transform of S1 and S2. Hence the birational map τ restricts to an isomorphism between S1 and S2, and by Oguiso's theorem these surfaces are not projectively equivalent.

Significance. If the identification is correct, this gives a classical-geometric interpretation of Oguiso's counterexample to the Matsumura–Monsky theorem in the exceptional case (n,d) = (3,4): the exotic Cremona isomorphism is an old object, the cubo-cubic transformation. The proof is direct and transparent: the key matrix identity M(x)·yᵗ = N(y)·xᵗ makes the identification of the strict transforms a matter of linear algebra, and the paper uses standard external results (Katz, Dolgachev, Oguiso) without circularity. The paper is short and contains no fitted parameters or heuristic steps. Its main value is conceptual: it shows that the 'exotic' phenomenon is already present in classical 19th-century birational geometry.

minor comments (4)

[Section 3] In Section 3, the matrices B_i = (a^k_{ij})_{j,k} are defined but never used; please remove them or explain their role.
[Section 3] The equality ~S1 = S = ~S2 is stated in one line; a short justification would help, e.g., that away from C the first three equations Q1 = Q2 = Q3 = 0 determine y uniquely from x, and Q4 = 0 is then equivalent to det M(x) = 0, so the strict transform of S1 coincides with S (and analogously for S2).
[Section 1] The phrase 'the linear system |3H − E| defines a morphism Ψ : X → P3 which can be shown to be of degree 1' is imprecise: the complete linear system defines a morphism (after the blow-up), and 'degree 1' should be read as 'birational onto its image'; please clarify.
[Section 3] The statement that the Laplace expansion with respect to the last row shows det(M(x)) = 0 on C is misleading: the vanishing follows immediately because the first three rows of M(x) are dependent for x ∈ C. Please rephrase.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: Oguiso's tau is identified with the cubo-cubic transformation by direct equations and standard graph constructions.

full rationale

The derivation is self-contained and non-circular. The paper fits no parameters, invokes no self-citation as load-bearing support, and does not rename a known result. Its reliance on prior work is limited to standard facts about cubo-cubic transformations (Dolgachev, Katz) and to Oguiso's theorem as an external input for the non-projective-equivalence of S1 and S2; neither constitutes a circular premise. The central identification is made by constructing the three bidegree (1,1) divisors Q1, Q2, Q3 from the rows of A(x) for a general genus-3 degree-6 curve C, so that V = Q1 ∩ Q2 ∩ Q3 is exactly the graph of the cubo-cubic transformation. The equality S1~ = S = S2~ is justified by the matrix identity M(x)·y^t = N(y)·x^t together with Oguiso's smoothness and isomorphism results, and the class computation -K_V = H1 + H2 with K_V = -4H1 + E yields H2 = 3H1 - E, forcing the same map. No step reduces to its own input by construction.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

No free parameters or invented entities appear. The argument uses standard algebraic geometry facts, a classical construction, and Oguiso's theorem as external input. The central claim is an identification, so the axioms listed are background theorems, not fitted assumptions.

assumptions (5)

standard math A general smooth genus 3 degree 6 curve C in P3 has ideal generated by the 3x3 minors of a 3x4 matrix of linear forms.
Invoked in Section 1 to define the cubo-cubic map; cited to [Ell75, Exemples 2].
standard math The graph of the map defined by those minors is the intersection of three divisors of bidegree (1,1) in P3 x P3 and has multidegree (3,3).
Used throughout Sections 1 and 3 to identify Oguiso's V with the graph; cited to [Dol12, Theorem 7.2.4].
standard math The blow-up of P3 along C resolves the cubo-cubic map and is isomorphic to the graph.
Gives the equality X = Q1 cap Q2 cap Q3; cited to [Kat87] and [Dol12, Example 7.2.6].
domain assumption Oguiso's theorem: the surfaces S1 and S2 constructed from a very general fourth divisor are Cremona isomorphic but not projectively equivalent.
The paper relies on this for the non-projectivity part and for the existence of the example; cited as [Ogu17, Theorem 1.5].
standard math K3 surfaces have nef canonical divisors, so a birational map between them extends to an isomorphism.
Used in Section 3 to pass from a birational restriction to an isomorphism.

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Cite this review

Pith. "Pith review of The cubo-cubic transformation and K3 surfaces." pith.science (2026). https://pith.science/paper/XHLBWEBK

@misc{pith2026190805548,
  author       = {Pith},
  title        = {Pith review of: The cubo-cubic transformation and K3 surfaces},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/XHLBWEBK}},
  note         = {Machine review of arXiv:1908.05548}
}

read the original 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.

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Reference graph

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Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.

[1] [1]

Determinantal hypersurfaces

Arnaud Beauville. Determinantal hypersurfaces. Michigan Math. J. , 48:39--64, 2000

work page 2000

[2] [2]

A M emoir on Q uartic S urfaces

Arthur Cayley. A M emoir on Q uartic S urfaces. Proc. Lond. Math. Soc. , 3:19--69, 1869/71

work page

[3] [3]

Quantum periods for 3-dimensional F ano manifolds

Tom Coates, Alessio Corti, Sergey Galkin, and Alexander Kasprzyk. Quantum periods for 3-dimensional F ano manifolds. Geom. Topol. , 20(1):103--256, 2016

work page 2016

[4] [4]

Dolgachev

Igor V. Dolgachev. Classical algebraic geometry - A modern view . Cambridge University Press, Cambridge, 2012

work page 2012

[5] [5]

Sur le sch\' e ma de H ilbert des vari\' e t\' e s de codimension 2 dans P e \`a c\^ o ne de C ohen- M acaulay

Geir Ellingsrud. Sur le sch\' e ma de H ilbert des vari\' e t\' e s de codimension 2 dans P e \`a c\^ o ne de C ohen- M acaulay. Ann. Sci. \' E cole Norm. Sup. (4) , 8(4):423--431, 1975

work page 1975

[6] [6]

The C ayley- O guiso automorphism of positive entropy on a K 3 surface

Dino Festi, Alice Garbagnati, Bert van Geemen, and Ronald van Luijk. The C ayley- O guiso automorphism of positive entropy on a K 3 surface. J. Mod. Dyn. , 7(1):75--97, 2013

work page 2013

[7] [7]

The cubo-cubic transformation of P ^3 is very special

Sheldon Katz. The cubo-cubic transformation of P ^3 is very special. Math. Z. , 195(2):255--257, 1987

work page 1987

[8] [8]

On the automorphisms of hypersurfaces

Hideyuki Matsumura and Paul Monsky. On the automorphisms of hypersurfaces. J. Math. Kyoto Univ. , 3:347--361, 1963/1964

work page 1963