CONGRUENCE MODULO 8 FOR REAL

ALGEBRAIC CURVES OF DEGREE 9

O.Ya. Viro, S.Yu. Orevkov

1. Introduction and statement of the result. Let A be a non-singular real algebraic curve of degree m in RP^{2}. Its connected components are embedded circles. Those of them whose complement in RP^{2} is not connected are called ovals. One says that an oval u lies inside an oval v if u is contained is the orientable component of the complement of v. A union of d ovals v_{1}, …, v_{d} such that v_{i} is inside v_{i+1}, 1 ≤ i < d, is called a nest of the depth d. An oval is called exterior if it does not lye inside any other oval; an oval is called empty if there is no other ovals inside it. An oval is called even if it is contained inside an even number of other ovals, and odd otherwise. Denote by p and n the number of even and odd ovals respectively. One says that A is an M-curve if it has the maximal possible number of connected components which equals M(m) = (m − 1)(m − 2)/2 + 1. If A has M(m) − i connected components then it is called an (M − i)-curve Let CA be the complexification of A. If CA \ A is not connected, A is a curve of type I; if CA \ A is connected then A is a curve of type II.

For curves of an even degree m = 2k, in some cases, the difference p − n satisfies congruences. For example, Gudkov-Rohlin congruence p − n ≡ k^{2} mod 8 for M-curves, Gudkov-Krahnov-Kharlamov congruence p−n ≡ k^{2} ±1 mod 8 for (M −1)-curves, Kharlamov-Marin congruence p − n ≢ k^{2} + 4 mod 8 for M-curves of type II, and Arnold congruence p − n ≡ k^{2} mod 4 for curves of type I. These statements do not extend to curves of odd degrees. So, for an M-curve of any odd degree 2k + 1 with k ≥ 3, the residue p − n mod 8 may take any values congruent to k mod 2. As far as we know, the following theorem is the first result of this kind.

Theorem 1. Let A be a curve of degree m = 2k + 1 = 4d + 1 which has 4 paiwise distinct nests of the depth d. Then

if A is an M-curve then  p − n ≡ −k  mod 8;  (1)
if A is an (M − 1)-curve then  p − n ≡ −k ± 1 mod 8;
if A is an (M − 2)-curve of type II then p − n ≢ −k + 4 mod 8;
if A is a curve of type I then  p − n ≡ −k  mod 4;

It is clear that (1) for d = 2 is equivalent to the fact that the number of exterior empty ovals of an M-curve of degree 9 with 4 nests is divisible by 4. This was conjectured by Korchagin [2]. Theorem 1 is obtained below (see Sect. 4) as a consequence of Kharlamov-Viro congruence [1] which generalizes the classical congruences to the case of singular curves of even degrees.

Typeset by AMS-TEX

2. Brown - van der Blij invariant. By a quadratic space we mean a triple (V, ∘, q) composed of a vector space V over the field Z_{2}, a bilinear form V ×V → Z_{2}, (x, y) ↦ x ∘ y, and a function q : V ↦ Z_{4} which is quadratic with respect to ∘ in the sense that q(x + y) = q(x) + q(y) + 2x ∘ y. A quadratic space is determined by its Gram matrix with respect to a base e_{1}, …, e_{n} of V, i.e. the matrix Q = (q_{ij}) where q_{ii} = q(e_{i}) and q_{ij} = e_{i} ∘ e_{j} for i ≠ j (the diagonal entries are defined mod 4, the others mod 2; note that q(x) ≡ x ∘ x mod 2). It is easy to see that by elementary changes of the base, one can put the Gram matrix to the block-diagonal form diag(d_{1}, …, d_{t})⊕Q_{1} ⊕⋯⊕Q_{s} where each block Q_{i} is either ( \begin{matrix}0 & 1 \\ 1 & 0 \end{matrix} ), or ( \begin{matrix}2 & 1 \\ 1 & 2 \end{matrix} ). If all d_{i} ≠ 2, we say that the form q is informative and in this case we define its Brown - van der Blij invariant B(q) = ∑ B(d_{i})+∑ B(Q_{i}) mod 8 where B(0) = 0,

B(1) = 1, B(−1) = −1, B ( \begin{matrix}0 & 1 \\ 1 & 0 \end{matrix} ) = 0, and B ( \begin{matrix}2 & 1 \\ 1 & 2 \end{matrix} ) = 4.

3. Kharlamov-Viro congruence for nodal curves. Let A be a curve in RP^{2} of degree 2k defined by f = 0 and let each of its singular points be the point of transverse intersection of two smooth real local branches. A is called an M-curve (a curve of type I) if the normalization of any its irreducible component is an M-curve (a curve of type I). A curve which is not of type I, is of type II. Let x_{1}, …, x_{s} be the singular points and Γ_{A} be the union of the connected components of A passing through them. Let b = 0 if RP_{+}^{2} = {f ≥ 0} is contractable in RP^{2} and b = (−1)^{k} otherwise.

Suppose that Γ_{A} is connected. Let us define a quadratic space (V, ∘, q) as follows. Let C_{1}, …, C_{r} be the oriented components of RP^{2} \ Γ_{A} on which f > 0 near Γ_{A}. Let (V_{0}, ∘, q_{0}) be the quadratic space with the orthogonal base e_{1}, …, e_{s} such that q_{0}(e_{1}) = ⋯ = q_{0}(e_{s}) = −1. Set c_{i} = ∑_{j∈α_{i}} e_{j} where {x_{j}}_{j∈α_{i}} are the singular points through which ∂C_{i} passes only once. In the cases when either Γ_{A} is contractible in RP^{2} or, as in Sect. 4, there is a branch of Γ_{A} (i.e. a smoothly immersed circle) which is non-contractible in RP^{2}, we define V ⊂ V_{0} as the subset generated by c_{1}, …, c_{r} and we set q = q_{0}|_{V}.

In the case when Γ_{A} is not contractible in RP^{2} but all its branches are, let us choose a simple closed curve in Γ_{A} which is not contractible in RP^{2}. Let (V′_{0}, ∘, q′_{0}) be the quadratic space with the base (e_{0}, …, e_{s}) which contains V_{0} as a quadratic subspace (q′_{0}|_{V_{0}} = q_{0}) and let q′_{0}(e_{0}) = (−1)^{k}, e_{0} ∘ e_{j} = 0 iff L ∼ 0 in H_{1}(RP_{+}^{2}, RP_{+}^{2} \ x_{j}). Let V ⊂ V′_{0} be the subspace generated by c_{1}, …, c_{r}, and e_{0} + ∑_{j∈α_{0}} e_{j} where α_{0} = {j | L ≁ 0 in H_{1}(RP_{−}^{2}, RP_{−}^{2} \ x_{j})}, and let q = q′_{0}|_{V}.

If Γ_{A} is not connected, we define (V, ∘, q) as the direct sum of quadratic spaces associated as above to each connected component of Γ_{A}.

Theorem 2. Suppose that each branch of A which is contractible in RP^{2} cuts other branches at n ≡ 0 mod 4 singular points and each branch which is not contractible in RP^{2}, at n ≡ (−1)^{k+1} mod 4 singular points. If A is an M-curve then χ(RP_{+}^{2}) ≡ k^{2} + B(q) + b mod 8 and also the corresponding analogues of Gudkov-Krahnov-Kharlamov, Kharlamov-Marin, and Arnold congruences take place.

Theorem 2 is a corollary of Theorem (3.B) on curves with arbitrary singularities from the paper by Kharlamov and Viro [1]. Theorem 2 is formulated here because there are mistakes in [1] in the discussion of the corresponding particular case (4.I), (4.J) of Theorem (3.B).

4 Proof of Theorem 1. Let us choose any three pairwise distinct nests of the depth d and a point inside the innermost oval of each of them. Theorem 1 follows from Theorem 2 applied to the union of A and the three straight lines passing through the three chosen points. Indeed, the union of the three chosen lines and the non-contractible branch of A divides RP^{2} into 4 triangles and 3 quadrangles (curvilinear). All ovals not belonging to the three chosen nests lye in the quadrangles (otherwise would exist a conic having too many intersections with A). Therefore, after the suitable choice of the sign, one has χ(RP_{+}^{2}) = χ(⋃ \overline{C}_{j}) + p′ − n′ where p′ and n′ are the numbers of even and odd ovals, not belonging to the three chosen nests. B(q) can be computed according to Sect. 2.

References

1. V.M. Kharlamov, O.Ya. Viro, Extension of the Gudkov-Rohlin congruence, Lect. Notes. Math. 1346 (1988), 357–406.

2. A.B. Korchagin, Construction of new M-curves of 9th degree, Lect. Notes. Math. 1524 (1991), 407–426.

St.Petersburg Branch of Steklov Math. Inst. and Upsala Univ.

Steklov Math. Inst. (Moscow) and Univ. Paul Sabatier (Toulouse)