Link theory and oval arrangements of real algebraic curves.

*(English)*Zbl 0923.14032
Topology 38, No. 4, 779-810 (1999); erratum 41, No. 1, 211-212 (2002).

This article is a significant contribution to the first part of Hilbert’s 16th problem, topology of real plane algebraic curves. It contains a new method of obtaining restrictions to the disposition of branches of a real algebraic curve in the plane. The idea of the method is as follows. The union \(\Lambda_p\) of the complex points of real straight lines through a point \(p\in{\mathbb{R}}\mathbb{P}^2\) is a three-dimensional manifold which cuts \({\mathbb{C}}\mathbb{P}^2\) into two parts. Given the complexification \(A\subset{\mathbb{C}}\mathbb{P}^2\) of a real nonsingular algebraic curve, the intersection \(\Lambda_p \cap A\) is a union of intersecting circuits. Pushing \(\Lambda_p\) into one of the components of \({\mathbb{C}}\mathbb{P}^2\backslash\Lambda_p\), one turns \(\Lambda_p\cap A\) into a link \(L\) in \(S^3\) bounding an embedded surface \(N\subset B^4\) (which is, in fact, a piece of \(A\)). The link theory applies in this situation and gives necessary conditions for the existence of \(N\) such that \(\partial N=L\), in particular, a rather strong restriction to \(N\) in terms of the Seifert form of \(L\) is provided by the Murasugi-Tristram inequality.

This stuff, completed with the quasipositivity of braids corresponding to real algebraic curves, results in a series of concrete statements on the arrangement of ovals of real plane curves. Among these results are an essential progress in the classification of affine M-curves of degree 6, projective reducible curves of degree 7, singular curves of degree 8 and nonsingular curves of degree 9, a new formula for complex orientations of a projective M-curve with a deep nest, a construction of a flexible M-curve of degree 8. The author provides a detailed description of an algorithm for computation of the Seifert form invariants of links and braids coming from real algebraic curves, and shows how it works in the above examples.

The reviewer was informed by the author that the proofs of corollary 1.8 and lemma 1.9 are not correct. So, the statement of corollary 1.8 should read as: follows: there does not exist an \(M\)-curve \(C\) of degree 9 with the real scheme\(\langle J \sqcup 2 \sqcup 1\langle 1 \sqcup 1 \langle 23 \rangle\rangle\rangle\).

This stuff, completed with the quasipositivity of braids corresponding to real algebraic curves, results in a series of concrete statements on the arrangement of ovals of real plane curves. Among these results are an essential progress in the classification of affine M-curves of degree 6, projective reducible curves of degree 7, singular curves of degree 8 and nonsingular curves of degree 9, a new formula for complex orientations of a projective M-curve with a deep nest, a construction of a flexible M-curve of degree 8. The author provides a detailed description of an algorithm for computation of the Seifert form invariants of links and braids coming from real algebraic curves, and shows how it works in the above examples.

The reviewer was informed by the author that the proofs of corollary 1.8 and lemma 1.9 are not correct. So, the statement of corollary 1.8 should read as: follows: there does not exist an \(M\)-curve \(C\) of degree 9 with the real scheme\(\langle J \sqcup 2 \sqcup 1\langle 1 \sqcup 1 \langle 23 \rangle\rangle\rangle\).

Reviewer: E.I.Shustin (Tel Aviv)

##### MSC:

14P25 | Topology of real algebraic varieties |

57M25 | Knots and links in the \(3\)-sphere (MSC2010) |

20F36 | Braid groups; Artin groups |