Progress in the Classification of Decomposable Curves
by
Grigorii M. Polotovskii
Nizhny Novgorod University

We give a survey of current situation in the investigation of plane real decomposable algebraic curves.

Let us consider decomposable (=reducible over R) curves C_n of degree n in the real projective plane RP² under the following conditions:

1. All irreducible components C_n1, ... , C_ns of C_n are M-curves.
2. C_n1, ... , C_ns are situated in general position, i.e., C_n has only nondegenera- te nodes.
3. The number of common real points of any two irreducible components C_ni, C_nj is maximal possible, i.e., it is equal n_i ×n_j.
4. For each irreducible component C_ni, exactly one of its branches contains all of these common points.

Our problem is:

Problem 1 Which isotopy types are realizable by a decomposable curve C_n in the real projective plane RP² for given n?

A natural approach to solve such problems is the following. We draw topological models, i.e., collections of smooth circles in RP² that may pretend to represent a decomposable curve C_n up to a homeomorphism, and for every such model we try to determine the extent to which this pretense is justified. In other words, the procedure consists of the following steps.

Step 1. Enumeration of all "pretending" models.

Step 2. Prohibitions, i.e., attempts to prove that the given model cannot be realized by any n-th degree curve.

Step 3. Constructions, i.e., attempts to realize the given model by a n-th degree curve.

We have succeeded in solving Problem 1 for the first non-trivial case n=6, see [1, 2]. These results turned out to be useful for a number of other problems related with Hilbert's 16th problem; a recent application was found in [3], some others are listed in Section III [4].

For n=7 Problem 1 turned out much more difficult and demanded the attraction of new methods. One of these methods, the method of prohibitions based on the link theory, was recently suggested by S.Yu.Orevkov [5]. Using this method Orevkov finished the classification in the case n₁=1, n₂=6 (``affine sexics'') which was started in 1988 [6]. Then application of the same method makes it possible to solve Problem 1 in the cases n₁=3, n₂=4 [7] and n₁=2, n₂=5 [8] when points of intersections lie on ovals. A number of examples show that this method of prohibitions is effective in other cases, for instance, when points of intersections lie on the odd branch of the odd degree component, or for the case n₁=n₂=1, n₃=5.

Recently an attempt was made to apply Viro's ``patchworking'' for constructi- ons (similarly to constructions of complete intersections in [9]) of arrangements of cubics and quartics with 12 common real points lying on the odd branch was made [10]. As a result 10 new arrangements were realized.

We can hope that application of mentioned above methods together with Orevkov's approaches from [7, 8] to constructions will give a possibility to finish Problem 1 for n=7.

References

1. Polotovskii G.M., A catalogue of M-decomposing curves of 6-th degree, Dokl. Akad. Nauk SSSR, 236:3 (1977), 548-551; English transl.: Soviet Math. Dokl. 18:5 (1977), 1241-1245.

2. Kuzmenko T.V., Polotovskii G.M., Classification of curves of degree 6 decomposing into a product of M-curves in general position, Amer. Math. Soc. Transl. (2), Vol.173 (1996), 165-177.

3. Mikhalkin G., Topological arrangement of curves of degree 6 on cubic surfaces in RP³, J. Algebraic Geom., 7 (1998), 219-237.

4. Polotovskii G.M., On the classification of decomposing plane real algebraic curves, Lect. Notes in Math., Vol. 1524 (1992), 52-74.

5. Orevkov S.Yu., Link theory and oval arrangements of real algebraic curves, Topology, 38:4 (1999), 779-810.

6. Korchagin A.B., Shustin E.I., Affine curves of degree 6 and smoothing of non-degenerate six-fold singular points, Izv. AN SSSR, ser. mat., Vol. 52 (1988), 1181-1199; English transl.: Math. USSR-Izvestia, Vol.33 (1989), 501-520.

7. Orevkov S.Yu., Polotovskii G.M., Projektive M-cubics and M-quartics in general position with a maximally intersecting pair of ovals, Algebra i analis (St.Petersburg Math. J), Vol.5 (1999) (to appear).

8. Orevkov S.Yu., it Projective conics and M-quintics in general position with a maximally intersecting pair of ovals (1999) (to appear).

9. Sturmfels B., Viro's Theorem for Complete Intersection, Analli della Scuola Normale Superiore di Pisa, XXI (1994), 377-386.

10. Korobeinikov A.N., Constructions of arrangements of a cubic and a quartic by patchworking (1999) (to prepare).

Date received: May 29, 1999

Copyright © 1999 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # cacy-20.