BEGIN:VCALENDAR VERSION:2.0 PRODID:researchseminars.org CALSCALE:GREGORIAN X-WR-CALNAME:researchseminars.org BEGIN:VEVENT SUMMARY:Giancarlo Urzua (Pontificia Universidad Católica de Chile) DTSTART:20220125T163000Z DTEND:20220125T173000Z DTSTAMP:20260423T022620Z UID:Geolis/74 DESCRIPTION:Title: What is the right combinatorics for spheres in K3 surfaces?\nby Gianca rlo Urzua (Pontificia Universidad Católica de Chile) as part of Geometria em Lisboa (IST)\n\n\nAbstract\nTogether with Javier Reyes\, in https://ar xiv.org/abs/2110.10629 we have been able to construct compact 4-manifolds $3\\mathbb{CP}^2\\#(19-K^2)\\overline{\\mathbb{CP}}^2$ with complex struct ures for $K^2=1\,2\,3\,4\,5\,6\,7\,8\,9$. The cases $K^2=7\,9$ are complet ely new in the literature\, and this finishes with the whole range allowed by the technique of Q-Gorenstein smoothing (rational blow-down). But one can go further: Is it possible to find minimal exotic $3\\mathbb{CP}^2\\#( 19-K^2)\\overline{\\mathbb{CP}}^2$ for $K^2\\geq10$? Here it would be much harder to prove the existence of complex structures\, but\, as a motivati on\, there is not even one example for $K^2 > 15$\, and very few for $10 \ \leq K^2 \\leq 15$ (see e.g. works by Akhmedov\, Park\, Baykur). In this t alk I will explain the constructions in connection with the geography of s pheres arrangements in $K3$ surfaces\, where the question of the title ari ses. We do not have an answer. So far we have been implementing what we kn ow in computer searches\, finding these very rare exotic surfaces for $K^2 =10\,11\,12$. This is a new and huge world which promises more findings\, we have explored very little.\n LOCATION:https://researchseminars.org/talk/Geolis/74/ END:VEVENT END:VCALENDAR