BEGIN:VCALENDAR VERSION:2.0 PRODID:researchseminars.org CALSCALE:GREGORIAN X-WR-CALNAME:researchseminars.org BEGIN:VEVENT SUMMARY:Alexander Degtyarev (Bilkent) DTSTART:20231215T124000Z DTEND:20231215T134000Z DTSTAMP:20260423T021153Z UID:OBAGS/39 DESCRIPTION:Title: L ines on singular quartic surfaces via Vinberg\nby Alexander Degtyarev (Bilkent) as part of ODTU-Bilkent Algebraic Geometry Seminars\n\n\nAbstrac t\nLarge configurations of lines (or\, more generally\, rational curves of low degree) on algebraic surfaces appear in various contexts\, but only in the case of cubic surfaces the picture is complete. Our principal goal is the classification of large configurations of lines on quasi-polarized K3-surfaces in the presence of singularities. To the best of our knowledge \, no attempt has been made to attack this problem from the lattice-theore tical\, based on the global Torelli theorem\, point of view\; some partial results were obtained by various authors using ``classical'' algebraic g eometry\, but very little is known. The difficulty is that\, given a polar ized N\\'eron--Severi lattice\, computing the classes of smooth rational c urves depends on the choice of a Weyl chamber of a certain root lattice\, which is not unique.\n\nWe show that this ambiguity disappears and the alg orithm becomes deterministic provided that sufficiently many classes of li nes are fixed. Based on this fact\, Vinberg's algorithm\, and a combinator ial version of elliptic pencils\, we develop an algorithm that\, in princi ple\, would list all extended Fano graphs. After testing it on octic K3-su rfaces\, we turn to the most classical case of simple quartics where\, pri or to our work\, only an upper bound of 64 lines (Veniani\, same as in the smooth case) and an example of 52 lines (the speaker) were known. We show that\, in the presence of singularities\, the sharp upper bound is indeed 52\, substantiating the long standing conjecture (by the speaker) that th e upper bound is reduced by the presence of smooth rational curves of lowe r degree.\n\nWe also extend the classification (I. Itenberg\, A.S. Sertöz \, and the speaker) of large configurations of lines on smooth quartics do wn to 49 lines. Remarkably\, most of these configurations were known befor e.\n\nThis project was conceived and partially completed during our joint stay at the Max-Planck-Institut f\\ür Mathematik\, Bonn. The speaker is p artially supported by TÜBİTAK project 123F111.\n LOCATION:https://researchseminars.org/talk/OBAGS/39/ END:VEVENT END:VCALENDAR