BEGIN:VCALENDAR VERSION:2.0 PRODID:researchseminars.org CALSCALE:GREGORIAN X-WR-CALNAME:researchseminars.org BEGIN:VEVENT SUMMARY:Alexander Degtyarev (Bilkent) DTSTART:20250228T124000Z DTEND:20250228T134000Z DTSTAMP:20260423T021258Z UID:OBAGS/59 DESCRIPTION:Title: S plit hyperplane sections on polarized K3-surfaces\nby Alexander Degtya rev (Bilkent) as part of ODTU-Bilkent Algebraic Geometry Seminars\n\n\nAbs tract\nI will discuss a new result which is an unexpected outcome\, follow ing a question by Igor Dolgachev\, of a long saga about smooth rational cu rves on (quasi-)polarized $K3$-surfaces. The best known example of a $K3$- surface is a quartic surface in space. A simple dimension count shows that a typical quartic contains no lines. Obviously\, some of them do and\, ac cording to B.~Segre\, the maximal number is $64$ (an example is to be work ed out). The key r\\^ole in Segre's proof (as well as those by other autho rs) is played by plane sections that split completely into four lines\, co nstituting the dual adjacency graph $K(4)$. A similar\, though less used\, phenomenon happens for sextic $K3$-surfaces in~$\\mathbb{P}^4$ (complete intersections of a quadric and a cubic): a split hyperplane section consis ts of six lines\, three from each of the two rulings\, on a hyperboloid (t he section of the quadric)\, thus constituting a $K(3\,3)$.\n\nGoing furth er\, in degrees $8$ and $10$ one's sense of beauty suggests that the graph s should be the $1$-skeleton of a $3$-cube and Petersen graph\, respectfu lly. However\, further advances to higher degrees required a systematic st udy of such $3$-regular graphs and\, to my great surprise\, I discovered t hat typically there is more than one! Even for sextics one can also imagin e the $3$-prism (occurring when the hyperboloid itself splits into two pla nes).\n\nThe ultimate outcome of this work is the complete classification of the graphs that occur as split hyperplane sections (a few dozens) and t hat of the configurations of split sections within a single surface (manag eable starting from degree $10$). In particular\, answering Igor's origina l question\, the maximal number of split sections on a quartic is $72$\, w hereas on a sextic in $\\mathbb{P}^4$ it is $40$ or $76$\, depending on th e question asked. The ultimate champion is the Kummer surface of degree~$1 2$: it has $90$ split hyperplane sections.\n\nThe tools used (probably\, n ot to be mentioned) are a fusion of graph theory and number theory\, sewn together by the geometric insight.\n LOCATION:https://researchseminars.org/talk/OBAGS/59/ END:VEVENT END:VCALENDAR