INDECOMPOSABLE CYCLES ON SPECIAL QUARTICS IN ℙ^{3}

JAYA N.IYER

Abstract. In this note we give an example of an indecomposable higher Chow cycle on a special family of quartics in ℙ^{3}. The example is obtained as an extension of a cycle in the higher Chow group CH^{2}(K, 1) of a singular Kummer surface.

1. Introduction

Let X be a quasi-projective variety over ℂ. S.Bloch has defined the higher Chow groups CH^{p}(X, n), for p, n ≥ 0, in [1]. There is a product map

CH^{p−m}(X, n − m) ⊗ CH^{1}(X, 1)^{⊗m} ⟶^{ϵ} CH^{p}(X, n).

The cycles in the group \frac{CH^{p}(X,n)}{Image(ϵ)} are called indecomposable cycles.

Existence of indecomposable cycles on a general quartic surface is known due to A.Collino and C.Oliva-C.Voisin ([3],[10]). There is some interest in finding indecomposable cycles in CH^{2}(X, 1) for degenerations of K3-quartic surfaces ([8],p.540-541) and on special quartics.

In this paper, we show that there is an indecomposable cycle on the singular Kummer’s quartic surface K(C) ⊂ ℙ^{3}, in CH^{2}(K(C), 1), which extends in a family of non-singular quartics in ℙ^{3}.

More precisely: Let C be a non-singular curve of genus 2 and J(C) denote the Jacobian variety. The singular Kummer surface is the quotient of J(C) by the inverse map and is naturally embedded as a quartic surface with 16-nodes in the linear system |2C| = ℙ^{3}. We observe that the cycle Z defined in CH^{2}(J(C), 1) by A.Collino ([2]) descends to the Kummer surface. There is an inclusion of quasi-projective varieties K ⊂ S ⊂ ℙ^{6} where K denotes the locus of Kummer surfaces and S −K is the locus of non-singular invariant quartics Q_{s} in ℙ^{3} (here dimS = 6 and dimK = 3). The invariance is under a certain subgroup H of the Heisenberg group Heis(2, 2) acting on ℙ^{3}. In fact, consider a larger family S ⊂ S̃ of quartics such that the coordinate hyperplane sections are rational nodal curves and S̃ − K parameterises non-singular quartics. Consider the universal family Q ⟶ S̃, Q ⊂ ProjOS̃[Y_{0}, Y_{1}, Y_{2}, Y_{3}].

0MPI, Vivatsgasse 7, Bonn 53111 Germany, email: jniyer@mpim-bonn.mpg.de

0Mathematics Classification Number: 14C15,14C25,14H10,14H28,14F42

Theorem 1.1. 1. There is a family of cycles Z_{s} in CH^{2}(Q_{s}, 1)_{ℚ}, supported on Q_{s} ∩ {Y_{0}Y_{1} = 0}, for s ∈ S̃ such that Z_{s} = W_{s}, for s ∈ K. Here W_{s} is Collino’s cycle on a Kummer surface.

2. For a general s ∈ S̃, the cycle Z_{s} is indecomposable.

2. Higher Chow groups

2.1. Let X be a quasi projective variety over ℂ and Δ^{n} = Spec\frac{ℂ[x_{0},··· ,x_{n}]}{(1−∑_{j} x_{j})}. Let Z^{p}(X, n) ⊂ Z^{p}(X×Δ^{n}) be the subgroup containing algebraic cycles of codimension p which intersects all faces X ×Δ^{m}, m < n in codimension at least p. Let δ_{i} : Z^{p}(X, n) ⟶ Z^{p}(X, n−1) be the restriction to the i^{th} codimension 1 face (obtained by setting t_{i} = 0), for 0 ≤ i ≤ n and δ = ∑_{i}(−1)^{i}δ_{i}. The homology of the complex {Z^{p}(X, •), δ} at • = n is the higher Chow group CH^{p}(X, n) as defined by S.Bloch ([1]).

Suppose X is a quasi-projective variety and G is a finite group acting on X. Let Y = \frac{X}{G} be the geometric quotient and π : X ⟶ Y be the quotient morphism. The action of G can be extended to the product X × Δ^{n}, by letting G act trivially on Δ^{n}. Then G acts on Z^{p}(X × Δ^{n}) and also on Z^{p}(X, n). We denote the subgroup of G-invariant cycles by Z^{p}(X, n)^{G}, for all n ≥ 0.

Consider the complex (A):

⟶ Z^{p}(X, n + 1) ⊗ ℤ[\frac{1}{|G|}] ⟶^{δ} Z^{p}(X, n) ⊗ ℤ[\frac{1}{|G|}] ⟶^{δ} Z^{p}(X, n − 1) ⊗ ℤ[\frac{1}{|G|}] ⟶

Applying the functor −^{G} (associating the subgroup of G-invariants) to (A), we obtain the complex (B):

⟶ Z^{p}(X, n + 1)^{G} ⊗ ℤ[\frac{1}{|G|}] ⟶^{δ} Z^{p}(X, n)^{G} ⊗ ℤ[\frac{1}{|G|}] ⟶^{δ} Z^{p}(X, n − 1)^{G} ⊗ ℤ[\frac{1}{|G|}] ⟶

Denote the homology by CH^{p}(X, n)^{G}.

Proposition 2.1. There is an isomorphism CH^{p}(Y, n) ⊗ ℤ[\frac{1}{|G|}] ⟶ CH^{p}(X, n)^{G}.

Proof: The proof is similar to [5],p.20-21, given for n = 0. Let W ∈ Z^{p}(X, n) be an irreducible cycle and G_{W} = {g ∈ G : g(W) = W } be the stabilizer. We denote the maps X × Δ^{n} ⟶^{π×Id} Y × Δ^{n} by π. For an irreducible cycle V ∈ Z^{p}(Y, n), set π^{∗}(V) = ∑ |G_{W} |.W, where the sum is over all the irreducible components W of π^{−1}V. This gives a map π^{∗} : Z^{p}(Y, n) ⟶ Z^{p}(X, n)^{G} which composed with the pushforward

INDECOMPOSABLE CYCLES ON SPECIAL QUARTICS IN ℙ^{3} 3

π_{∗} : Z^{p}(X, n) ⟶ Z^{p}(Y, n), is multiplication by deg(π) = |G|. Hence with ℤ[\frac{1}{|G|}]- coefficients, we have an isomorphism Z^{p}(Y, n) ⊗ ℤ[\frac{1}{|G|}] ≃ Z^{p}(X, n)^{G} ⊗ ℤ[\frac{1}{|G|}]. Using (B), we obtain the isomorphism CH^{p}(Y, n) ⊗ ℤ[\frac{1}{|G|}] ≃ CH^{p}(X, n)^{G}. □

Remark 2.2. If X is non-singular, CH^{1}(X, 1) = ℂ^{∗} ([1],p.287). Hence CH^{1}(X, 1) has a trivial action by G and CH^{1}(Y, 1) = ℂ^{∗}.

3. Invariant quartics

3.1. The singular Kummer Surface in ℙ^{3}. Consider a non-singular projective curve C of genus 2. Let f : C ⟶ ℙ^{1} be the hyperelliptic involution branched at the points 0, 1, ∞, a_{1}, a_{2}, a_{3} on ℙ^{1}. Let p and q be the Weierstrass points on the curve C such that f(p) = 0 and f(q) = ∞ and J(C) denote the Jacobian variety. Consider the Abel-Jacobi embeddings u_{p}, u_{q} : C ↪ J(C), u_{p}(x) = x − p and u_{q}(x) = x − q and let C_{p} and C_{q} denote the respective images.

Consider the linear system |2C| on J(C) and the associated morphism ϕ : J(C) ⟶ |2C|^{∗} = ℙ^{3}. It is well-known that the image of ϕ is the Kummer surface K(C) = \frac{J(C)}{<i>} of degree 4 with 16 nodes. Here i denotes the inverse map on J(C). The hyperelliptic involution on C induces the inverse map on J(C) and the group generated by the Weierstrass points on C_{p} is the subgroup J(C)_{2} of 2-torsion points on J(C). This implies that the curves C_{p} and C_{q} are symmetric divisors, i.e., i(C_{p}) = C_{p}, i(C_{q}) = C_{q} and the translation of C_{p} by elements in J(C)_{2} give 16 symmetric curves C_{j} on J(C). We thus conclude that the images R_{j} of C_{j}, 1 ≤ j ≤ 16 are non-singular rational curves on K(C) under the quotient map ϕ : J(C) ⟶ K(C) ⊂ ℙ^{3}.

3.2. Invariant quartics in ℙ^{3}. Consider the Heisenberg group Heis(2, 2) = ℂ^{∗} × (\frac{ℤ}{2ℤ})^{2} × (\frac{ℤ}{2ℤ})^{∗ 2}. Here (\frac{ℤ}{2ℤ})^{∗ 2} denotes the character group. The group law is defined as follows: (α, x, l).(β, y, m) = (α.β.m(x), x + y, l.m). Let σ_{1} = (1, 1, 0, 0, 0), σ_{2} = (1, 0, 1, 0, 0), τ_{1} = (1, 0, 0, 1, 0), τ_{2} = (1, 0, 0, 0, 1) ∈ Heis(2, 2) which act on the coordinates δ_{a,b}, a, b ∈ (\frac{ℤ}{2ℤ})^{2} of ℙ^{3} as follows:

\begin{matrix}σ_{1} : δ_{a,b} ↦ δ_{a−1,b} & σ_{2} : δ_{a,b} ↦ δ_{a,b−1} \\ τ_{1} : δ_{a,b} ↦ (−1)^{a}δ_{a,b} & τ_{2} : δ_{a,b} ↦ (−1)^{b}δ_{a,b} \end{matrix}

Denote X_{0} = δ_{0,0}, X_{1} = δ_{1,0}, X_{2} = δ_{0,1}, X_{3} = δ_{1,1}.

Consider the new coordinates on ℙ^{3}:

\begin{matrix}Y_{0} = −X_{0} − X_{1} − X_{2} − X_{3}, & Y_{1} = −X_{0} + X_{1} + X_{2} − X_{3}, \\ Y_{2} = −X_{0} + X_{1} − X_{2} + X_{3}, & Y_{3} = −X_{0} − X_{1} + X_{2} + X_{3}. \end{matrix}

Then σ_{1}(Y_{0}, Y_{1}, Y_{2}, Y_{3}) = (Y_{1}, Y_{0}, −Y_{3}, −Y_{2}) and τ_{1}(Y_{0}, Y_{1}, Y_{2}, Y_{3}) = (Y_{2}, Y_{3}, Y_{0}, Y_{1}).

4 JAYA N.IYER

A quartic in ℙ^{3} invariant under the subgroup H generated by σ_{1} and τ_{1} is given by the equation :

Q = μ_{0}(Y_{0}^{2}Y_{1}^{2} + Y_{2}^{2}Y_{3}^{2}) + μ_{1}(Y_{0}^{2}Y_{2}^{2} + Y_{1}^{2}Y_{3}^{2}) + μ_{2}(Y_{0}^{2}Y_{3}^{2} + Y_{1}^{2}Y_{2}^{2}) + μ_{3}(Y_{0}Y_{1} + Y_{2}Y_{3})(−Y_{0}Y_{2} + Y_{1}Y_{3}) + μ_{4}(Y_{0}Y_{1} − Y_{2}Y_{3})(Y_{0}Y_{3} − Y_{1}Y_{2}) + μ_{5}(Y_{0}Y_{2} + Y_{1}Y_{3})(Y_{0}Y_{3} + Y_{1}Y_{2}) + μ_{6}Y_{0}Y_{1}Y_{2}Y_{3} = 0, for (μ_{0}, ⋯, μ_{6}) ∈ ℙ^{6}.

Let G_{0} = Q ∩ {Y_{0} = 0} and G_{1} = Q ∩ {Y_{1} = 0}.

a. Suppose Q is a singular Kummer surface. Then it is defined by the equation

λ_{1}^{2}(Y_{0}^{2}Y_{1}^{2} + Y_{2}^{2}Y_{3}^{2}) + λ_{2}^{2}(Y_{0}^{2}Y_{2}^{2} + Y_{1}^{2}Y_{3}^{2}) + λ_{3}^{2}(Y_{0}^{2}Y_{3}^{2} + Y_{1}^{2}Y_{2}^{2}) + 2λ_{1}λ_{2}(Y_{0}Y_{1} + Y_{2}Y_{3})(−Y_{0}Y_{2} + Y_{1}Y_{3}) + 2λ_{1}λ_{3}(Y_{0}Y_{1} − Y_{2}Y_{3})(Y_{0}Y_{3} − Y_{1}Y_{2}) + 2λ_{2}λ_{3}(Y_{0}Y_{2} + Y_{1}Y_{3})(Y_{0}Y_{3} + Y_{1}Y_{2}) + λ_{0}^{2}Y_{0}Y_{1}Y_{2}Y_{3} = 0, for some (λ_{0}, ⋯, λ_{3}) ∈ ℙ^{3}.

In this case G_{0} and G_{1} are doubled conics ([7],p.298-301). Moreover, G_{0} ∩ G_{1} = {2c, 2d}, where c = (0, 0, 1, 0), d = (0, 0, 0, 1).

b. Suppose Q is non-singular.

Then one easily checks that G_{0} and G_{1} are irreducible plane quartics having 3 nodes. The coordinate points b = (0, 1, 0, 0), c = (0, 0, 1, 0), d = (0, 0, 0, 1) are the nodes on G_{0} and the points a = (1, 0, 0, 0), c, d are the nodes on G_{1}. Moreover, G_{0} ∩ G_{1} = {2c, 2d}.

Let S ⊂ ℙ^{6} be the locus which parametrizes the Kummer surfaces and the non-singular H-invariant quartics. More generally, consider the locus S̃ ⊂ P Sym^{4}(H^{0}(ℙ^{3}, O(1))) satisfying:

1. S ⊂ S̃

2. for any s ∈ S̃ − K, the corresponding quartic surface Q_{s} is non-singular

3. the hyperplane sections G_{0} = Q_{s} ∩ {Y_{0} = 0} and G_{1} = Q_{s} ∩ {Y_{1} = 0} are rational nodal curves and div(\frac{Y_{3}}{Y_{2}} _{|G_{0}}) + div(\frac{Y_{2}}{Y_{3}} _{|G_{1}}) = 0 as a cycle on Q_{s}.

4. A Higher Chow cycle on the invariant quartics

Let K_{j} denote Quillen’s K-theory of coherent sheaves and K_{j,X} denote the corresponding Zariski sheaf on a variety X of dimension d. If X is non-singular, then it is known that CH^{k}(X, 1) ≃ H_{Zar}^{k−1}(X, K_{k,X}) ([9], Corollary 5.3). Using the Gersten-Quillen complex, an element of CH^{k}(X, 1) is written as a finite sum ∑_{i} W_{i} ⊗ h_{i}, where W_{i} ⊂ X are irreducible subvarieties of codimension k − 1 and h_{i} ∈ ℂ(W_{i})^{∗} satisfies ∑_{i} div(h_{i}) = 0 as a cycle on X.

Definition 4.1. A higher cycle W ∈ CH^{k}(X, 1) is said to be decomposable if its class lies in the image of the product map

CH^{k−1}(X) ⊗ CH^{1}(X, 1) ⟶ CH^{k}(X, 1).

INDECOMPOSABLE CYCLES ON SPECIAL QUARTICS IN ℙ^{3} 5

A cycle W is called indecomposable if it is not decomposable and we denote the group of indecomposable cycles by CH^{k}(X, 1)_{ind}.

4.1. Regulator indecomposable cycles. We recall some definitions from the survey [6].

For a subring A ⊂ ℝ, A(k) = (2πi)^{k}A ⊂ ℂ and the Deligne cohomology group H_{D}^{j}(X, A(k)) is the hypercohomology of the complex

A(k)_{D} : A(k) ⟶ O_{X} ⟶ Ω_{X}^{1} ⟶ ...Ω_{X}^{k−1}.

The short exact sequence of complexes

0 ⟶ Ω_{X}^{<k}[−1] ⟶ A(k)_{D} ⟶ A(k) ⟶ 0

induces a long exact sequence:

⟶ H^{2k−2}(X, A(k)) ⟶ \frac{H^{2k−2}(X,ℂ)}{F^{k}H^{2k−2}(X,ℂ)} ⟶ H_{D}^{2k−1}(X, A(k)) ⟶ H^{2k−1}(X, A(k)) ⟶. Hence there is a short exact sequence:

0 ⟶ \frac{H^{2k−2}(X, ℂ)}{F^{k}H^{2k−2}(X, ℂ) + H^{2k−2}(X, A(k))} ⟶ H_{D}^{2k−1}(X, A(k)) ⟶ T^{k} ⟶ 0.

Here T^{k} = H^{2k−1}(X, A(k)) ∩ F^{k}H^{2k−1}(X, ℂ) = H^{2k−1}(X, A(k))_{torsion}.

Remark 4.2. With ℝ-coefficients, we have ([6],p.203)

H_{D}^{2k−1}(X, ℝ(k)) = \frac{H^{2k−2}(X, ℂ)}{F^{k}H^{2k−2}(X, ℂ) + H^{2k−2}(X, ℝ(k))}
≃ H^{2k−1}(X, ℝ(k − 1)) ∩ F^{k−1}H^{2k−2}(X, ℂ)
=: H^{k−1,k−1}(X, ℝ(k − 1)).

There is a cycle class map (or Chern class map or regulator map) c_{k,1} : CH^{k}(X, 1)_{ℚ} ⟶ H_{D}^{2k−1}(X, ℚ(k)). Extending the coefficients to ℝ and given a cycle W ∈ CH^{k}(X, 1)_{ℚ} there is a class c_{k,1}(W) ∈ H_{D}^{2k−1}(X, ℝ(k)) ≃ H^{k−1,k−1}(X, ℝ(k − 1)). Explicitly, it is the class of the current

ω ↦ (2π\sqrt{−1})^{k−1−d} ∑_{i} ∫_{W_{i}−W_{i}^{sing}} ωlog|h_{i}|,

for ω ∈ H^{d−k+1,d−k+1}(X, ℝ(d − k + 1)).

Definition 4.3. A cycle W ∈ CH^{k}(X, 1)_{ℚ} is said to be (real) regulator indecomposable if there exists a differential form

ω ∈ (Hdg^{k−1}(X)_{ℝ})^{⊥} ⊂ H^{d−k+1,d−k+1}(X, ℝ(d − k + 1))

such that c_{k,1}(W)(ω) ≠ 0. Here Hdg^{k−1}(X) is the image under the cycle class map \frac{CH^{k−1}(X)}{CH^{k−1}(X)_{hom}} ⟶ H^{2k−2}(X, ℤ).

Remark 4.4. It follows that if W is regulator indecomposable then W is indecomposable.

4.2. A criterion for regulator indecomposability. Let π : X ⟶ Y be a smooth projective morphism of relative dimension d. We assume X and Y are non-singular and Y is a curve. Let W ∈ CH^{k}(X, 1)_{ℚ} be a family of cycles over Y, i.e., W restricts to a cycle W_{y} ∈ CH^{k}(X_{y}, 1)_{ℚ} on a fibre X_{y}.

Proposition 4.5. Suppose the cycle W_{x} is regulator indecomposable, for some x ∈ Y. Then for a general y ∈ Y, the cycle W_{y} ∈ CH^{k}(X_{y}, 1)_{ℚ} is regulator indecomposable.

Proof: Let c_{k,1}(W) be the regulator image in the rational Deligne cohomology H_{D}^{2k−1}(X, ℚ(k)). Consider the sheaf R^{k−1} = \frac{R^{2k−2}π_{∗}Ω_{X/Y}^{<k}}{R^{2k−2}π_{∗}ℚ} on the curve Y. Since the class c_{k,1}(W) restricts to the cycle class c_{k,1}(W_{y}) ∈ H_{D}^{2k−1}(X_{y}, ℚ(k)), for y ∈ Y, the class c_{k,1}(W) induces a holomorphic section ν ∈ H^{0}(Y, R^{k−1}). Extending the coefficients to ℝ, we write R^{k−1} = \frac{R^{2k−2}π_{∗}(Ω_{X/Y}^{<k} )}{Im(R^{2k−2}π_{∗}ℝ)}. Tensoring the sheaf R^{k−1} with the real analytic structure sheaf of Y we obtain R_{ℝ}^{k−1} as a real analytic vector bundle on Y. Moreover, ν corresponds to an analytic section ν_{ℝ} of the bundle R_{ℝ}^{k−1}.

There is a map

H^{d−k+1,d−k+1}(X_{y}, ℝ(d − k + 1)) ⟶ ℝ(d)

given by the class c_{k,1}(W_{y}) ∈ H_{D}^{2k−1}(X_{y}, ℝ(k)) ≃ H^{k−1,k−1}(X_{y}, ℝ(k − 1)) as a current. Hence there is a map

R_{ℝ}^{d−k+1} ⟶ ℝ(d)

given by the section ν_{ℝ} (here R_{ℝ}^{d−k+1} is defined as above).

Since the cycle W_{x} is regulator indecomposable for some x ∈ Y, there is a cohomology class ω ∈ (Hdg^{k−1}(X_{x})_{ℝ})^{⊥} ⊂ H^{d−k+1,d−k+1}(X_{x}, ℝ(d − k + 1)) such that c_{k,1}(W_{x})(ω) ≠ 0. Now the cohomology class ω extends as a family of classes {ω_{y}} in some analytic neighbourhood U of x ∈ Y. By semi-continuity, a general fibre X_{y} has rank(Hdg^{k−1}(X_{y})) ≤ rank(Hdg^{k−1}(X_{x})). Hence the class ω_{y} ∈ (Hdg^{k−1}(X_{y})_{ℝ})^{⊥}, for y ∈ U (after shrinking U if necessary). Since ν_{ℝ} is an analytic section ν_{ℝ}(W_{y})(ω_{y}) ≠ 0, for y ∈ Y and general. □

INDECOMPOSABLE CYCLES ON SPECIAL QUARTICS IN ℙ^{3} 7

4.3. A higher Chow cycle. With notations as in 3.1, A.Collino ([2]) defined a higher Chow cycle Z = C_{p} ⊗ f_{p} + C_{q} ⊗ f_{q} in CH^{2}(J(C), 1) = H^{1}(J(C), K_{2}) where f_{p} (resp. f_{q}) is the hyperelliptic involution on C_{p} with div(f_{p}) = 2(0) − 2(p − q) (resp. on C_{q} with div(f_{q}) = 2(p − q) − 2(0)). Here C_{p} ∩ C_{q} = {0, p − q} ⊂ J(C).

In fact, we consider any two symmetric curves C_{j} and C_{k} on J(C). Since the selfintersection number (C^{2}) = 2 and C_{j} ∩ C_{k} is invariant under i and σ (here t_{σ}^{∗}C_{j} = C_{k}, for some σ ∈ J(C)_{2}), C_{j} ∩ C_{k} = {a, a + σ}, for some a ∈ J(C)_{2}. Consider the higher cycle Z = C_{j} ⊗ f_{j} + C_{k} ⊗ f_{k} ∈ CH^{2}(J(C), 1) constructed as above.

Consider the push-forward map

ϕ_{∗} : CH^{2}(J(C), 1) ⟶ CH^{2}(K(C), 1).

Then ϕ_{∗}(Z) = 2.W where W = R_{j} ⊗ z_{j} + R_{k} ⊗ z_{k}, for some rational curves R_{j} = ϕ(C_{j}) and R_{k} = ϕ(C_{k}) as in 3.1. Here R_{j} ∩ R_{k} = {r, s} and z_{j} (resp. z_{k}) is the parameter on R_{j} ≃ ℙ^{1} with div(z_{j}) = (r) − (s) (resp. on R_{k} with the opposite divisor).

Remark 4.6. 1. The cycle Z ∈ CH^{2}(J(C), 1)^{i} and using the isomorphism in Proposition 2.1, can be identified with the cycle W ∈ CH^{2}(K(C), 1)_{ℚ}.

2. As pointed out by the referee, if F ⊂ J(C) consists of a 2-torsion point on J(C) and E ⊂ J(C)′ is the exceptional divisor on the blow-up J(C)′ at the 2-torsion point then we have the localization sequences

⟶ CH^{r−2}(F, 1) ⟶ CH^{r}(J(C), 1) ⟶ CH^{r}(J(C) − F, 1) ⟶ CH^{r−2}(F) ⟶

and

⟶ CH^{r−1}(E, 1) ⟶ CH^{r}(J(C)′, 1) ⟶ CH^{r}(J(C)′ − E, 1) ⟶ CH^{r−1}(E) ⟶

For r = 2, since CH^{0}(F, 1) = 0, CH^{1}(E, 1) = ℂ^{∗} and J(C)−F ≃ J(C)′−E, we obtain an isomorphism CH^{2}(J(C), 1)_{ind} ≃ CH^{2}(J(C)′, 1)_{ind}. In particular, successive blow-ups at all the 2-torsion points give an isomorphism CH^{2}(J(C), 1)_{ind} ≃ CH^{2}(J̃(C), 1)_{ind}. Now again using Proposition 2.1 and equivariance of products under the i-action, we get isomorphisms CH^{2}(J(C), 1)_{ind}^{i} ≃ CH^{2}(K(C), 1)_{ind} and CH^{2}(K(C), 1)_{ind} ≃ CH^{2}(K̃(C), 1)_{ind} with rational coefficients. (Here J̃(C) denotes the blow-up of J(C) at all the 2-torsion points on J(C) and K̃(C) = \frac{J̃(C)}{<i>}).

Let W̃ denote the pullback of the cycle W to the desingularization K̃(C) of K(C). By [2] and Remark 4.6, W̃ is indecomposable. The following lemma shows it is (real) regulator indecomposable as well.

8 JAYA N.IYER

Lemma 4.7. The cycle W̃ ∈ CH^{2}(K̃(C)), 1)_{ℚ} is regulator indecomposable for a generic Jacobian surface.

Proof: Consider the blow-up map f : J̃(C) ⟶ J(C) and the commutative diagram:

CH^{2}(J(C), 1)_{ℚ} ⟶ H_{D}^{3}(J(C), ℝ(1))
↓ f^{∗}  ↓ f^{∗}
CH^{2}(J̃(C), 1)_{ℚ} ⟶ H_{D}^{3}(J̃(C), ℝ(1))

Since the cycle Z ∈ CH^{2}(J(C), 1)_{ℚ} is (real) regulator indecomposable for a generic curve C ([3],Proposition 1.3), there is a form ω ∈ Hdg^{1}(J(C))_{ℝ}^{⊥} ⊂ H^{1,1}(J(C), ℝ), such that c_{2,1}(Z)(ω) ≠ 0. Since f is a blow-up map, the pull-back cohomology classes still satisfy f^{∗}ω ∈ Hdg^{1}(J̃(C))_{ℝ}^{⊥} ⊂ H^{1,1}(J̃(C), ℝ(1)) and f^{∗}(c_{2,1}Z)(f^{∗}ω) ≠ 0. Hence the cycle f^{∗}Z is regulator indecomposable. The cycle class map c_{2,1} : CH^{2}(J̃(C), 1)_{ℚ} ⟶ H_{D}^{3}(J̃(C), ℝ(2)) is i-equivariant for the action of i on J̃(C). Restricting to i-invariant cycles gives a cycle class map CH^{2}(J̃(C), 1)_{ℚ}^{i} ⟶ H_{D}^{3}(J̃(C), ℝ(2))^{i} ≃ H^{1,1}(J̃(C), ℝ(1))^{i}. Now f^{∗}Z ∈ CH^{2}(J̃(C), 1)_{ℚ}^{i} and descends to W̃ ∈ CH^{2}(K̃(C)), 1)_{ℚ}. Since i acts trivially on the even cohomologies of J̃(C) we obtain that H_{D}^{3}(K̃(C)), ℝ(2)) = H_{D}^{3}(J̃(C), ℝ(2)) and it follows that W̃ is regulator indecomposable. □

With notations as in 3.2, we have a family of higher cycles:

A1. A family π : Q ↪ ℙ3 ; S̃ = ProjOS̃[Y_{0}, Y_{1}, Y_{2}, Y_{3}] of quartic surfaces over S̃ and Q is non-singular.

A2. A family of subvarieties G_{0}, G_{1} ↪ Q over S̃, cut out by the hyperplane sections {Y_{0} = 0} and {Y_{1} = 0} on Q.

A3. A family of cycles Z = G_{0} ⊗ \frac{Y_{3}}{Y_{2}} + G_{1} ⊗ \frac{Y_{2}}{Y_{3}} ∈ CH^{2}(Q, 1)_{ℚ} over S̃.

Proposition 4.8. For a general s ∈ S̃, the cycle Z_{s} is regulator indecomposable (hence indecomposable).

Proof: We consider the pull-back of the family Q ⟶ S̃ over a non-singular curve Y such that the generic fibre over Y is non-singular. Since a singular fibre (i.e a Kummer surface) has only rational double point singularities, there is a resolution of singularities π : Q̃ ⟶ Ỹ of the family Q ⟶ Y ([11]). This means that there is a commutative diagram

\begin{matrix}Q̃ & ⟶^{g} & Q^{H} \\ ↓ π & & ↓ \\ Ỹ & ⟶ & Y \end{matrix}

such that gỹ : Q̃ỹ ⟶ Q_{y} is a resolution of singularities for each ỹ ∈ Ỹ and π is a smooth projective morphism.

INDECOMPOSABLE CYCLES ON SPECIAL QUARTICS IN ℙ^{3} 9

Consider the commutative diagram:

CH^{2}(Q, 1)_{ℚ} ⟶^{c_{2,1}} H_{D}^{3}(Q, ℝ(1))
↓ g^{∗}  ↓ g^{∗}
CH^{2}(Q̃, 1)_{ℚ} ⟶^{c_{2,1}} H_{D}^{3}(Q̃, ℝ(1))

Let Z̃ = g^{∗}Z ∈ CH^{2}(Q̃, 1)_{ℚ}. Suppose Q_{y_{0}} is a singular fibre and y′ ∈ Ỹ lies over y_{o} ∈ Y. Then by Lemma 4.7, the cycle Z̃_{y′} is regulator indecomposable. Now by Proposition 4.5, for a general y ∈ Ỹ, the cycle Z̃_{y} is regulator indecomposable. □

Acknowledgements We thank A.Collino, H.Esnault and S.Muller-Stach for useful conversations and communications on this topic. We also thank J.Nagel, B.van Geemen for a conversation on semistability and F.Catanese for pointing out [11]. We also owe to the referee for useful remarks, incorporated here. This work was done at University of Essen and support from DFG‘Complex Geometry’ is gratefully acknowledged.

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