BEGIN:VCALENDAR VERSION:2.0 PRODID:researchseminars.org CALSCALE:GREGORIAN X-WR-CALNAME:researchseminars.org BEGIN:VEVENT SUMMARY:Alexander Degtyarev (Bilkent) DTSTART:20231013T124000Z DTEND:20231013T134000Z DTSTAMP:20260423T052641Z UID:OBAGS/30 DESCRIPTION:Title: S ingular real plane sextic curves without real points\nby Alexander Deg tyarev (Bilkent) as part of ODTU-Bilkent Algebraic Geometry Seminars\n\n\n Abstract\n(joint with Ilia Itenberg)\nIt is a common understanding that an y reasonable geometric question about K3\n-surfaces can be restated and so lved in purely arithmetical terms\, by means of an appropriately defined h omological type. For example\, this works well in the study of singular co mplex sextic curves in P2 or quartic surfaces in P3 (see [1\,2])\, as we ll 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]).\n\nWe show that the situation can be repaired if the curves in question have empty real part or\, more generall y\, have no real singular points\; then\, one can indeed confine oneself t o the homological types consisting of the exceptional divisors\, polarizat ion\, and real structure.\n\nStill\, the resulting arithmetical problem is not quite straightforward\, but we manage to solve it and obtain a satisf actory classification in the case of empty real part\; it matches all know n results obtained by an alternative purely geometric approach. In the gen eral case of smooth real part\, we also have a formal classification\; how ever\, establishing a correspondence between arithmetic and geometric inva riants (most notably\, the distribution of ovals among the components of a reducible curve) still needs a certain amount of work.\n\nThis project wa s conceived and partially completed during our joint stay at the Max-Planc k-Institut für Mathematik\, Bonn. The speaker is partially supported by T ÜBİTAK project 123F111.\n\nREFERENCES\n\n[1]. Ayşegül Akyol and Alex D egtyarev\, Geography of irreducible plane sextics\, Proc. Lond. Math. Soc. (3) 111 (2015)\, no. 6\, 13071337. MR 3447795\n\n[2]. Çisem Güneş Akt aş\, Classi\ncation of simple quartics up to equisingular deformation\, H iroshima Math. J. 47 (2017)\, no. 1\, 87112. MR 3634263\n\n[3]. I. V. Ite nberg\, Curves of degree 6 with one nondegenerate double point and groups of monodromy of nonsingular curves\, Real algebraic geometry (Rennes\, 199 1)\, Lecture Notes in Math.\, vol. 1524\, Springer\, Berlin\, 1992\, pp. 2 67288. MR 1226259\n\n[4]. V. M. Kharlamov\, On the classi\ncation of nons ingular surfaces of degree 4 in RP3\n with respect to rigid isotopies\, Fu nktsional. Anal. i Prilozhen. 18 (1984)\, no. 1\, 4956. MR 739089\n\n[5]. Sébastien Moriceau\, Surfaces de degré 4 avec un point double non dég énéré dans l'espace projectif réel de dimension 3\, Ph.D. thesis\, 200 4.\n\n[6]. V. V. Nikulin\, Integer symmetric bilinear forms and some of th eir 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)\n LOCATION:https://researchseminars.org/talk/OBAGS/30/ END:VEVENT END:VCALENDAR