(joint with Ilia Itenberg)
It is a common understanding that any reasonable geometric question about K3-surfaces can be restated and solved in purely arithmetical terms, by means of an appropriately defined homological type. For example, this works well in the study of singular complex sextic curves in $\mathbb{P}^2$ or quartic surfaces in $\mathbb{P}^3$ (see [1,2]), as well as in that of smooth real ones (see [4,6]). However, when the two are combined (both singular and real curves or surfaces), the approach fails as the `"obvious'' concept of homological type does not fully reflect the geometry (cf., e.g., [3] or [5]).
We show that the situation can be repaired if the curves in question have empty real part or, more generally, have no real singular points; then, one can indeed confine oneself to the homological types consisting of the exceptional divisors, polarization, and real structure.
Still, the resulting arithmetical problem is not quite straightforward, but we manage to solve it and obtain a satisfactory classification in the case of empty real part; it matches all known results obtained by an alternative purely geometric approach. In the general case of smooth real part, we also have a formal classification; however, establishing a correspondence between arithmetic and geometric invariants (most notably, the distribution of ovals among the components of a reducible curve) still needs a certain amount of work.
This project was conceived and partially completed during our joint stay at the Max-Planck-Institut für Mathematik, Bonn. The speaker is partially supported by TÜBİTAK project 123F111.
REFERENCES
[1]. Ayşegül Akyol and Alex Degtyarev, Geography of irreducible plane sextics, Proc. Lond. Math. Soc. (3) 111 (2015), no. 6, 13071337. MR 3447795
[2]. Çisem Güneş Aktaş, Classi cation of simple quartics up to equisingular deformation, Hiroshima Math. J. 47 (2017), no. 1, 87112. MR 3634263
[3]. I. V. Itenberg, Curves of degree 6 with one nondegenerate double point and groups of monodromy of nonsingular curves, Real algebraic geometry (Rennes, 1991), Lecture Notes in Math., vol. 1524, Springer, Berlin, 1992, pp. 267288. MR 1226259
[4]. V. M. Kharlamov, On the classi cation of nonsingular surfaces of degree 4 in RP3 with respect to rigid isotopies, Funktsional. Anal. i Prilozhen. 18 (1984), no. 1, 4956. MR 739089
[5]. Sébastien Moriceau, Surfaces de degré 4 avec un point double non dégénéré dans l'espace projectif réel de dimension 3, Ph.D. thesis, 2004.
[6]. V. V. Nikulin, Integer symmetric bilinear forms and some of their geometric applications, Izv. Akad. Nauk SSSR Ser. Mat. 43 (1979), no. 1, 111177, 238, English translation: Math USSR-Izv. 14 (1979), no. 1, 103167 (1980). MR 525944 (80j:10031)
