VARIETIES OF PLANES ON INTERSECTIONS OF THREE QUADRICS

BRENDAN HASSETT AND YURI TSCHINKEL

Abstract. We study the geometry of spaces of planes on smooth complete intersections of three quadrics, with a view toward rationality questions.

1. Introduction

This note studies the geometry of smooth complete intersections of three quadrics X ⊂ ℙ^{n}, with a view toward rationality questions over nonclosed fields k. We review what is known over ℂ:

• X is irrational for n ≤ 6 [Bea77];

• X may be either rational or irrational for n = 7, and the rational ones are dense in moduli [HPT18];

• X is always rational for n ≥ 8 [Tju75, Cor. 5.1]. The analysis in higher dimensions relies on the geometry of planes in X. Indeed, when X contains a plane P defined over k then projection from P gives a birational map

π_{P} : X −−→^{∼} ℙ^{n−3}.

This leads us to study the variety of planes F_{2}(X) ⊂ Gr(3, n+1). When n ≥ 12, the geometry of these varieties gives a quick and uniform proof of rationality over finite fields and function fields of complex curves (see Theorem 1).

We are therefore interested in the intermediate cases n = 8, 9, 10, 11, and especially in n = 8 and 9. For generic X ⊂ ℙ^{8}, the variety F_{2}(X) is finite of degree 1024 (see Proposition 5); we explore the geometry of the associated configurations of planes in ℙ^{8}. We then focus most on the case n = 9. Here the variety F_{2}(X) is a threefold of general type with complicated geometry – we analyze its numerical invariants.

Our original motivation was to understand certain singular complete intersections of three quadrics in ℙ^{9} associated with universal torsors over degree 4 del Pezzo surfaces fibered in conics over ℙ^{1} [CTS87]. Conjectures of Colliot-Thélène and Sansuc predict that such torsors are rational when they admit a point, over number fields. Rationality of torsors has significant arithmetic applications, e.g., to proving the uniqueness of the Brauer–Manin obstruction to the Hasse principle and weak approximation. It has geometric consequences as well, e.g., the construction of new examples of nonrational but stably rational threefolds over ℂ. The geometry of smooth intersections, presented here, turned out to be quite rich and interesting on its own.

Date: July 2020.

Here is a road map of the paper: Section 2 presents uniform proofs of rationality for high-dimensional cases. Section 3 is devoted to determinantal presentations of plane curves arising as degeneracy loci of the net of quadrics. In Sections 4 and 5, we turn to numerical invariants of the variety of planes. Our main results concern the computation of degrees and cohomology of F_{2}(X), for X a smooth intersection of three quadrics, in ℙ^{8} and ℙ^{9}.

Acknowledgments: The first author was partially supported by NSF grants 1551514 and 1701659, and the Simons Foundation; the second author was partially supported by NSF grant 1601912. We are grateful to Nicolas Addington for suggestions on the Schubert calculus computations in Section 4 and to Borys Kadets for briefing us on his upcoming work with Hashimoto [HK20].

2. Uniform rationality in high dimensions

Theorem 1. Let k be a finite field or the function field of a complex curve. Suppose that X ⊂ ℙ^{n}, n ≥ 12, is a smooth complete intersection of three quadrics. Then X is rational.

Proof. The results of [DM98] show that for X generic over k̄, the variety F_{2}(X) ⊂ Gr(3, n + 1) is smooth and connected of the expected dimension 3n − 24. The adjunction formula implies that F_{2}(X) has canonical class

K_{F_{2}(X)} = (12 − n − 1)σ_{1};

here σ_{1} is the hyperplane class from the Plücker embedding – see Section 4 for details. In particular, F_{2}(X) is Fano for n ≥ 12.

Suppose k is a function field. When F_{2}(X) is smooth of the expected dimension, we have F_{2}(X)(k) ≠ ∅, by the Graber-Harris-Starr theorem [GHS03]. What if F_{2}(X) is singular or fails to have the expected dimension? However, we still have rational points, by an argument of Starr: Suppose that X_{∘} → B is a family of varieties over a complex projective curve B corresponding to a morphism B → Hilb to the appropriate Hilbert scheme. There exists a one-parameter family of curves C_{t}, meeting the locus over which F_{2} is smooth of the expected dimension, such that C_{0} contains B as an irreducible component. (Take the C_{t} to be complete intersection curves in Hilb.) For the induced families X_{t} → C_{t}, the fibrations F_{2}(X_{t}) → C_{t} admit sections for most t. This remains true as t → 0; restricting to B gives a section of F_{2}(X_{∘}) → B.

A similar argument holds over finite fields, using Esnault’s Theorem [Esn03] and the specialization version due to Fakhruddin and Rajan [FR05]. □

3. Determinantal representations

3.1. Recollection of invariants. We recall formulas that may be obtained from Appendix I of Hirzebruch’s Topological Methods, specifically, [Hir66, Th. 22.1.1].

Proposition 2. Suppose that n = 2m. Then we have

χ(X) = −4m(m − 1)

and

h^{m−1,m−2}(X) = h^{m−2,m−1}(X) = \binom{2m}{2} − 1 = 2m^{2} − m − 1.

The other Hodge numbers of weight 2m − 3 vanish.

Proposition 3. Suppose that n = 2m − 1. Then we have

χ(X) = 4m(m − 1)

and

h^{m−1,m−3}(X) = h^{m−3,m−1}(X) = \binom{m − 1}{2}, h^{m−2,m−2}(X) = 3m^{2}−3m+2.

The odd case follows from [O’G86] and [Las89]; the even case from [Bea77]. For reference, we give a tabulation of nontrivial middle Hodge numbers

n Hodge numbers
4  5 5
5  1 20 1
6  14 14
7  3 38 3
8  27 27
9  6 62 6
10  44 44

3.2. Interpretation of cohomology. Let X ⊂ ℙ^{n} be a smooth complete intersection of three quadrics containing a plane P. It follows that n ≥ 7 and X is of special moduli when n = 7, as the expected dimension of the Fano variety of planes is

expdim F_{2}(X) = 3(n − 2) − 3 · 6 = 3n − 24.

This case is studied in depth in [HPT18].

Projection from P induces a birational map X −−→^{∼} ℙ^{n−3}. The center Y of the inverse has dimension n − 5 and admits a fibration in quadric hypersurfaces of relative dimension n − 7 over ℙ^{2}. This is instrumental in computing and interpreting the cohomology of X.

Suppose n is odd. Then Y is fibered in even-dimensional quadrics and its cohomology is governed by the associated double cover

S → ℙ^{2},

branched over the degeneracy locus D, and the associated Brauer class η ∈ Br(S)[2], when n ≥ 9. It is possible to interpret F_{2}(X) via moduli spaces of vector bundles over S [Bho86].

When n is even, Y is fibered in odd-dimensional quadrics, degenerate over a plane curve D of degree n + 1. The cohomology is obtained via a Prym construction arising from the associated double cover of D.

3.3. Equations of the center. Given a vector space or bundle V, ℙ(V) denotes the projective space of lines in V.

In general, write

E := O_{ℙ^{n−3}}^{3} ⊕ O_{ℙ^{n−3}}(−1) ↪ O_{ℙ^{n−3}}^{n+1}

so that ℙ(E) = Bl_{P} (ℙ^{n}) with bundle morphism ϖ : ℙ(E) → ℙ^{n−3}. Quadrics containing P correspond to elements of

Γ(O_{ℙ(E)}(1) ⊗ ϖ^{∗}O_{ℙ^{n−3}}(1)) = Γ(E^{∗}(1)),

and complete intersections of three such forms correspond to linear transformations

O_{ℙ^{n−3}}^{3} → E^{∗}(1) ≃ O_{ℙ^{n−3}}(1)^{3} ⊕ O_{ℙ^{n−3}}(2).

We are interested in their degeneracy loci, whose equations are given by minors of

(1)  (\begin{matrix}L_{11} & L_{12} & L_{13} \\ L_{21} & L_{22} & L_{23} \\ L_{31} & L_{32} & L_{33} \\ Q_{1} & Q_{2} & Q_{3} \end{matrix})

where the L_{ij} are linear and the Q_{i} are quadratic; maximal minors yield one cubic equation and three quartic equations.

Let Y ⊂ ℙ^{n−3} denote the subscheme given by the maximal minors, which generically has codimension two. The (n + 1)-dimensional linear series Γ(I_{Y} (4)) induces a birational map

ℙ^{n−3} −−→ X ⊂ ℙ^{n}

inverse to π_{P}.

The determinantal equations (1) for Y yield a rational map

ϕ : Y −−→ ℙ^{2},

assigning to each rank-two matrix its one-dimensional kernel. This fails to be defined along the locus Z ⊂ Y where the matrix (1) has rank one. See Section 3.5 for more information about its geometry.

We shall analyze the cohomology of X using the cohomology of natural resolutions of Y, expressed via ϕ as quadric bundles over ℙ^{2}.

3.4. Linear algebra. We start with some linear algebra on generic m × 3 (m ≥ 3) matrices L = (L_{ij}). Consider the stratification by rank

R_{1} := ℙ^{m−1} × ℙ^{2} ⊂ R_{2} ⊂ ℙ^{3m−1}

where dim(R_{2}) = 2m + 1. Over R_{1}, we may write

L_{ij} = u_{i}v_{j}, i = 1, …, m, j = 1, 2, 3.

We obtain two small resolutions of R_{2} by keeping track of just the kernel or the image of L, respectively. For example, we may consider

R̂_{2} ⊂ ℙ_{L_{ij}}^{3m−1} × ℙ_{x_{j}}^{2}

given by the equations

L (\begin{matrix}x_{1} \\ x_{2} \\ x_{3} \end{matrix}) = 0.

The exceptional locus Ê ⊂ R̂_{2} is a ℙ^{1}-bundle over R_{1} – for each rank one matrix we extract the one-dimensional subspaces of its kernel. It has codimension m − 1 in R̂_{2}.

3.5. Generic behavior. Assume that n = 11 and the L_{ij} are linearly independent.

Consider the locus where the upper 3×3 matrix has rank at most one; here we may write L_{ij} = u_{i}v_{j}, i, j = 1, 2, 3. The remaining equations involving the Q_{i} take the form

u_{1}v_{2}Q_{1} − u_{1}v_{1}Q_{2} = u_{2}v_{2}Q_{1} − u_{2}v_{1}Q_{2} = ⋯ = 0.

Writing q_{i} = Q_{i}(u_{1}v_{1}, …, u_{3}v_{3}) and dividing by the u_{i}, we obtain

v_{2}q_{1} − v_{1}q_{2} = v_{3}q_{1} − v_{1}q_{3} = v_{2}q_{3} − v_{3}q_{2} = 0.

The indeterminacy locus for the kernel map ϕ is a surface Z ⊂ ℙ^{2} ×ℙ^{2}.

Resolving the kernel map ϕ : Y −−→ ℙ^{2} yields a small resolution Ŷ → Y with center Z. Write the kernel of (1) as [x_{1}, x_{2}, x_{3}] so that

[L_{12}L_{23} − L_{13}L_{23}, −L_{11}L_{23} + L_{13}L_{23}, L_{11}L_{22} − L_{12}L_{21}] ∼ [x_{1}, x_{2}, x_{3}]

etc. The rank two matrices with this kernel correspond to a codimensionthree linear subspace

Λ_{[x_{1},x_{2},x_{3}]} ≃ ℙ^{5} ⊂ ℙ^{8}.

The 3 × 3 minors involving the Q_{i} are all proportional to

Q_{1}x_{1} + Q_{2}x_{2} + Q_{3}x_{3}.

Hence

ϕ : Ŷ → ℙ^{2}

is a quadric bundle of relative dimension four.

3.6. Degenerate cases in small dimension. n = 5: Suppose we have a complete intersection of three quadrics in ℙ^{5} containing a plane

X = P ∪_{B} U.

Here π_{P} : U → ℙ^{2} is birational, realizing U = Bl_{Y} (ℙ^{2}), where

ϕ : Y = {y_{1}, …, y_{9}} ↪ ℙ^{2}

is a generic collection of nine points. The imbedding

U ↪ ℙ^{5}

is via quartics vanishing at those nine points.

Write g for the hyperplane class on ℙ^{2} and its restriction to plane curves. Note there is a canonical cubic curve W ⊃ Y where the determinant of linear forms vanishes. For each divisor

Σ ≡_{W} 4g − Y

there is a unique quartic form – modulo the defining equation of W – cutting out Σ ∪ Y.

n = 6: Now consider a complete intersection of three quadrics in ℙ^{6} containing a plane

P ⊂ X ⊂ ℙ^{6}.

The threefold X is a nodal Fano threefold, with six singularities along P. Note that X depends on 21 parameters, codimension six in the parameter space of all complete intersections.

In this case, we have

Y ⊂ W ⊂ ℙ^{3},

where W is a smooth cubic surface, Y is a smooth curve with

deg(Y) = 9, genus(Y) = 9,

residual to a twisted cubic Σ ⊂ W in the complete intersection of W and a quartic. Write

Pic(W) = ⟨L, E_{1}, E_{2}, E_{3}, E_{4}, E_{5}, E_{6}⟩ , L = [Σ],

where the E_{i} are pairwise disjoint (−1)-classes with L · E_{i} = 0. Then we have

[Y] = 11L − 4E_{1} − … − 4E_{6}

and the residual twisted cubic to Σ, with divisor class

5L − 2E_{1} − … − 2E_{6},

realizes Y is a septic plane curve with six nodes. The corresponding linear series induces ϕ : Y → ℙ^{2}.

The intermediate Jacobian of a minimal resolution X̃ → X is isomorphic to the Jacobian of Y.

n = 7: The simplest smooth case, obtained by taking a codimensionfour linear section of the generic case. For generic choices of the linear section, Z = ∅ and ϕ : Y → ℙ^{2} is a double cover branched over a special octic plane curve. Thus we have

Y ⊂ W := {det(L_{ij}) = 0} ⊂ ℙ^{4},

where W is a cubic threefold with six ordinary double points [HT10]. The surface Y has Picard group

\begin{aligned}& K & h \\ K & 2 & 7 \\ h & 7 & 9 \end{aligned}

where K is the canonical class – pulled back from ℙ^{2} – and h is associated with the embedding

Y ↪ W ↪ ℙ^{4}.

There is a second morphism Y → ℙ^{2} associated with the divisor 2h−K because

χ(2h − K) = \frac{(2h − K)^{2} − (2h − K)K}{2} + 4

= \frac{36 − 28 + 2 − 14 + 2}{2} + 4 = 3.

The residual intersection to Y in the complete intersection of W and a quartic hypersurface is a cubic scroll Σ. By [HT10], Y admits two families of such scrolls, each parametrized by ℙ^{2}. An adjunction computation shows that

Σ ∩ Y ≡_{Y} 2h − K.

Remark 4. The lattice ⟨K, h⟩ has discriminant −31. Consider the lattice

⟨g^{2}, P ⟩ ⊂ H^{4}(X, ℤ)

under the intersection pairing, where g is the hyperplane class. A Chern-class computation gives P^{2} = 4 whence

\begin{aligned}& g^{2} & P \\ g^{2} & 8 & 1 \\ P & 1 & 4 \end{aligned} ,

which has discriminant 31. The birational parametrization of X induces an isomorphism of Hodge structures

H^{2}(Y, ℤ)(−1) ⊃ ⟨K, h⟩^{⊥} ≃ ⟨g^{2}, P ⟩^{⊥} ⊂ H^{4}(X, ℤ).

n = 8: Here the L_{ij} are linear forms on ℙ^{5} so the determinant cubic is singular along an elliptic normal curve. We again have Z = ∅ and the center Y ⊂ ℙ^{5} of the mapping to ℙ^{8} is a conic bundle over ℙ^{2}. The equations of Ŷ ⊂ ℙ^{5} × ℙ^{2} may be written:

(2)  L_{i1}x_{1} + L_{i2}x_{2} + L_{i3}x_{3} = 0, Q_{1}x_{1} + Q_{2}x_{2} + Q_{3}x_{3} = 0.

The linear forms induce

0 → K → O_{ℙ^{2}}^{⊕6} → O_{ℙ^{2}}(1)^{⊕3} → 0

and the quadratic form induces a symmetric map

K → K^{∗}(1).

The degeneracy C ⊂ ℙ^{2} has degree 9 since

c_{1}(K^{∗}(1)) − c_{1}(K) = 9c_{1}(O_{ℙ^{2}}(1)).

We count parameters using for equations of type (2):

• the linear terms depend on 45 parameters, the dimension of Gr(3, Γ(O_{ℙ^{5}×ℙ^{2}}(1, 1)));

• the quadratic term depends on 43 paramaters, the dimension of the projectivization of

Γ(O_{ℙ^{5}×ℙ^{2}}(2, 1))/ ⟨ linear terms⟩;

• coordinate changes on ℙ^{5} × ℙ^{2} account for 43 parameters.

Thus we are left with a total of 46 parameters – the number of moduli of a plane curve of degree 9. We know that C comes with a distinguished double cover C̃ → C but even after fixing this we obtain a total of 1024 determinantal representations (see Proposition 5).

n = 9: Here the L_{ij} are linear forms on ℙ^{6}. This allows us to use the normal form

(\begin{matrix}A_{1} & A_{2} & A_{3} \\ A_{5} & A_{1} & A_{4} \\ A_{6} & A_{7} & A_{1} \end{matrix}) (\begin{matrix}x_{1} \\ x_{2} \\ x_{3} \end{matrix}) = 0.

The rank-one locus is a degree-six del Pezzo surface. The equations of Ŷ ⊂ ℙ^{6} × ℙ^{2} may be written:

A_{1}x_{1} + A_{2}x_{2} + A_{3}x_{3} = A_{5}x_{1} + A_{1}x_{2} + A_{4}x_{3} = A_{6}x_{1} + A_{7}x_{2} + A_{1}x_{3} =0
Q_{1}x_{1} + Q_{2}x_{2} + Q_{3}x_{3} =0.

The linear forms induce

0 → K → O_{ℙ^{2}}^{⊕7} → O_{ℙ^{2}}(1)^{⊕3} → 0

and the quadratic form induces a symmetric map

K → K^{∗}(1)

with degeneracy C ⊂ ℙ^{2} of degree 10.

Fixing C and the double cover C̃ → C, the various determinant representations as above are parametrized by the planes in X, i.e., by the threefold F_{2}(X).

4. Schubert calculus of the variety of planes

Now assume that the variety of planes

F_{2}(X) ⊂ Gr(3, n + 1)

is smooth of the expected dimension 3n − 24.

Consider the canonical exact sequence over Gr(3, n + 1)

0 → S → V ⊗ O → Q → 0.

The defining equations

X = {F_{1} = F_{2} = F_{3} = 0}, F_{1}, F_{2}, F_{3} ∈ Sym^{2}(V^{∗}),

induce sections f_{1}, f_{2}, f_{3} ∈ Sym^{2}(S^{∗}) so that

F_{2}(X) = {f_{1} = f_{2} = f_{3} = 0}.

We compute Chern classes. We use the notation of [GH94, pp. 197 ff.], e.g., the Chern classes of S are the Schubert cycles

c(S) = 1 − σ_{1} + σ_{1,1} − σ_{1,1,1}.

Computing via the ‘SchurRings’ or ‘Schubert2’ packages of Macaulay2 [GS], we obtain

[F_{2}(X)] = 512(σ_{9,6,3}+2σ_{9,5,4}+2σ_{8,7,3}+6σ_{8,6,4}+4σ_{8,5,5}+4σ_{7,7,4}+8σ_{7,6,5}+2σ_{6,6,6}).

The paper [DM98] explores these formulas more systematically; our formula extends the tabulation on [DM98, p. 563]. See also [Jia12] for more information on Noether-Lefschetz questions for Fano schemes.

Proposition 5. When n = 8 and X generic, the variety F_{2}(X) ⊂ Gr(3, 9) has dimension zero and

[F_{2}(X)] = 1024[point].

Question 6. Describe the Galois action arising in this case. To what extent is it governed by the Galois representation on the intermediate Jacobian?

Frank Sottile and his collaborators (see, e.g., [HRS18]) are developing computational approaches to Galois groups of enumerative questions. Computations with Taylor Brysiewicz indicate the group in this case might be smaller than the symmetric group 𝔖_{1024} but a full computation appears difficult with existing techniques and computational resources.

Recently, Hashimoto and Kadets [HK20] analyzed Galois groups of zero-dimensional Fano varieties of linear subspaces in complete intersections using the classification of multiply transitive groups. Except for a few exceptional cases – like cubic surfaces and even-dimensional complete intersections of two quadrics – the Galois group always contains the alternating group. Thus in our situation the Galois group is either the alternating group 𝔄_{1024} or 𝔖_{1024}.

Remark 7. We propose to construct these examples synthetically: Fix seven generic planes

P_{1}, …, P_{7} ⊂ ℙ^{8}

which depend up to projectivity on 18 · 7 − 80 = 46 parameters. Write

Π = P_{1} ⊔ P_{2} ⊔ ⋯ ⊔ P_{7}

and note that

h^{0}(I_{Π}(2)) = h^{0}(O_{ℙ^{8}}(2)) − 7 · 6 = 3.

Thus there is a unique complete intersection of three quadrics X ⊃ Π which we expect is smooth. How do we construct the other 1017 planes of F_{2}(X)?

We focus next on the case n = 9 where F_{2}(X) ⊂ Gr(3, 10) is a threefold with class

[F_{2}(X)] = 512(4σ_{7,7,4} + 8σ_{7,6,5} + 2σ_{6,6,6})

so we have

deg(F_{2}(X)) = 11264 = 11 · 2^{10}.

Since K_{F_{2}(X)} = 2σ_{1} we have

K_{F_{2}(X)}^{3} = 11 · 2^{13}.

Proposition 8. [DM98, Th. 3.4] When F_{2}(X) is smooth of the expected dimension we have

h^{0,1}(F_{2}(X)) = h^{1,0}(F_{2}(X)) = 0.

Consider the rank 18 vector bundle

E = Sym^{2}(S^{∗})^{⊕3}

so there exists a section s ∈ Γ(Gr(3, 10), E) with F_{2}(X) = {s = 0}. The exact sequence

0 → T_{F_{2}(X)} → T_{Gr(3,10)}|F_{2}(X) → N_{F_{2}(X)/ Gr(3,10)} → 0

and the interpretation of the normal bundle as E|F_{2}(X) allows us to compute all the Chern classes of T_{F_{2}(X)}:

c_{1}(T_{F_{2}(X)}) = −2σ_{1}
c_{2}(T_{F_{2}(X)}) = 8σ_{2} − 3σ_{1,1}
c_{3}(T_{F_{2}(X)}) = −20σ_{3} − σ_{2,1} + 8σ_{1,1,1}.

A computation with the ‘SchurRings’ or ‘Schubert2’ packages of

Macaulay2 gives

χ(O_{F_{2}(X)}) = c_{1}(T_{F_{2}(X)})c_{2}(T_{F_{2}(X)})/24 = −2816 = −2^{8} · 11

and

χ(F_{2}(X)) = c_{3}(T_{F_{2}(X)}) = −36, 864 = −2^{12} · 3^{2}.

The Hirzebruch-Riemann-Roch formula gives

χ(O_{F_{2}(X)}(1)) = 0,

which is to be expected, as Serre duality gives

h^{i}(O_{F_{2}(X)}(1)) = h^{3−i}(ω_{F_{2}(X)}(−1)) = h^{3−i}(O_{F_{2}(X)}(1)).

Similarly, we obtain χ(O_{F_{2}(X)}(2)) = 2816. We also get

χ(O_{F_{2}(X)}(3)) = h^{0}(O_{F_{2}(X)}(3)) = 16, 896 = 2^{9} · 3 · 11,

where the first equality is Kodaira vanishing. Thus we have enough data to extract the Hilbert polynomial of F_{2}(X)

χ(O_{F_{2}(X)}(m)) = \frac{2^{7} · 11}{3}(m − 1)(5(m − 1)^{2} − 2).

Hirzebruch-Riemann-Roch allows us to compute χ(Ω_{F_{2}(X)}^{1}) as well

χ(Ω_{F_{2}(X)}^{1}) = 15, 616 = 2^{8} · 61.

5. Koszul computations

5.1. General set-up. Let E denote a vector bundle and s ∈ Γ(E) a section such that the degeneracy locus Z = {s = 0} has codimension equal to the rank R of E. Then we have a resolution

0 → ⋀^{R} E^{∗} → ⋀^{R−1} E^{∗} → ⋯ → E^{∗} → O → O_{Z} → 0

and the Koszul complex

0 → ⋀^{R} E^{∗} → ⋀^{R−1} E^{∗} → ⋯ → E^{∗} → O → 0.

The arrow

ι_{s} : ⋀^{n} E^{∗} → ⋀^{n−1} E^{∗}

is contraction by s. The hypercohomology spectral sequence gives

E_{1}^{p,q} = H^{q}(⋀^{−p} E^{∗}) ⇒ H^{p+q}(O_{Z}),

where p = −R, …, 0.

Following [Man91], we may use this to compute the cohomology of degeneracy loci over Grassmannians 𝔾(n, d − 1) = Gr(n + 1, V). Given a d-dimensional vector space V and integers λ = (λ_{1}, …, λ_{k}) with λ_{1} ≥ λ_{2} ≥ ⋯ ≥ λ_{k} and λ_{k} = 0 for k > d, let Γ^{λ}V denote the associated Schur functor. We observe the convention Γ^{λ}V = 0 for sequences of integers λ failing the decreasing or vanishing conditions.

We keep track of the decompositions using the dictionary between Schubert calculus and tensor products

Γ^{λ}V ⊗ Γ^{μ}V = ∑_{ν} c_{λ,μ}^{ν}Γ^{ν}V,

where the multiplicies are defined by

σ_{λ}σ_{μ} = ∑_{ν} c_{λ,μ}^{ν}σ_{ν}.

5.2. Planes in intersections of three quadrics in ℙ^{9}. Consider the bundle

E = Sym^{2}(S^{∗})^{⊕3}

and s ∈ Γ(E) with F_{2}(X) = {s = 0}. We have the associated Koszul resolution

0 → ⋀^{18} E^{∗} → ⋯ → ⋀^{r} E^{∗} → ⋯ → E^{∗} → O → O_{F_{2}(X)} → 0.

The terms of the Koszul complex decompose into direct sums of products

⋀^{e_{1}} Sym^{2}(S) ⊗ ⋀^{e_{2}} Sym^{2}(S) ⊗ ⋀^{e_{3}} Sym^{2}(S), e_{1} + e_{2} + e_{3} = r.

The hypercohomology spectral sequence gives

E_{1}^{p,q} = H^{q}(⋀^{−p} E^{∗}) ⇒ H^{p+q}(O_{F_{2}(X)}),

where p = −18, …, 0.

Recall the Weyl character formula for GL(V) representations:

dim Γ^{λ}(V) = ∏_{i<j} \frac{λ_{i} − λ_{j} + j − i}{j − i}.

Assume S has rank three then

rank Γ^{(a_{1},a_{2},a_{3})}S = (a_{1} − a_{2} + 1)(a_{1} − a_{3} + 2)(a_{2} − a_{3} + 1)/2.

We observe the dictionary

Sym^{2}(S) ∼ σ_{2}
⋀^{2} Sym^{2}(S) ∼ σ_{3,1}
⋀^{3} Sym^{2}(S) ∼ σ_{3,3} + σ_{4,1,1}
⋀^{4} Sym^{2}(S) ∼ σ_{4,3,1}
⋀^{5} Sym^{2}(S) ∼ σ_{4,4,2}
⋀^{6} Sym^{2}(S) ∼ σ_{4,4,4}

whence

Sym^{2}(S) ≃ Γ^{(2)}(S)
⋀^{2} Sym^{2}(S) ≃ Γ^{(3,1)}(S)
⋀^{3} Sym^{2}(S) ≃ Γ^{(3,3)}(S) ⊕ Γ^{(4,1,1)}(S)
⋀^{4} Sym^{2}(S) ≃ Γ^{(4,3,1)}(S)
⋀^{5} Sym^{2}(S) ≃ Γ^{(4,4,2)}(S)
⋀^{6} Sym^{2}(S) ≃ Γ^{(4,4,4)}(S)

Note also [Jia12, Cor. 1.3] that

H^{j}(Gr(3, 10), Γ^{λ}S) = 0

whenever j is not divisible by 7 and for all but one value of j.

We tabulate the representations Γ^{a}S, a = (a_{1}, a_{2}, a_{3}) appearing in each ⋀^{r} E^{∗}:

r decomposition
0 Γ^{(0,0,0)}
1 3Γ^{(2)}
2 3Γ^{(4)} + 6Γ^{(3,1)} + 3Γ^{(2,2)}
3 Γ^{(6)} + 8Γ^{(5,1)} + 9Γ^{(4,2)} + 10Γ^{(4,1,1)} + 10Γ^{(3,3)} + 8Γ^{(3,2,1)} + Γ^{(2,2,2)}
4 3Γ^{(7,1)} + 9Γ^{(6,2)} + 15Γ^{(6,1,1)} + 18Γ^{(5,3)} + 24Γ^{(5,2,1)} + 6Γ^{(4,4)}

+33Γ^{(4,3,1)} + 9Γ^{(4,2,2)} + 15Γ^{(3,3,2)}

5 3Γ^{(8,2)} + 6Γ^{(8,1,1)} + 9Γ^{(7,3)} + 24Γ^{(7,2,1)} + 18Γ^{(6,4)} + 54Γ^{(6,3,1)}

+21Γ^{(6,2,2)} + 6Γ^{(5,5)} + 48Γ^{(5,4,1)} + 54Γ^{(5,3,2)} + 39Γ^{(4,4,2)} + 30Γ^{(4,3,3)}

6 Γ^{(9,3)} + 8Γ^{(9,2,1)} + 8Γ^{(8,4)} + 27Γ^{(8,3,1)} + 19Γ^{(8,2,2)} + 9Γ^{(7,5)}

+64Γ^{(7,4,1)} + 62Γ^{(7,3,2)} + 10Γ^{(6,6)} + 53Γ^{(6,5,1)} + 117Γ^{(6,4,2)} + 56Γ^{(6,3,3)} +46Γ^{(5,5,2)} + 88Γ^{(5,4,3)} + 38Γ^{(4,4,4)}

7 3Γ^{(10,3,1)} + 6Γ^{(10,2,2)} + 3Γ^{(9,5)} + 24Γ^{(9,4,1)} + 27Γ^{(9,3,2)} + 6Γ^{(8,6)}

+42Γ^{(8,5,1)} + 93Γ^{(8,4,2)} + 36Γ^{(8,3,3)} + 3Γ^{(7,7)} + 48Γ^{(7,6,1)} + 132Γ^{(7,5,2)} +144Γ^{(7,4,3)} + 66Γ^{(6,6,2)} + 138Γ^{(6,5,3)} + 114Γ^{(6,4,4)} + 60Γ^{(5,5,4)}

8 3Γ^{(11,3,2)} + 9Γ^{(10,5,1)} + 24Γ^{(10,4,2)} + 9Γ^{(10,3,3)} + 3Γ^{(9,7)} + 24Γ^{(9,6,1)}

+75Γ^{(9,5,2)} + 72Γ^{(9,4,3)} + 24Γ^{(8,7,1)} + 102Γ^{(8,6,2)} + 168Γ^{(8,5,3)} + 99Γ^{(8,4,4)} +69Γ^{(7,7,2)} + 168Γ^{(7,6,3)} + 213Γ^{(7,5,4)} + 96Γ^{(6,6,4)} + 75Γ^{(6,5,5)}

9 Γ^{(12,3,3)} + 9Γ^{(11,5,2)} + 8Γ^{(11,4,3)} + 9Γ^{(10,7,1)} + 36Γ^{(10,6,2)} + 63Γ^{(10,5,3)}

+28Γ^{(10,4,4)} + Γ^{(9,9)} + 8Γ^{(9,8,1)} + 63Γ^{(9,7,2)} + 128Γ^{(9,6,3)} + 142Γ^{(9,5,4)} +28Γ^{(8,8,2)} + 142Γ^{(8,7,3)} + 216Γ^{(8,6,4)} + 146Γ^{(8,5,5)} + 146Γ^{(7,7,4)} +160Γ^{(7,6,5)} + 20Γ^{(6,6,6)}

The terms in red contribute to the cohomology [Jia12, Th. 1.2]. The representations for r > 9 may be read off via duality

⋀^{r} E^{∗} = det(E^{∗}) ⊗ ⋀^{18−r} E, Γ^{(a_{1},a_{2},a_{3})} × Γ^{(12−a_{3},12−a_{2},12−a_{1})} → Γ^{(12,12,12)}.

r decomposition
18 Γ^{(12,12,12)}
17 3Γ^{(12,12,10)}
16 3Γ^{(12,12,8)} + 6Γ^{(12,11,9)} + 3Γ^{(12,10,10)}
15 Γ^{(12,12,6)} + 8Γ^{(12,11,7)} + 9Γ^{(12,10,8)} + 10Γ^{(11,11,8)} + 10Γ^{(12,9,9)} + 8Γ^{(11,10,9)}

+Γ^{(10,10,10)}

14 3Γ^{(12,11,5)} + 9Γ^{(12,10,6)} + 15Γ^{(11,11,6)} + 18Γ^{(12,9,7)} + 24Γ^{(11,10,7)} + 6Γ^{(12,8,8)}

+33Γ^{(11,9,8)} + 9Γ^{(10,10,8)} + 15Γ^{(10,9,9)}

13 3Γ^{(12,10,4)} + 6Γ^{(11,11,4)} + 9Γ^{(12,9,5)} + 24Γ^{(11,10,5)} + 18Γ^{(12,8,6)} + 54Γ^{(11,9,6)}

+21Γ^{(10,10,6)} + 6Γ^{(12,7,7)} + 48Γ^{(11,8,7)} + 54Γ^{(10,9,7)} + 39Γ^{(10,8,8)} + 30Γ^{(9,9,8)}

12 Γ^{(12,9,3)} + 8Γ^{(11,10,3)} + 8Γ^{(12,8,4)} + 27Γ^{(11,9,4)} + 19Γ^{(10,10,4)} + 9Γ^{(12,7,5)}

+64Γ^{(11,8,5)} + 62Γ^{(10,9,5)} + 10Γ^{(12,6,6)} + 53Γ^{(11,7,6)} + 117Γ^{(10,8,6)} +56Γ^{(9,9,6)} + 46Γ^{(10,7,7)} + 88Γ^{(9,8,7)} + 38Γ^{(8,8,8)}

11 3Γ^{(11,9,2)} + 6Γ^{(10,10,2)} + 3Γ^{(12,7,3)} + 24Γ^{(11,8,3)} + 27Γ^{(10,9,3)} + 6Γ^{(12,6,4)}

+42Γ^{(11,7,4)} + 93Γ^{(10,8,4)} + 36Γ^{(9,9,4)} + 3Γ^{(12,5,5)} + 48Γ^{(11,6,5)} + 132Γ^{(10,7,5)} +144Γ^{(9,8,5)} + 66Γ^{(10,6,6)} + 138Γ^{(9,7,6)} + 114Γ^{(8,8,6)} + 60Γ^{(8,7,7)}

10 3Γ^{(10,9,1)} + 9Γ^{(11,7,2)} + 24Γ^{(10,8,2)} + 9Γ^{(9,9,2)} + 3Γ^{(12,5,3)} + 24Γ^{(11,6,3)}

+75Γ^{(10,7,3)} + 72Γ^{(9,8,3)} + 24Γ^{(11,5,4)} + 102Γ^{(10,6,4)} + 168Γ^{(9,7,4)}

+99Γ^{(8,8,4)} + 69Γ^{(10,5,5)} + 168Γ^{(9,6,5)} + 213Γ^{(8,7,5)} + 96Γ^{(8,6,6)} + 75Γ^{(7,7,6)}

And then we record those contributing cohomology in degree j, using [Jia12, Th. 1.2]. Note that the sequence

(−1, −2, −3, −4, −5, −6, −7, a_{1} − 8, a_{2} − 9, a_{3} − 10)

must have no repeating integers as these yield ‘singular’ weights. Thus we must have a_{1} ≥ 8; if a_{1} = 8 then a_{2} = 1, 0, etc.

j weights a contributing
0 (0, 0, 0)
7 no contributions in ⋀^{4} E^{∗}
(8, 1, 1)
no contributions in ⋀^{6} E^{∗}
14 (9, 9, 0)
(10, 9, 1), (9, 9, 2)
(10, 10, 2), (11, 9, 2)
no contributions in ⋀^{12} E^{∗}
21 (10, 10, 10)
(12, 10, 10)
(12, 12, 10)
(12, 12, 12)

The degree 7 cohomology contributes through

H^{7}(⋀^{5} E^{∗}) = H^{7}((Γ^{(8,1,1)}S)^{⊕6}) ≃ (Γ^{(1,1,1,1,1,1,1,1,1,1)}V)^{⊕6} ≃ (det(V))^{⊕6} ≃ ℂ^{6}.

Here V is the standard 10-dimensional representation.

Proposition 9. We have

H^{0}(O_{F_{2}(X)}) = ℂ, H^{1}(O_{F_{2}(X)}) = 0.

For higher degrees, we have

ℂ^{6} ≃ E_{1}^{−5,7} = E_{2}^{−5,7} = ⋯ = E_{∞}^{−5,7} = H^{2}(O_{F_{2}(X)}).

Consequently, we deduce that

h^{3}(O_{F_{2}(X)}) = 2816 + 1 + 6 − 0 = 2823.

Proof. The only term in degree zero is E_{1}^{(0,0)} and there are no terms in degree one, which gives the degree 0 and 1 cohomology; see also Prop. 8 and [DM98]. The spectral sequence is supported at the following values of (p, q):

(0, 0), (−5, 7), (−9, 14),(−10, 14), (−11, 14),
(−15, 21), (−16, 21),(−17, 21), (−18, 21).

All the values after (−5, 7) have degree p + q ≥ 3 and p ≤ −5, thus do not receive arrows from E_{r}^{−5,7}. And clearly there are no maps from the degree (0, 0) term, abutting to Γ(O_{F_{2}(X)}). This yields the equalities asserted above. □

The degree 14 cohomology contributes through

H^{14}(⋀^{11} E^{∗}) = H^{14}((Γ^{(10,10,2)}S)^{⊕6}) ⊕ (Γ^{(11,9,2)}S)^{⊕3}))
= (Γ^{(3,3,2,2,2,2,2,2,2,2)}V)^{⊕6} ⊕ (Γ^{(4,2,2,2,2,2,2,2,2)}V)^{⊕3}
≃ (ℂ^{45})^{6} ⊕ (ℂ^{55})^{3} ≃ ℂ^{435}
H^{14}(⋀^{10} E^{∗}) = H^{14}((Γ^{(9,9,2)}S)^{⊕9}) ⊕ (Γ^{(10,9,1)}S)^{⊕3}))
= (Γ^{(2,2,2,2,2,2,2,2,2,2)}V)^{⊕9} ⊕ (Γ^{(3,2,2,2,2,2,2,2,2,1)}V)^{⊕3}
≃ (ℂ)^{9} ⊕ (ℂ^{99})^{3} ≃ ℂ^{306}
H^{14}(⋀^{9} E^{∗}) = H^{14}(Γ^{(9,9)}S)
= Γ^{(2,2,2,2,2,2,2,2,2,0)}V ≃ ℂ^{55}

The degree 21 cohomology contributes through:

H^{21}(⋀^{18} E^{∗}) = H^{21}(Γ^{12,12,12}S)
= Γ^{(5,5,5,3,3,3,3,3,3,3)}V ≃ ℂ^{4950}
H^{21}(⋀^{17} E^{∗}) = H^{21}((Γ^{12,12,10}S)^{⊕3})
= (Γ^{(5,5,3,3,3,3,3,3,3,3)}V)^{⊕3} ≃ (ℂ^{825})^{3} ≃ ℂ^{2475}
H^{21}(⋀^{16} E^{∗}) = H^{21}((Γ^{12,10,10}S)^{⊕3})
= (Γ^{(5,3,3,3,3,3,3,3,3,3)}V)^{⊕3} ≃ (ℂ^{55})^{3} ≃ ℂ^{165}
H^{21}(⋀^{15} E^{∗}) = H^{21}(Γ^{10,10,10}S)
= Γ^{(3,3,3,3,3,3,3,3,3)}V ≃ ℂ

Remark 10. Standard properties of spectral sequences imply

−2816 = χ(O_{F_{2}(X)}) = ∑(−1)^{p+q}H^{q}(⋀^{−p} E^{∗}) = ∑(−1)^{a+p}H^{7a}(⋀^{−p} E^{∗}).

We compute

−2816 = 1 + 6 + (−55 + 306 − 435) + (−4950 + 2475 − 165 + 1), thus everything checks. Question 11. The simplest structure on the spectral sequence would be to have exactness in all degrees where there is no contribution to cohomology, i.e., degeneration at E_{2}. Thus the only nonzero terms in the second page would be

E_{2}^{−18,21} ≃ ℂ^{4950−2475+165−1} = ℂ^{2639}, E_{2}^{−11,14} ≃ ℂ^{435−306+55} = ℂ^{184}, as well as

E_{2}^{−5,7} ≃ ℂ^{6}, E_{2}^{0,0} ≃ ℂ.

Does this degeneration occur? It would imply a nontrivial filtration on the holomorphic three-forms of F_{2}(X). Proposition 12. The Hodge diamond of F_{2}(X) is

1

0  0
6  62  6
2823  15684  15684  2823

Proof. We recall that • The Picard rank ρ(F_{2}(X)) = 1 when X is very general [Jia12,

Th. 0.3].

• Let Σ → ℙ^{2} denote the double cover branched along degeneracy curve; there exists an isometry on primitive cohomology

H^{6}(X, ℤ)_{prim} ↪ H^{2}(Σ, ℤ)(−2)_{prim},

realizing the former as an index-two sublattice of the latter [O’G86, Th. 0.1].

The latter observation gives the Hodge numbers of X displayed in Section 3.1.

The incidence correspondence

Z
↙^{f}  ↘^{g}
X  F_{2}(X)

induces Abel-Jacobi maps

α_{1} : g_{∗}f^{∗} : H^{6}(X, ℤ) → H^{2}(F_{2}(X), ℤ)(−2),
α_{2} : f_{∗}g^{∗} : H^{4}(F_{2}(X), ℤ) → H^{6}(X, ℤ)(1).

Letting

L : H^{2}(F_{2}(X), ℤ) → H^{4}(F_{2}(X), ℤ)

denote intersection by the hyperplane class σ_{1}, the composition

α_{2} ∘ L ∘ α_{1} : H^{6}(X, ℤ) → H^{6}(X, ℤ)

forces the cohomology of X and F_{2}(X) to be tightly intertwined. Such constructions are used to establish Grothendieck’s version of the Hodge conjecture for varieties with sparse Hodge diamond; see for example [Voi10].

For our immediate purpose, we can apply the Main Theorem of [Shi04] to conclude that the cylinder map α_{1} injects the primitive cohomology H^{6}(X, ℤ)_{prim} into H^{2}(F_{2}(X), ℤ). Since we computed

h^{0,2}(F_{2}(X)) = 6

in Proposition 9, we conclude that

H^{2}(F_{2}(X), ℚ)_{prim} ≃ H^{6}(X, ℚ)(2)_{prim} ⊕ ℚ(−1)^{N}

for some N. The last summand is of Hodge-Tate type. However, N = 0 by Jiang’s result on the Picard group. □

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