BEGIN:VCALENDAR VERSION:2.0 PRODID:researchseminars.org CALSCALE:GREGORIAN X-WR-CALNAME:researchseminars.org BEGIN:VEVENT SUMMARY:Kirsten Wickelgren (Duke) DTSTART:20210706T160000Z DTEND:20210706T170000Z DTSTAMP:20260423T021350Z UID:eAKTS/24 DESCRIPTION:Title: C ounts of rational curves on Del Pezzo surfaces enriched in bilinear forms< /a>\nby Kirsten Wickelgren (Duke) as part of electronic Algebraic K-theory Seminar\n\nLecture held in 979 0634 7355.\n\nAbstract\nDel Pezzo surfaces here are smooth projective surfaces with\nample anticanonical bundle\, in cluding $\\mathbb{P}^2\, \\mathbb{P}^1 \\times \\mathbb{P}^1$\, and cubic\ nsurfaces. By imposing the condition that a rational curve of fixed\ndegre e passes through an appropriate number of points\, the number of\nsuch cur ves is finite. Over the complex numbers\, these counts are\nindependent of the generic choice of points. This invariance of number\nfails over the r eals\, but there is a beautiful method of Welschinger\nto correct this. It is a feature of $\\mathbb{A}^1$-homotopy theory that analogous\nreal and complex results can indicate the presence of a common\ngeneralization\, va lid over a general field. For $\\mathbbA^1$-connected Del Pezzo\nsurfaces under appropriate hypotheses\, we give counts of rational\ncurves valued i n the group completion GW(k) of symmetric\,\nnon-degenerate\, bilinear for ms over k\, which are again independent of\nthe generic choice of points. By replacing the positive integer count\nwith such a bilinear form\, one r ecords information about the field of\ndefinition of the rational curve an d the tangent directions at its\nnodes. We compute some low degree example s\, including on the Del Pezzo\nsurfaces listed above. This is joint work with Jesse Kass\, Marc\nLevine\, and Jake Solomon.\n LOCATION:https://researchseminars.org/talk/eAKTS/24/ END:VEVENT END:VCALENDAR