TOPOLOGICAL ARRANGEMENT OF CURVES OF DEGREE 6 ON CUBIC SURFACES IN ℝP^{3}

G.MIKHALKIN

Abstract. A quadric in ℝP^{3} cuts a curve of degree 6 on a cubic surface in ℝP^{3}. The papers classifies the nonsingular curves cut in this way on non-singular cubic surfaces up to homeomorphism.

Two issues new in the study related to the first part of the 16th Hilbert problem appear in this classification. One is the distribution of the components of the curve between the components of the non-connected cubic surface which turns out to depend on the patterns of arrangements (see Theorem 1). The other is presence of positive genus among the components of the complement and genusrelated restrictions (see Theorems 3 and 4).

1. Introduction

Let ℝB be a real algebraic cubic surface in ℝP^{3} and let ℝQ be a real algebraic quadric surface in ℝP^{3}. This just means that for some homogeneous polynomials f, deg(f) = 3, and g, deg(g) = 2, surfaces ℝB and ℝQ are given by the following equations

ℝB = {[x_{0} : x_{1} : x_{2} : x_{3}] ∈ ℝP^{3} | f(x_{0}, x_{1}, x_{2}, x_{3}) = 0}
ℝQ = {[x_{0} : x_{1} : x_{2} : x_{3}] ∈ ℝP^{3} | g(x_{0}, x_{1}, x_{2}, x_{3}) = 0}.

Let ℂB and ℂQ be the complexifications of surfaces ℝB and ℝQ

ℂB = {[x_{0} : x_{1} : x_{2} : x_{3}] ∈ ℂP^{3} | f(x_{0}, x_{1}, x_{2}, x_{3}) = 0}
ℂQ = {[x_{0} : x_{1} : x_{2} : x_{3}] ∈ ℂP^{3} | g(x_{0}, x_{1}, x_{2}, x_{3}) = 0}.

We suppose that ℂB and ℂQ are nonsingular and intersect transversely along a curve ℂA. The intersection ℂA = ℂB ∩ ℂQ is a smooth curve in ℂP^{3}, its real part ℝA is ℝB ∩ ℝQ.

Consider now the diffeomorphism type of ℂB, ℂQ, ℂA. It is well-known that ℂQ is diffeomorphic to ℂP^{1} × ℂP^{1}, ℂB is diffeomorphic to ℂP^{2}#6\overline{ℂP}^{2} and we can construct ℂB from ℂP^{2} by blowing up 6 points not sitting on the same conic. The curve ℂA is the proper transform of a plane curve of degree 6 with 6 ordinary double points in the blowup points so the genus of ℂA is 4.

Research at MSRI is supported in part by NSF grant DMS-9022140

Consider now the diffeomorphism type of ℝB, ℝQ, ℝA. It is well-known that a quadric ℝQ is a hyperboloid (i.e. diffeomorphic to ℝP^{1} × ℝP^{1}), an ellipsoid (i.e. diffeomorphic to S^{2}) or empty. It is also well-known that a cubic ℂB is diffeomorphic to #(2n + 1)ℝP^{2} ≈ ℝP^{2}#nT^{2}, where n ≤ 3 or ℝP^{2} ⊔ S^{2}. Since ℝA is smooth, every component of ℝA is diffeomorphic to S^{1}. Note that the embedding of this S^{1} into ℝB is two-sided, the sides of ℝA are defined by the sign of polynomial g (since deg g is even).

Since the genus of ℂA is 4, the Harnack inequality implies that the number of components of ℝA is at most 5 (ℂA/ conj is a connected surface and, therefore, the number of components of its boundary is at most 2−χ(ℂA/ conj) = 2− \frac{1}{2}χ(ℂA) = 5).

The topological classification of pairs (ℝQ, ℝA) was done by Gudkov [3] in the case where ℝQ is a hyperboloid and by Gudkov and Shustin [6] in the case where ℝQ is an ellipsoid (see [10] for the classification of flexible curves).

This paper gives the topological classification of pairs (ℝB, ℝA). Two new phenomena appear in this classification — the arrangement of ℝA with respect to the components of ℝB in the case when ℝB is not connected and the arrangement of the components of ℝA with respect to the handles of ℝB in the case when ℝB is diffeomorphic to a projective plane with handles. These possibilities for arrangement turn out to be controlled by the degree of the curve.

The topological type of a pair (ℝB, ℝA) is also called the type of topological arrangement of ℝA in ℝB or the real scheme of ℝA in ℝB.

Acknowledgment. A part of the work was done during my stay at the Institute for Advanced Study (supported by the NSF grant DMS 9304580) and at the Max-Planck-Institut.

2. Curves on cubic surfaces of positive Euler characteristic

There are two diffeomorphism types of cubic surfaces of positive Euler characteristic in ℝP^{3}:

ℝP^{2} ⊔ S^{2} and ℝP^{2}.

We consider them separately since if ℝB is one of these surfaces than any component of ℝA bounds a disk in ℝB and even the notations for the topological type of (ℝB, ℝA) are simpler in this case.

2.1. Notations. Recall the system of codes for topological arrangements of a collection of two-sided circles (or ovals) in ℝP^{2} introduced by Viro [15]. Every oval separates ℝP^{2} into a disk (caller the interior of the oval) and a Möbius band. Two ovals are called disjoint if their interiors are disjont. The system of codes is inductive, n disjoint ovals are encoded by < n > and if we add a new oval containing a collection < A > in its interior then the code of the new collection is < 1<A>>.

To make the same system suitable for encoding of topological arrangements of embedded curves in S^{2} it suffices to fix a point in S^{2} in the complement of the curve — then the interior of an oval is defined as the component of the complement of the oval not containing the fixed point. Such a code determines uniquely a topological type of curve in S^{2}, but because of ambiguity in fixing a point there is more than one code corresponding to the same topological type. We choose codes minimizing the number of ovals in the interior of other ovals. To encode types of topological arrangement of ℝA into a cubic surface diffeomorphic to ℝP^{2} ⊔ S^{2} we combine systems for ℝP^{2} and S^{2}.

Parameters α and β in the theorems are nonnegative integer numbers.

2.2. Classification.

Theorem 1. Topological arrangement of a nonsingular algebraic curve of degree 6 on a nonsingular cubic surface diffeomorphic to ℝP^{2} ⊔ S^{2} is of one of the following types:

a: < α ⊔ 1<1>>_{ℝP^{2}} ⊔ < ∅ >_{S^{2}}, α ≤ 3,
b: < 1<α>>_{ℝP^{2}} ⊔ < ∅ >_{S^{2}}, 2 ≤ α ≤ 4,
c: < α >_{ℝP^{2}} ⊔ < β >_{S^{2}}, α + β ≤ 5,
d: < 1<1<1>>>_{ℝP^{2}} ⊔ < ∅ >_{S^{2}},
e: < 1<1>>_{ℝP^{2}} ⊔ < 1 >_{S^{2}},
f: < ∅ >_{ℝP^{2}} ⊔ < 1 ⊔ 1<1>>_{S^{2}} .

Each of the 31 types listed above is realizable by a nonsingular algebraic curve of degree 6.

Theorem 2. Topological arrangement of a nonsingular algebraic curve of degree 6 on a nonsingular cubic surface diffeomorphic to ℝP^{2} is of one of the following types:

a: < α ⊔ 1<β >>, α + β ≤ 4,

b: < 1<1<1>>>,
c: < ∅ >.

Each of the 17 types listed above is realizable by a nonsingular algebraic curve of degree 6.

3. Curves on cubic surfaces of negative Euler characteristic

There are three diffeomorphism types of cubic surfaces of negative Euler characteristic in ℝP^{3}:

ℝP^{2}#3T^{2}, ℝP^{2}#2T^{2} and ℝP^{2}#T^{2}.

3.1. Notations. We define

B_{+} = {x ∈ ℝB | g(x) ≥ 0}, B_{−} = {x ∈ ℝB | g(x) ≤ 0}.

Then ∂B_{+} = ∂B_{−} = ℝA. Note that at least one of B_{+}, B_{−} is non-orientable since

χ(ℝB) ≡ 1 (mod 2).

Changing the sign of g we may assume that

χ(B_{+}) ≡ b_{0}(ℝA) (mod 2)
χ(B_{−}) ≡ 1 + b_{0}(ℝA) (mod 2),

where b_{0} stands for the number of components (the 0-dimensional Betti number). Then B_{−} is always nonorientable.

Note that the topological type of (ℝB, ℝA) is determined by the topological types of B_{+} and B_{−} if the number of non-disk components is not more than one for both B_{+} and B_{−} (since then there is only one way to glue B_{+} and B_{−} along the boundary to obtain a connected ℝB). In this case we code the topology of ℝA on ℝB with

< j ⊔ F, n ⊔ G >

where j is the number of disks in B_{+}, n is the number of disks in B_{−}, F is the topological type of the only non-disk component of B_{+} (if any) and G is the only nondisk component of B_{−}. We denote the topological type of the connected orientable surface of genus g with gT^{2} and that of the non-orientable one with gℝP^{2}. Notation F_{k} stands for the result of puncturing of a closed surface F k times.

Lemma 1 implies that if B_{+} or B_{−} has more than one non-disk components then the topological type of (ℝB, ℝA) is still determined by the topological types of B_{+}, B_{−} and ℝB. If ℝB ≈ ℝP^{2}#nT^{2}, n = 1, 2, 3 then the code of (ℝB, ℝA) is

< 1 ⊔ nT_{2}^{2}, S_{2}^{2} ⊔ ℝP_{1}^{2} >

if the core of the annulus is null-homologous and

< (n − 1)T_{3}^{2}, S_{2}^{2} ⊔ ℝP_{1}^{2} >

if not.

3.2. Classification.

Theorem 3. Topological arrangement of a nonsingular algebraic curve of degree 6 on a nonsingular cubic surface diffeomorphic to ℝP^{2}#3T^{2} is of one of the following types:

1. < α ⊔ S_{β+γ}^{2}, β ⊔ (9 − 2γ)ℝP_{α+γ}^{2} >,

β = 0  β = 1  β = 2 β = 3
α = 0 γ = 1, 2, 3, 4 γ = 1, 2, 3, 4 γ = 1, 2, 3 γ = 2
α = 1 γ = 1, 2, 3, 4 γ = 1, 2, 3  γ = 2
α = 2 γ = 1, 2, 3  γ = 1, 2
α = 3  γ = 1, 2  γ = 1

2. < α ⊔ T_{β+γ}^{2}, β ⊔ (7 − 2γ)ℝP_{α+γ}^{2} >,

β = 0 β = 1 β = 2 β = 3 β = 4
α = 0 γ = 1, 2, 3 γ = 1, 2, 3 γ = 1, 2, 3 γ = 1, 2 γ = 1
α = 1 γ = 1, 2, 3 γ = 1, 2, 3 γ = 1, 2
α = 2 γ = 1, 2, 3 γ = 1, 2  γ = 1
α = 3 γ = 1, 2
α = 4  γ = 1

3. < α ⊔ 2T_{β+γ}^{2}, β ⊔ (5 − 2γ)ℝP_{α+γ}^{2} >,

β = 0 β = 1 β = 2 β = 3
α = 0 γ = 1, 2 γ = 1, 2 γ = 1, 2 γ = 1, 2
α = 1 γ = 1, 2 γ = 1, 2 γ = 1, 2 γ = 1
α = 2 γ = 1, 2 γ = 1, 2
α = 3 γ = 1, 2 γ = 1

4. < α ⊔ 3T_{β+1}^{2}, β ⊔ ℝP_{α+1}^{2} >,

\begin{matrix}β = 0 & β = 1 & β = 2 & β = 3 & β = 4 \\ α = 0, 1, 2, 3 & α = 0 & α = 0 & α = 0 & α = 0 \end{matrix}

5. < α ⊔ 2ℝP_{β+γ}^{2}, β ⊔ (7 − 2γ)ℝP_{α+γ}^{2} >,

β = 0 β = 1 β = 2 β = 3 β = 4
α = 0 γ = 1, 2, 3 γ = 1, 2, 3 γ = 1, 2, 3 γ = 1, 2 γ = 1
α = 1 γ = 1, 2, 3 γ = 1, 2, 3 γ = 1, 2 γ = 1
α = 2 γ = 1, 2, 3 γ = 1, 2
α = 3 γ = 1, 2  γ = 1
α = 4  γ = 1

6. < α ⊔ 4ℝP_{β+γ}^{2}, β ⊔ (5 − 2γ)ℝP_{α+γ}^{2} >,

β = 0 β = 1 β = 2 β = 3 β = 4
α = 0 γ = 1, 2 γ = 1, 2 γ = 1, 2 γ = 1, 2 γ = 1
α = 1 γ = 1, 2 γ = 1, 2 γ = 1, 2 γ = 1
α = 2 γ = 1, 2 γ = 1, 2 γ = 1
α = 3 γ = 1, 2 γ = 1
α = 4 γ = 1

7. < α ⊔ 6ℝP_{β+1}^{2}, β ⊔ ℝP_{α+1}^{2} >,

\begin{matrix}β = 0 & β = 1 & β = 2 & β = 3 & β = 4 \\ α = 0, 1, 2, 3, 4 & α = 0, 1 & α = 0, 1 & α = 0, 1 & α = 0 \end{matrix}

8. < 1 ⊔ 3T_{2}^{2}, S_{2}^{2} ⊔ ℝP_{1}^{2} >, < 2T_{3}^{2}, S_{2}^{2} ⊔ ℝP_{1}^{2} >

9. < ∅, 7ℝP^{2} >. Each of the 157 types listed above is realizable by a nonsingular algebraic curve of degree 6.

Theorem 4. Topological arrangement of a nonsingular algebraic curve of degree 6 on a nonsingular cubic surface diffeomorphic to ℝP^{2}#2T^{2} is of one of the following types:

1. < α ⊔ S_{β+γ}^{2}, β ⊔ (7 − 2γ)ℝP_{α+γ}^{2} >, α + β + γ ≤ 5, γ = 1, 2, 3,

2. < α ⊔ T_{β+γ}^{2}, β ⊔ (5 − 2γ)ℝP_{α+γ}^{2} >, α + β + γ ≤ 5, γ = 1, 2,

3. < α ⊔ 2T_{β+1}^{2}, β ⊔ ℝP_{α+1}^{2} >, α + β ≤ 4, α ≠ 4,

4. < α ⊔ 2ℝP_{β+γ}^{2}, β ⊔ (5 − 2γ)ℝP_{α+γ}^{2} >, α + β + γ ≤ 5, γ = 1, 2,

5. < α ⊔ 4ℝP_{β+1}^{2}, β ⊔ ℝP_{α+1}^{2} >, α + β ≤ 4,

6. < 1 ⊔ 2T_{2}^{2}, S_{2}^{2} ⊔ ℝP_{1}^{2} >, < T_{3}^{2}, S_{2}^{2} ⊔ ℝP_{1}^{2} >

7. < ∅, 5ℝP^{2} >. Each of the 113 types listed above is realizable by a nonsingular algebraic curve of degree 6.

Theorem 5. Topological arrangement of a nonsingular algebraic curve of degree 6 on a nonsingular cubic surface diffeomorphic to ℝP^{2}#T^{2} is of one of the following types:

1. < α ⊔ S_{β+γ}^{2}, β ⊔ (5 − 2γ)ℝP_{α+γ}^{2} >, α + β + γ ≤ 5, γ = 1, 2,

2. < α ⊔ T_{β+1}^{2}, β ⊔ ℝP_{α+1}^{2} >, α + β ≤ 4,

3. < α ⊔ 2ℝP_{β+1}^{2}, β ⊔ ℝP_{α+1}^{2} >, α + β ≤ 4,

4. < 1 ⊔ T_{2}^{2}, S_{2}^{2} ⊔ ℝP_{1}^{2} >, < S_{3}^{2}, S_{2}^{2} ⊔ ℝP_{1}^{2} >

5. < ∅, 3ℝP^{2} >. Each of the 58 types listed above is realizable by a nonsingular algebraic curve of degree 6.

4. Auxiliary lemma

Lemma 1 (cf. [1]). B_{+} contains not more than one component of non-positive Euler characteristic. If B_{−} containes more then one component of non-positive Euler characteristic then B_{−} ≈ S_{2}^{2} ⊔ ℝP_{1}^{2} and the cores of S_{2}^{2} and ℝP_{1}^{2} of B_{−} form a cycle dual to w_{1}(ℝB).

Remark. Of course, if ℝA is empty and ℝB is not S^{2} ⊔ ℝP^{2} then all components of positive Euler characteristic are disks.

Proof. Consider the double covering ℂX of ℂB branched along ℂA. The complex conjugation of ℂB can be lifted in two different ways to an involution on ℂY (which differ by the covering automorphism). The fixed point set Y_{+} and Y_{−} of these two liftings c_{+} and c_{−} are double branched coverings of B_{+} and B_{−}. Note that Y_{+} and Y_{−} are orientable (even though B_{−} is never orientable and B_{+} may be non-orientable) since w_{1}(ℝB) is induced by the plane section in ℝP^{3}.

The homology vector space H_{2}(ℂY; ℝ) splits into the orthogonal direct sum of the subspaces H_{2}^{+}(ℂY; ℝ) and H_{2}^{−}(ℂY; ℝ) invariant under actions of both c_{+} and c_{−}, where the intersection form of ℂY is positive definite on H_{2}^{+}(ℂY; ℝ) and negative definite on H_{2}^{−}(ℂY; ℝ). An easy computation shows that

dim H_{2}^{+}(ℂY; ℝ) = 3

(ℂY is a K3-surface). Furthermore, H_{2}^{+}(ℂY; ℝ) splits under the action of c_{+} and c_{−} into (−1)-eigenspaces E_{c_{+}}^{−1} and E_{c_{−}}^{−1} and (+1)-eigenspaces E_{c_{+}}^{+1} and E_{c_{−}}^{+1}. Since c_{+} ∘c_{−} is the covering authomorphism

dim(E_{c_{+}}^{+1} ∩ E_{c_{−}}^{−1}) = 1
dim(E_{c_{+}}^{−1} ∩ E_{c_{−}}^{+1}) = 1.

Note that c_{−} reverses orientation of Y_{+}. Therefore if the linear combination of the homology classes realized by components of Y_{+} is in H_{2}^{+}(ℂY; ℝ) then it is in (E_{c_{+}}^{+1}∩E_{c_{−}}^{−1}). Similarily, if the linear combination of the homology classes realized by components of Y_{−} is in H_{2}^{+}(ℂY; ℝ) then it is in (E_{c_{+}}^{−1} ∩ E_{c_{−}}^{+1}).

Note that the self-intersection number of a component of Y_{+} (Y_{−}) in ℂY is equal to (−1) times its Euler characteristic since the multiplication by i provides an orientationreversing isomorphism between the tangent and the normal bundles of Y_{+} (Y_{−}) in ℂY. Therefore, the number of the components of Y_{+} (Y_{−}) with non-positive Euler characteristic is not greater then dim(E_{c_{+}}^{+1} ∩ E_{c_{−}}^{−1}) = 1 (dim(E_{c_{+}}^{−1} ∩ E_{c_{−}}^{+1}) = 1) plus the number of linear relations on the ℝ-homology classes of these components.

Since H_{1}(ℂY) = 0 the universal coefficients theorem implies that the number of linear relations on the ℝ-homology classes of the components of Y_{+} (Y_{−}) with non-positive Euler characteristic is not more than the number of corresponding ℤ_{2}- relations. The Smith exact sequence (see e.g. appendix in [16]) assures that the only possible non-trivial linear relation on the Z_{2}-homology classes of Y_{+} (Y_{−}) is the sum of all of them. Therefore the maximal number of the ℝ-relations is 1 and it can be reached only if all the components of Y_{+} (Y_{−}) are of non-positive Euler characteristic. Furthermore, if at least one of them is of positive Euler characteristic then they are linearily independent over ℝ since the only possible ℝ-relation (the sum of them taken with non-zero coefficients) intersects that component non-trivially.

Therefore, if there is more than one component of B_{+} (B_{−}) of non-positive Euler characteristic then B_{+} (B_{−}) consists of two components of zero Euler characteristic. In addition Y_{+} (Y_{−}) is ℤ_{2}-homologous to zero in ℂY. The latter implies that ℝA is ℤ_{2}-homologous to zero in ℂA. Indeed, if there exists a loop in ℂA which intersects ℝA once then a disk bounded by this loop in ℂB (recall that π_{1}(ℂB) = 0) is a closed surface which intersects Y_{+} (Y_{−}) in odd number of points. This implies in turn that the number of components of ℝA is of the opposite pairity with the genus of ℂA since ℝA separates ℂA into two diffeomorphic complex conjugate orientable surfaces.

There are two types of connected surfaces of zero Euler characteristic with boundary — annuli and Möbius bands. The above discussion implies that the only possibility for the surface B_{+} (B_{−}) to have more than one component of non-positive Euler characteristic is to consist of an annulus and a Möbius band. By our convention (see 3.1) this surface is not B_{+}.

If the Euler characteristic of ℝB is positive (ℝB ≈ S^{2} ⊔ ℝP^{2} or ℝB ≈ ℝP^{2}) then the proof of the Lemma is finished since w_{1}(ℝB) is the only non-trivial element in H_{1}(ℝB; ℤ_{2}). To finish the proof for ℝB of negative Euler characteristic we consider the birational equivalence β : ℂB → ℂP^{2} which is the blowup in 6 points. The image of ℝA under β is a curve ℝC of degree 6 in ℝP^{2}. which has singularities of multiplicity 2 at the points of blowup. We have to prove that the image of the union of the cores of the annulus and the Möbius band is dual to w_{1}(RP^{2}) and contains all points of blowup.

We deduce it from the Bezout theorem. The cores are disjoint so if the image of the cores is not dual to w_{1}(ℝP^{2}) then the image of the each core bounds a disk in ℝP^{2}. Pick a point in each of these disks which does not belong to the image of B_{−} and draw a line through these points. This line intersects ℝC at least in 8 points which contradicts to the Bezout theorem. Therefore, the image of exactly one of the cores bounds a disk. If there is a blowup point on ℝP^{2} which does not belong to the image of either of the cores then we draw a line through that blowup point and a point on the disk which does not belong to B_{−}. This line intersects ℝC at least in 6 points other than the blowup point (which is of multiplicity 2).

5. Proof of Theorem 1

The proof of Theorem 1 (as well as the proofs of Theorems 2, 3, 4 and 5) consists of two parts — the part ”Restrisctions” where we prove that the topological arrangements not listed in the theorem are not realizable and the part ”Constructions” where we construct the curves with the listed topological arrangements.

5.1. Restrictions. Recall (see section 1) that by the Harnack inequality the maximal number of components of ℝA is 5. Lemma 1 implies that unless (ℝB, ℝA) is of type d of Theorem 1 there is no more than one component of ℝA which does not bound a disk in ℝB disjoint from the other components of ℝA. If this component belongs to the sphere component of ℝB then Lemma 1 implies that (ℝB, ℝA) is of type < ∅ >_{ℝP^{2}} ⊔ < 1⊔1<1 >>_{S^{2}}. If this component belongs to the plane component of ℝB then Lemma 1 implies that there is no more than one component of ℝA on the sphere component of ℝB.

Thus we have only to prove that the following arrangements do not appear

< α ⊔ 1<β >>_{ℝP^{2}} ⊔ < ∅ >_{S^{2}}, if α > 0, β > 1,
< α ⊔ 1<β >>_{ℝP^{2}} ⊔ < 1 >_{S^{2}}, if β > 0, unless α = 0, β = 1.

Lemma 5.1. ℝB is of type Irel, i.e. ℝB is dual to the second Stiefel-Whitney class w_{2}(ℂB) of ℂB.

Proof. The sphere component of ℝB bounds a ball in ℝP^{3}. Let x be a point inside of that ball. Then any line in ℝP^{3} passing through x intersects the sphere component of ℝB in 2 points. But it also intersects the plane component of ℝB because of homology reasons. Therefore, ℝB is the full inverse image of ℝP^{2} under the projection ℂB → ℂP^{2} from x. The lemma now follows since ℝP^{2} is dual to w_{2}(ℂP^{2}).

This lemma allows us to apply the results of [11]. By Theorem 1 and Addendum 1 of [11] ℝA bounds such a surface B_{1} in ℝB that

χ(B_{1}) ≡ 5 (mod 8), if the number of components of ℝA is 5, χ(B_{1}) ≡ 4 or 6 (mod 8), if the number of components of ℝA is 4.

Therefore if < α ⊔ 1<β >>_{ℝP^{2}} ⊔ < 1 >_{S^{2}} is realisable then

2 + α − β ≡ 5 (mod 8), if α + β = 3,
2 + α − β ≡ 4 or 6 (mod 8), if α + β = 2.

Neither of these congruences is possible if β > 0.

To apply the congruence for χ(B_{1}) to < α ⊔ 1 < β >>_{ℝP^{2}} ⊔ < ∅ >_{S^{2}} we have to consider two possibilities for B_{1} — the sphere component of ℝB is either contained in B_{1} or not. We get

1 + α − β ≡ 5 (mod 8) or 3 + α − β ≡ 5 (mod 8), if α + β = 4,
1 + α − β ≡ 4 or 6 or 3 + α − β ≡ 4 or 6, if α + β = 3.

This is not possible if α > 0 and β > 1.

5.2. Constructions. To construct < 3⊔1<1>>_{ℝP^{2}} ⊔ < ∅ >_{S^{2}} we perturb a union of two plane sections of ℝB.

Lemma 5.2. There exists a plane section ℝH of ℝB which consists of two components which both belong to the plane component of ℝB.

Figure 1. Construction of ℝH.

Figure 2. < 3 ⊔ 1<1>>_{ℝP^{2}} ⊔ < ∅ >_{S^{2}}.

Proof. Consider a generic plane P in ℝP^{3} tangent to the plane component of ℝB. The intersection P ∩ℝB is a singular curve with an ordinary double point. Varying P a little we can make ℝH = P ∩ ℝB non-singular and non-connected (see Fig. 1).

Let l = 0 be the equation of the plane in ℝP^{3} intersecting ℝH in three distinct points. Let ℝH′ ⊂ ℝB be the curve defined by equation h + ϵl = 0, where h is the equation of ℝH and ϵ > 0 is small. Perturbing ℝH ∪ ℝH′ we get the topological arrangement < 3⊔1<1>>_{ℝP^{2}} ⊔ < ∅ >_{S^{2}} (see Fig. 2). The same technique produces < 2 ⊔ 1 < 1 >>_{ℝP^{2}} ⊔ < ∅ >_{S^{2}} and < 1 ⊔ 1 < 1 >>_{ℝP^{2}} ⊔ < ∅ >_{S^{2}} if l = 0 intersects ℝH in 2 points (one is a tangent point) and in 1 point respectively. This constructs the type a of Theorem 1 with the exception of < 1<1>>_{ℝP^{2}} ⊔ < ∅ >_{S^{2}}.

We construct < 1 < 1 >>_{ℝP^{2}} ⊔ < ∅ >_{S^{2}}, < 1 < 1 < 1 >>>_{ℝP^{2}} ⊔ < ∅ >_{S^{2}} (type d), < 1 < 1 >>_{ℝP^{2}} ⊔ < 1 >_{S^{2}} (type e) and < ∅ >_{ℝP^{2}} ⊔ < 1 ⊔ 1 < 1 >>_{S^{2}} (type f) on the cubic surface ℝB ⊂ ℝP^{3} given in the affine coordinates (x, y, z) in ℝ^{3} ⊂ ℝP^{3} by equation

x^{2} + y^{2} = z(z^{2} − 1).

The required topological arrangements are cut on ℝB by equations (z − C_{1})(z − C_{2})(z − C_{3}) = 0 where we vary constants C_{1}, C_{2} and C_{3} (see Fig.3).

To construct types b and c we perturb the union of a plane P ⊂ ℝP^{3} and an ellipsoid ℝQ ⊂ ℝP^{3} such that ℝH = P ∩ ℝQ is a non-singular non-empty conic. If p = 0 is the equation of the plane P and q = 0 is the equation of the ellipsoid ℝQ

Figure 3. Curves on x^{2} + y^{2} = z(z^{2} − 1).

Figure 4. Construction of the types b and c of Theorem 1.

then ℝB is given by equation

pq + ϵs = 0,

where s is an auxiliary cubic polynomial which cuts a non-singular curve ℝS ⊂ ℝQ not intersecting P ∩ ℝQ and ϵ ≠ 0 is small.

Let ℝA = ℝB∩ℝQ. The topological type of (ℝB, ℝA) depends only on the mutual position of ℝH and ℝS in ℝQ and the sign of ϵ (see Fig. 4). The curve ℝH splits ℝQ into 2 disks Q_{1} and Q_{2}. For the arrangements of type c we need to construct for any α, β, α + β ≤ 5, a curve ℝS such that ℝS ∩ Q_{1} bounds α disjoint disks in Q_{1} and ℝS ∩ Q_{2} bounds β disjoint disks in Q_{2}. For the arrangements of type 1.b we need to construct for any 2 ≤ α ≤ 4 a curve ℝS such that ℝS ∩ Q_{1} has a component which bounds a disk in Q_{1} which contains all other components of ℝS (see Fig. 5).

To construct ℝS ⊂ ℝQ consisting of 5 components we petrurb the union of 3 plane sections of ℝQ which intersect as shown on Fig. 6. To construct ℝS ⊂ ℝQ consisting of 4, 3, 2, 1 or 0 components we just make some of the plane sections (or their intersections) on Fig. 6 imaginary. It is easy to find all the needed positions of

Figure 5. Auxiliary curves for the types c and b of Theorem 1.

Figure 6. Construction of ℝS.

Figure 7. Different arrangements of a quartic and a conic.

the plane section ℝH directly from Fig. 6.

6. Proof of Theorem 2

All the needed restrictions for Theorem 2 are provided by Lemma 1. To construct the curves we obtain the cubic surface ℝB as the real part of the blowup β : ℂB → ℂP^{2} in 6 imaginary points of ℂP^{2} (these 6 points come in 3 complex conjugate pairs and they do not belong to the same conic). Thus to construct a curve ℝA ⊂ ℝB of degree 6 it suffices to construct a plane curve ℝC ⊂ ℝP^{2} such that the singular points of its complexifications ℂC are 6 imaginary ordinary double points which do not belong to the same conic. It is easy to construct a degree 6 curve decomposing to the union of a quartic and a conic transverse to each other and with no real intersection points for every topological arrangement listed in Theorem 2 (see Fig. 7). This curve has 8 imaginary ordinary double points which belong to the same conic. To get ℝC we perturb this curve so that 2 double points disappear and the other 6 survive and move to a general position (cf. [5]).

7. Proof of Theorem 3

7.1. Restrictions. Lemma 1 implies that if B_{−} contains more than one component of non-positive Euler characteristic then (ℝB, ℝA) is of type 8 of Theorem 3. Otherwise the type of (ℝB, ℝA) is

< α ⊔ S_{β+γ}^{2}, β ⊔ kℝP_{α+γ}^{2} >,
< α ⊔ jT_{β+γ}^{2}, β ⊔ kℝP_{α+γ}^{2} >,
or < α ⊔ 2jℝP_{β+γ}^{2}, β ⊔ kℝP_{alpha+γ}^{2} >,
where α ≥ 0, β ≥ 0, γ ≥ 1, α + β + γ ≤ 5.

The number k is determined by the equality χ(B_{+}) + χ(B_{−}) = χ(ℝB) = −5. We get k = 9 − 2j − 2γ (we assume that j = 0 in the first case). Since k ≥ 0 we obtain

γ < 4 − j.

Therefore to get all the restrictions for Theorem 3 we have to prove that the following 22 types are not realisable.

• < α ⊔ S_{β+γ}^{2}, β ⊔ (9 − 2γ)ℝP_{α+γ}^{2} >, where (α, β, γ) = (0, 3, 1), (0, 4, 1), (1, 2, 1), (1, 3, 1), (2, 2, 1) or (4, 0, 1). These types are not realisable because of the Rokhlin ([13]) and Kharlamov-Gudkov-Krakhnov ([8], [4]) congruences

χ(B_{+}) = α − β + 1 ≡ 3 (mod 8), if α + β = 4 and γ = 1,
χ(B_{+}) = α − β + 1 ≡ 2 or 4 (mod 8), if α + β = 3 and γ = 1.

We apply the congruences in the form of Theorem 7.1 of [11] which is more convenient for our application (note that it is the condition that γ = 1 which implies the hypothesis e = 0 of the theorem).

• < α ⊔ T_{β+γ}^{2}, β ⊔ (7 − 2γℝP_{α+γ}^{2} >, where (α, β, γ) = (1, 3, 1) or (3, 1, 1). These types are not realizable since Theorem 7.4.d of [11] implies that

α − β − 1 ≡ 3 (mod 4), if α + β = 4 and γ = 1.

• < α ⊔ 2T_{β+γ}^{2}, β ⊔ (5 − 2γ)ℝP_{α+γ}^{2} >, where (α, β, γ) = (0, 4, 1), (2, 2, 1) or (4, 0, 1). These types are not realizable since Theorem 7.4.d of [11] implies that

α − β − 3 ≡ 3 (mod 4), if α + β = 4 and γ = 1.

• < α ⊔ 3T_{β+1}^{2}, β ⊔ ℝP_{α+1}^{2} >, where (α, β) = (4, 0), (1, 1), (2, 1), (3, 1), (1, 2), (2, 2) or (1, 3). These types are not realizable since Theorem 1 of [11] implies

χ(B_{−}) ≡ 3 + B (mod 8),

where B is the Brown invariant of the Guillou-Marin form of ℂA/ conj ∪B_{−}. But B ≡ 1 (mod 8) if ℝA is an M-curve and B ≡ 0 or 2 (mod 8) if ℝA is an (M-1)-curve (cf. Addenda 1.a, 1.b of [11]) since the only class in H_{1}(B_{−}; ℤ_{2}) which does not vanish in H_{1}(ℝB; ℤ_{2}) is the class dual to w_{1}(ℝB) and the value of the Guillou-Marin form of ℂA/ conj ∪B_{−} on such a class is 1. Therefore

β − α ≡ 4 (mod 8), if α + β = 4 and γ = 1,
β − α ≡ 3 or 5 (mod 8), if α + β = 3 and γ = 1.

This rules out the pairs (α, β) = (2, 1), (3, 1), (1, 2), (2, 2), (1, 3). In a similar way Addendum 1.c of [11] implies that (α, β) = (1, 1) is not realizable by a curve of type II (i.e. if ℂA/ conj is non-orientable) since β − α ≡ 0 (mod 8) in this case.

To see that (α, β) = (4, 0) is not realizable we apply a version of the Rokhlin complex orientation formula [14]. Note that B_{+} in this case is a disjoint union of 4 disks and a punctured sphere with 3 handles and therefore F = ℂA/ conj ∪B_{+} ⊂ ℂB/ conj ≈ S^{4} (see [9]) is orientable. Therefore F.F = 0 but on the other hand F.F = \frac{1}{2}ℂB.ℂB − 2χ(B_{+}) = 6 + 2 = 8. A similar argument shows that (α, β) = (1, 1) is not realizable by a curve of type I (i.e. if ℂA/ conj is orientable). In this case B_{+} is a disjoint union of a disk and a sphere with 3 handles punctured two times and B_{−} is a disjoint union of a disk and a projective plane punctured two times. Denote the disk component of B_{−} by D_{−} the disk component of B_{+} by D_{+} and the union of D and the non-disk component of B_{+} by E. Note that G = ℂA/ conj ∪D_{+} ∪ E ∪ D_{−} is a ℤ-cycle in ℂB/ conj ≈ S^{4} and thus G.G = 0. If the orientations of ℝA = ∂ℂA/ conj = ∂B_{+} induced from ℂA/ conj and from B_{+} agree then G.G = \frac{1}{2}ℂB.ℂB − 2(χ(D_{+}) + χ(E) + χ(D_{−})) + 4 = 16 ≠ 0. If the orientations do not agree then G.G = \frac{1}{2}ℂB.ℂB − 2(χ(D_{+}) + χ(E) + χ(D_{−})) − 4 = 8 ≠ 0.

• < 2 ⊔ 2ℝP_{3}^{2}, 2 ⊔ 5ℝP_{3}^{2} >. This type is not realisable because of Theorem 7.4.c of [11], since χ(B_{+}) = −1 and B_{+} is not orientable.

• < α ⊔ 6ℝP_{β+1}^{2}, β ⊔ ℝP_{α+1}^{2} >, where (α, β) = (2, 1), (3, 1) or (2, 2). The pair (2, 1) is ruled out by Theorem 7.4.c of [11] applied to B_{−} since χ(B_{−}) = β − α = −1 and B_{−} is not orientable. The pairs (3, 1) and (2, 2) are ruled out by Theorem 7.4.b of [11].

7.2. Constructions. Similarily to the proof of Theorem 2 we construct the curves for Theorem 3 perturbing the union of a quartic and a conic. This time we have to use a quartic and a conic which intersect in at least 6 real points since we need 6 points in ℝP^{2} to blow up to get the cubic surface diffeomorphic to 7ℝP^{2}. We use Polotovskii’s catalog [12] which lists arrangements of a conic and a quartic which intersect in 8 real points. Then we smooth 2 out of the 8 points so that the other 6 move to a general position with respect to each other (cf. [5]). One curve in Polotovskii’s catalog may produce more than one type for Theorem 3 because of the ambiguity in the choice of the 2 points to smooth and in (real) smoothing of these points.

Example 7.1. The curve encoded by (12387456)[0] in Polotovskii’s catalog [12] produces < 3⊔S_{2}^{2}, 5ℝP_{5}^{2} >, < 4⊔T_{1}^{2}, 5ℝP_{5}^{2} >, < 3⊔T_{2}^{2}, 3ℝP_{5}^{2} >, < 3⊔2ℝP_{2}^{2}, 1⊔5ℝP_{4}^{2} > and < 3 ⊔ 2ℝP_{2}^{2}, 3ℝP_{5}^{2} > out of M-curves (see Fig. 8) and < 3 ⊔ T_{1}^{2}, 5ℝP_{4}^{2} > and < 3 ⊔ 2ℝP_{1}^{2}, 5ℝP_{4}^{2} > out of (M − 1)-curves.

The catalog [12] lists only arrangements of M-decomposing curves thus the quartic in the union consists of 4 ovals. But if an oval of the quartic is disjoint from the conic then we can remove it so that the resulting arrangement is still realizable by a decomposing curve of degree 6. This means that if a triple (α, β, γ) is obtained from a decomposing curve then (α′, β′, γ) with α′ < α and β′ < β can be also obtained in this way. In the following tables we indicate the code of a union of a conic and a

TOPOLOGY OF CURVES OF DEGREE SIX ON CUBICS 15

Figure 8. Curves produced by (12387456)[0]

quartic in [12] for arrangements in Theorem 3 with maximal (α, β).

1. < α ⊔ S_{β+γ}^{2}, β ⊔ (9 − 2γ)ℝP_{α+γ}^{2} >,

β = 0  β = 1  β = 2  β = 3
γ = 4  γ = 3  γ = 2
α = 0  (12)(34)(56)(78) (1867)(3452)[2] (18276543)[3]
γ = 1

*

γ = 4 γ = 3 γ = 2
α = 1 (12)(34)(56)(78) (1678)(2345)[1] (18723456)[2]
γ = 3  γ = 2
α = 2 (145678)(23)[0]  (12345678)[1]
γ = 2  γ = 1
α = 3 (12387456)[0] (12345678)[1]

2. < α ⊔ T_{β+γ}^{2}, β ⊔ (7 − 2γ)ℝP_{α+γ}^{2} >,

16 G.MIKHALKIN

β = 0  β = 1  β = 2  β = 3  β = 4
γ = 3  γ = 2  γ = 1
α = 0  (1876)(2345)[2] (18743256)[3] (18276543)[3]
γ = 3  γ = 2
α = 1  (187654)(23)[1] (18723456)[2]
γ = 3 γ = 2 γ = 1
α = 2 (1867)(3452)[2] (18765234)[1] (18723456)[2]
γ = 2
α = 3 (12387456)[0]
γ = 1
α = 4 (12387456)[0]

3. < α ⊔ 2T_{β+γ}^{2}, β ⊔ (5 − 2γ)ℝP_{α+γ}^{2} >,

β = 0 β = 1 β = 2 β = 3
γ = 2
α = 0  (18743256)[3]
γ = 2  γ = 1
α = 1  (18765432)[2] (18234765)[2]
γ = 2
α = 2  (18765234)[1]
γ = 2  γ = 1
α = 3 (18276543)[3] (18765234)[1]

4. < α ⊔ 3T_{β+1}^{2}, β ⊔ ℝP_{α+1}^{2} >,

\begin{matrix}β = 0 & β = 4 \\ α = 3 & α = 0 \\ (18276543)[3] & (18234567)[3] \end{matrix}

5. < α ⊔ 2ℝP_{β+γ}^{2}, β ⊔ (7 − 2γ)ℝP_{α+γ}^{2} >,

β = 0 β = 1 β = 2 β = 3 β = 4
γ = 3  γ = 2 γ = 1
α = 0  (1867)(3452)[2] (18276543)[3]  *
γ = 3  γ = 2  γ = 1
α = 1  (1845)(23)(67)[0] (18723456)[2] (18723456)[2]
γ = 3  γ = 2
α = 2 (1845)(3672)[0]  (18432765)[1]
γ = 2  γ = 1
α = 3 (12387456)[0]  (12387456)[0]
γ = 1
α = 4 (16254378)[0]

6. < α ⊔ 4ℝP_{β+γ}^{2}, β ⊔ (5 − 2γ)ℝP_{α+γ}^{2} >,

β = 0 β = 1 β = 2 β = 3 β = 4
γ = 2  γ = 1
α = 0  (18437625)[3] (18437625)[3]
γ = 2  γ = 1
α = 1  (18432765)[1] (18437625)[3]
γ = 2  γ = 1
α = 2  (18765234)[1] (18765234)[1]
γ = 2  γ = 1
α = 3 (18276345)[0] (18276345)[0]
γ = 1
α = 4 (18276345)[0]

7. < α ⊔ 6ℝP_{β+1}^{2}, β ⊔ ℝP_{α+1}^{2} >,

\begin{matrix}β = 0 & β = 3 & β = 4 \\ α = 4 & α = 1 & α = 0 \\ * & (18765432)[2] & (18234567)[3] \end{matrix}

To construct the arrangements marked by * in the table we start from a plane curve of degree 6 and collapse 6 of its empty ovals to points using [7]. To construct < 2ℝP_{5}^{2}, 4⊔5ℝP_{1}^{2} > and < 4⊔6ℝP_{1}^{2}, ℝP_{5}^{2} > we use the Gudkov curve < 5⊔1<5>> constructed in [2]. To get < 2ℝP_{5}^{2}, 4 ⊔ 5ℝP_{1}^{2} > we collapse the 5 inner ovals and one of the outer ovals; to construct < 4 ⊔ 6ℝP_{1}^{2}, ℝP_{5}^{2} > we collapse the 5 outer ovals and one of the inner ovals (see Fig. 9). To construct < S_{3}^{2}, 2 ⊔ 7ℝP_{1}^{2} > we collapse the 6 outer ovals of < 6 ⊔ 1<2>>.

8. < 1 ⊔ 3T_{2}^{2}, S_{2}^{2} ⊔ ℝP_{1}^{2} >, < 2T_{3}^{2}, S_{2}^{2} ⊔ ℝP_{1}^{2} > To construct the plane curves of degree 6 with 6 double points for these arrangements we start from the union of a two-component cubic in ℝP^{2} and its ”parallel” copy which intersect in 9 disjoint point and smooth 3 out of the 9 nodes of the union (see Fig. 10).

Figure 9. < 2 ⊔ ℝP_{5}^{2}, 4 ⊔ 5ℝP_{1}^{2} > and < 4 ⊔ 6ℝP_{1}^{2}, ℝP_{5}^{2} >.

Figure 10. < 1 ⊔ 3T_{2}^{2}, S_{2}^{2} ⊔ ℝP_{1}^{2} > and < 2T_{3}^{2}, S_{2}^{2} ⊔ ℝP_{1}^{2} >.

Figure 11. < 1 ⊔ S_{4}^{2}, 3 ⊔ 5ℝP_{2}^{2} > and < 2 ⊔ 2T_{3}^{2}, 2 ⊔ ℝP_{3}^{2} >.

8. Proof of Theorem 4 and Theorem 5

8.1. Restrictions. Lemma 1 implies that unless (ℝB, ℝA) is of type 6 of Theorem 4 or of type 4 of Theorem 5 it is of type

< α ⊔ S_{β+γ}^{2}, β ⊔ kℝP_{α+γ}^{2} >,
< α ⊔ jT_{β+γ}^{2}, β ⊔ kℝP_{α+γ}^{2} >,
or < α ⊔ 2jℝP_{β+γ}^{2}, β ⊔ kℝP_{α+γ}^{2} >,
where α ≥ 0, β ≥ 0, γ ≥ 1, α + β + γ ≤ 5.

In the case of Theorem 4 k = 7 − 2j − 2γ ≥ 0 and in the case of Theorem 5 k = 5 − 2j − 2γ ≥ 0 since χ(B_{+}) + χ(B_{−}) = χ(ℝB). To finish the restrictions for Theorem 4 and Theorem 5 we have to prove that < 4 ⊔ 2T_{1}^{2}, ℝP_{5}^{2} > is not realizable (for Theorem 4). Note that F = ℂA/ conj ∪B_{+} ⊂ ℂB/ conj ≈ \overline{ℂP}^{2} (see [9]) is orientable and therefore F.F ≤ 0 (\overline{ℂP}^{2} is negative definite). On the other hand F.F = \frac{1}{2}ℂB.ℂB − 2χ(B_{+}) = 6 − 2 = 4.

8.2. Constructions. If < j ⊔F #T^{2}, n⊔G > or < j ⊔F, n⊔G#T^{2} > is constructed in 7.2 out of Polotovskii’s catalog then we can construct < j ⊔ F, n ⊔ G > out of a similar pair of a conic and a quartic which intersect in 6 real points. Considering similar pairs of a conic and a quartic which intersect in 4 real points produces the arrangements for Theorem 5. A similar modification works for < 1⊔2T_{2}^{2}, S_{2}^{2} ⊔ℝP_{1}^{2} > and < T_{3}^{2}, S_{2}^{2} ⊔ ℝP_{1}^{2} >. This produces all the arrangements for Theorem 4 except for < 1 ⊔ S_{4}^{2}, 3 ⊔ 5ℝP_{2}^{2} > and < 2 ⊔ 2T_{3}^{2}, 2 ⊔ ℝP_{3}^{2} >. To construct them we perturb a union of a quartic and two lines as shown on Fig. 11.

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Department of Mathematics, University of Toronto, Toronto, Ont. M5S 3G3, Canada

Current address: Mathematical Science Research Institute, 1000 Centennial Dr., Berkeley, CA 94720, USA

E-mail address: mihalkin@math.toronto.edu, grisha@msri.org