The two extremal rays of some Hyper-K\"ahler fourfolds

Bert van Geemen; Federica Galluzzi

The two extremal rays of some Hyper-K\"ahler fourfolds

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Pith's one-line read A Hyper-Kähler fourfold of K3[2] type with two divisorial contractions has one of only seven possible pairs of contraction types.

desk verdict A complete, explicit classification of pairs of divisorial contractions on K3[2] fourfolds; the main theorem is sound and the tables are useful. read the letter →

arxiv 2501.09578 v2 pith:AMBEC5F6 submitted 2025-01-16 math.AG

Federica Galluzzi , Bert Van Geemen This is my paper · ORCID

classification math.AG MSC 14J4214J2814E3011D09

keywords Hyper-KählerfourfoldsK3[2]typedivisorialcontractionsconicbundlesmovableconePellequationFanovarietyofcubicfourfoldlatticeembeddings

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

This paper answers, for Hyper-Kähler fourfolds of K3[2] type with Picard rank two, which pairs of divisorial contractions can coexist. The contractions contract conic bundles over K3 surfaces, and prior work lists five types, conventionally H, M1, M3, B0, and B1. The paper proves that when a fourfold has two contractions and $|\det(\mathrm{Pic}(X))|$ is not a square, the pair must be one of exactly seven combinations, not the a priori fifteen. Five of these are the double types T+T, and the only mixed pairs are H+M3 and H+B1, with parity conditions dictating when the mixed pairs occur. The result pins down the second extremal ray by a single Pell equation and gives the concrete lists for Hilbert squares and Fano varieties of cubic fourfolds.

What carries the argument

The argument runs on the rank-two Picard lattice $\mathrm{Pic}(X) = \mathbb{Z}H \oplus \mathbb{Z}\tau$ with $q(H) = 2d$, $q(\tau) = -2$, whose primitive embedding into the cohomology lattice $\Lambda_{\mathrm{K3}^{[2]}}$ has five isometry classes. Each class is a type of exceptional divisor, a conic bundle over a K3 surface: H, M1, M3, B0, B1. The second extremal ray is forced by the minimal positive solution $(a,b)$ of the Pell equation $x^2 - d y^2 = 1$: the other contraction is defined by $H' = aH - bd\tau$ and $\tau' = bH - a\tau$. The divisibilities of $\tau$ and $\tau'$ in $H^2(X,\mathbb{Z})$, together with the residue of $d$ modulo 4 and the parity of $a$ and $b$, select which of the five types the two rays have. A proposition drawn from [Dr] certifies that any conic bundle embedded in a fourfold appears as the exceptional fiber of a divisorial contraction, which is how the tables for Hilbert squares and Fano fourfolds are read.

What would settle it

Exhibit, or find by lattice search, a projective K3[2]-type fourfold of Picard rank two with a divisorial contraction, $|\det(\mathrm{Pic}(X))|$ not a square, and a pair of extremal-ray types not among M1+M1, B0+B0, H+H, B1+B1, H+B1, M3+M3, H+M3 — for instance H+M1 — or an H+B1 pair with $d \equiv 0 \pmod 8$ and $b$ even; either would refute Theorem 3.1.

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Extended reading notes

Core claim

Let $X$ be a projective Hyper-Kähler fourfold of K3[2] type with Picard rank two, admitting a divisorial contraction, and with $|\det(\mathrm{Pic}(X))|$ not a square. Then the movable cone has two extremal rays that both define divisorial contractions, and the pair of types is exactly: M1+M1 when $|\det(\mathrm{Pic}(X))| \equiv 1 \pmod 4$; B0+B0 when the discriminant group of the transcendental lattice is not cyclic; and, when that group is cyclic with $|\det(\mathrm{Pic}(X))| = 4d$, H+H, B1+B1, M3+M3, H+B1, or H+M3 according to $d$ modulo 4, the divisibilities of the two $(-2)$-classes, and the parity of the minimal Pell solution $(a,b)$. The mixed pair H+B1 occurs exactly when $d \equiv 0 \pmod 8$ and $b$ is odd, and H+M3 occurs exactly when $a$ is even. The theorem turns the apparent fifteen possibilities into seven.

Load-bearing premise

The whole list rests on the earlier five-type classification of embeddings of $\langle 2d\rangle \oplus \langle -2\rangle$ into $\Lambda_{\mathrm{K3}^{[2]}}$ being complete and on the explicit embedding representatives quoted from [vGK] being correct; if a sixth embedding type exists or one representative is wrong, a pair could be missing from the seven cases.

Editorial extensions

If this is right

For a Hilbert square $S^{[2]}$ with $S$ of degree $e$, one contraction is always of type H, and the second ray's type is H, M3, or B1; the paper's Table 2 lists these types for $e = 2,\dots,32$.
For Fano varieties of cubic fourfolds with rank-two Picard lattice, only four pairs occur: H+H, H+M3, M3+M3, and B1+B1.
In the mixed cases H+M3 and H+B1, the fourfold is birational to a Hilbert square $S^{[2]}$ for a K3 surface $S$.
When the determinant condition forces an isotropic second ray instead, the fourfold has a Lagrangian fibration rather than a second contraction; the non-square assumption in the theorem is exactly what rules this out.
In the degree-14 cubic fourfold case, the second exceptional divisor of type M3 is identified with the known scroll over the K3 surface, and the paper presents this identification as new.

Reading between the lines

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

Editorial extension: the same Pell-equation mechanism likely extends to contractions on higher-dimensional K3$^{[n]}$-type manifolds with Picard rank two, though the conic-bundle classification used here is specific to fourfolds.
Editorial extension: the parity criteria suggest a closed congruence rule for Hilbert squares, where the second type should change with $e \bmod 8$ and the parity of the minimal Pell solution; checking all admissible $e$ would be a quick computational confirmation.
Editorial extension: the seven-case list can be stress-tested by an exhaustive lattice search over primitive rank-two sublattices of $\Lambda_{\mathrm{K3}^{[2]}}$ using the five embedding normal forms, recomputing each second ray from the Pell solution; any pair outside the list would expose a missing case.

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Formalized claims in Lean

Claim #1: Let $X$ be a projective Hyper-Kähler fourfold of K3[2] type with Picard rank two, admitting a divisorial contraction, and with $|\det(\mathrm{Pic}(X))|$ not a square. Then the movable cone has two extremal rays that both define divisorial contractions, and the pair of types is exactly: M1+M1 when $|\det(\mathrm{Pic}(X))| \equiv 1 \pmod 4$; B0+B0 when the discriminant group of the transcendental la

/-- @claim 1 Let $X$ be a projective Hyper-Kähler fourfold of K3[2] type with Picard rank two, admitting a divisorial contraction, and with $|\det(\mathrm{Pic}(X))|$ not a square. Then the movable cone has two extremal rays that both define divisorial contractions, and the pair of types is exactly: M1+M1 when $|\det(\mathrm{Pic}(X))| \equiv 1 \pmod 4$; B0+B0 when the discriminant group of the transcendental la -/
def central_claim : Prop :=

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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

Bert — the short version: this paper does what it says. It classifies all possible pairs of extremal rays that define divisorial contractions on a K3[2] fourfold with Picard rank two and non-square discriminant. The list is short — seven cases — and the proof is explicit. I went through the delicate parity step in case (c) of Theorem 3.1 and it checks out; the B1 embedding normal form quoted from [vGK] gives the claimed divisibility condition.

What is genuinely new here is the pair classification, not the individual contraction types. The five types were already in [vGK] and Debarre–Macrì. The new content is the seven-case list, the parity conditions that govern the mixed cases H+M3 and H+B1, and the restriction to four pairs for Fano varieties of cubic fourfolds. Tables 2 and 3 are concrete and consistent with the Pell data — I spot-checked several rows. For the Hilbert square table, the conditions reduce to parity of b, and the e=16 example (H+B1, BOSS bundle) is a nice touch.

The main soft spot is the same one the reader flagged: Theorem 3.1 takes as input the completeness of the five-type classification of embeddings of <2d> + <-2> into Λ_K3[2], quoted from [vGK], a paper co-authored by the second author. That is a real dependency, but it is not a hidden one — the authors state it plainly and [vGK] is published and peer-reviewed. The present paper does not reprove the embedding classification, which is fine; it would be a lot of extra length for little gain. The e=30 example depends on an external conjecture, but the authors flag it and it is not needed for the main theorem. Minor complaint: the wording around Theorem 4.4(d) is a bit tangled, but the argument is clear enough.

I did not find a load-bearing error. The Pell equation argument in Proposition 2.4 is clean, and the use of [BM] for the movable cone is standard. The paper is aimed at people working on hyperkähler birational geometry or cubic fourfolds; they will want these tables.

Recommendation: send to a serious referee. This deserves a proper peer review, and it should be accepted after minor revision — the only real request I would make is that the dependence on [vGK] be stated even more explicitly in the introduction, and that the e=30 conjecture dependence be kept visibly separate, which it already is.

Referee Report

0 major / 8 minor

Summary. This paper classifies the possible types of the two extremal rays of the movable cone of a projective hyper-Kähler fourfold of K3[2] type with Picard rank two, under the assumptions that the fourfold admits a divisorial contraction and that the determinant of the Picard lattice is not a square. Theorem 3.1 shows that the pair of types is one of M1+M1, B0+B0, H+H, B1+B1, H+B1, M3+M3, or H+M3, with parity conditions on the minimal Pell solution governing the mixed cases. The proof reduces each case to divisibility computations in explicit lattice embeddings quoted from Debarre–Macrì and van Geemen–Kapustka. The paper then specializes to Hilbert squares of K3 surfaces and to Fano varieties of cubic fourfolds, providing tables of examples.

Significance. The result is a useful, explicit classification with a clear computational structure. The reduction of the geometry to the minimal solution of a Pell equation is elegant, and the tables for S[2] and Fano fourfolds will be a convenient reference. I checked the load-bearing computations in the proof of Theorem 3.1, including the B1 embedding τ = (1,-d)_1 + 2(1,d/4)_2 + δ: since the two copies of U are orthogonal, q(τ) = -2d + 2d - 2 = -2, and the parity criterion for the divisibility of τ' = bH - aτ is correct. The main caveat is that the classification is conditional on the completeness and correctness of the five-type classification and the embedding normal forms in [vGK] and [DM]; these are cited but not reproved here. That dependency is explicit and does not, in my assessment, affect the internal soundness of the paper.

minor comments (8)

[Table 3] In the row e = 74, the entry (6,1) in the (a,b) column solves x^2 - 37 y^2 = -1, not the positive Pell equation x^2 - 37 y^2 = 1; the minimal positive solution is (73,12). Since no contractions occur for e = 74, the entry is not used in the analysis, but it should be corrected or clearly labeled as a negative-Pell solution.
[Abstract and Introduction] The phrase "There are exactly seven cases" could be read as asserting that each of the seven pairs occurs. Theorem 3.1 establishes that these are the only possible pairs; if realizability of every case is not intended, I suggest rephrasing to "at most seven cases" or adding a sentence clarifying which cases are known to occur.
[Theorem 4.4(e)] The proof uses the symmetry (x,y) ↦ (x,-y) of the Picard lattice and concludes that the other extremal ray has the same type. It should be stated explicitly that this conclusion concerns the situation where both extremal rays are divisorial contractions; for e ≡ 0 mod 6 this is automatic when a (-2)-class exists (as shown in the proof of part (d)), but the point is not stated.
[Section 4.4(d)] There is a typo in "equvalently", which should be "equivalently".
[Section 1.5] The expression "H62(X,Z)" is a typo for "H^2(X,Z)".
[Section 1.6] The notation "Pic( X) ⁄= ZH ⊕ Zτ" should use the standard symbol "≠".
[Table 2] The heading "Divisorial contractions on a Hilbert square of S[2]" is redundant; it should be "Divisorial contractions on S[2]".
[Proposition 2.4] In the proof, the display "τ 3H′" should be typeset as τ^3 H′ for clarity.

Circularity Check

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No circularity: Theorem 3.1's seven-case list follows from external lattice-embedding results and explicit Pell/divisibility computations, not from its own conclusion.

full rationale

The claimed derivation is not circular. Theorem 3.1 is proved from the standard movable-cone description, Pell-equation arithmetic, and the five-type embedding classification of Debarre-Macrì and van Geemen-Kapustka. Proposition 2.4 derives the second extremal ray as τ' = bH − aτ and H' = aH − bdτ from the Pell equation; this is an independent computation, not an assumption of the target seven-case list. In cases (c) and (e) the proof computes the divisibility of τ' from explicit representatives quoted from [vGK, Props. 3.4, 3.5, 3.8]; those representatives are prior published, parameter-free classification data whose stated assumptions (embeddings of <2d> ⊕ <−2> into Λ_K3[2]) do not include the seven-case conclusion. Although [vGK] is co-authored by van Geemen, the citation is independent external evidence rather than a self-referential loop. The theorem's assumption that |det(Pic(X))| is not a square only excludes a Lagrangian fibration for the second extremal ray; the conclusion that both extremal rays define contractions follows from the cited movable-cone results and is not a restatement of the input. The one legitimate risk is correctness or verification of the quoted embedding normal forms and the completeness of the five-type list; that is a mathematical correctness concern, not circularity. No equation in the paper is equivalent by construction to the theorem's conclusion.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

The paper introduces no free parameters fitted to data and no invented entities. The central results rest on four prior results: the five-type classification, the explicit embedding representatives, the conic-bundle realization of exceptional divisors, and the movable-cone boundary description; these are listed as domain assumptions because they are quoted rather than proved in this paper.

assumptions (4)

domain assumption The five-type classification of embeddings of <2d>+<-2> into Lambda_K3[2], and the induced five types of divisorial contractions H, M1, M3, B0 and B1, is complete.
Quoted from [DM], [De, Thm. 3.32] and [vGK, Table 1]; the paper builds on this classification rather than reproving it.
domain assumption The explicit embedding normal forms for types H, B1 and M3 in [vGK, Props. 3.4, 3.5 and 3.8] are correct.
These representatives are used in the proof of Theorem 3.1(c,e) to translate parity of Pell solutions into divisibility of tau and tau'.
domain assumption For a Picard rank two K3[2] fourfold with a divisorial contraction, the exceptional divisor is a conic bundle over a K3 surface and the contraction is one of the five classified types.
Taken from [vGK, Thm. 3.3] and [BM]; this is the bridge from lattice classification to the geometry of conic bundles.
domain assumption The movable cone description of Markman and Debarre applies: a (-2)-class boundary ray gives a divisorial contraction, an isotropic boundary ray gives a Lagrangian fibration, and the minimal Pell solution gives the neighboring extremal ray.
Used in Section 2 and Proposition 2.4; cited to [Ma1], [Ma2] and [De, Thm. 3.16].

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Pith. "Pith review of The two extremal rays of some Hyper-K\"ahler fourfolds." pith.science (2026). https://pith.science/paper/AMBEC5F6

@misc{pith2026250109578,
  author       = {Pith},
  title        = {Pith review of: The two extremal rays of some Hyper-K\"ahler fourfolds},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/AMBEC5F6}},
  note         = {Machine review of arXiv:2501.09578}
}

read the original abstract

We consider projective Hyper-K\"ahler manifolds of dimension four that are deformation equivalent to Hilbert squares of K3 surfaces. In case such a manifold admits a divisorial contraction, the exceptional divisor is a conic bundle over a K3 surface. A classification of lattice embeddings implies that there are five types of such conic bundles. In case the manifold has Picard rank two and has two (birational) divisorial contractions we determine the types of these conic bundles. There are exactly seven cases. For the Fano varieties of cubic fourfolds there are only four cases and we provide examples of these.

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Works this paper leans on

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Pith tools

Reviewed August 10, 2026 · model on record in the stance chip above.

[1] [2]

N.\ Addington, R.\ Thomas, Hodge theory and derived categories of cubic fourfolds , Duke Math.\ J.\ 163 (2014) 1885--1927

work page 2014

[2] [1]

N.\ Addington, On two rationality conjectures for cubic fourfolds , Math.\ Res.\ Lett.\ 23 (2016) 1--13

work page 2016

[3] [3]

A.\ Alzati, F.\ Russo, Some extremal contractions between smooth varieties arising from projective geometry , Proc.\ London Math.\ Soc.\ 89 (2004) 25–53

work page 2004

[4] [4]

A.\ Bayer, B.\ Hassett, Y.\ Tschinkel, Mori cones of holomorphic symplectic varieties of K3 type , Ann.\ Sci.\ \'Ec.\ Norm.\ Sup\'er.\ 48 (2015) 941--950

work page 2015

[5] [5]

A.\ Bayer, A.\ Kuznetsov, E.\ Macr\`i, Mukai models of Fano varieties , arXiv:2501.16157

arXiv

[6] [6]

MMP for moduli of sheaves on K3s via wall-crossing: nef and movable cones, Lagrangian fibrations , Invent.\ Math.\ 198 (2014) 505--590

A.\ Bayer, E.\ Macr\` . MMP for moduli of sheaves on K3s via wall-crossing: nef and movable cones, Lagrangian fibrations , Invent.\ Math.\ 198 (2014) 505--590

work page 2014

[7] [7]

I Math.\ 301 (1985), no

A.\ Beauville, R.\ Donagi, La vari\'et\'e des droites d'une hypersurface cubique de dimension 4 , C.R.\ Acad.\ Sci.\ Paris S\'er. I Math.\ 301 (1985), no. 14, 703--706

work page 1985

[8] [8]

M.\ Beltrametti, M.\ Schneider, A.J.\ Sommese, Threefolds of degree 9 and 10 in P^5 , Math.\ Ann.\ 288 (1990) 413–-444

work page 1990