BEGIN:VCALENDAR VERSION:2.0 PRODID:researchseminars.org CALSCALE:GREGORIAN X-WR-CALNAME:researchseminars.org BEGIN:VEVENT SUMMARY:Andres Fernandez Herrero (Cornell University) DTSTART:20220128T003000Z DTEND:20220128T020000Z DTSTAMP:20260423T024623Z UID:UCSBsga/32 DESCRIPTION:Title: Intrinsic construction of moduli spaces via affine Grassmannians\nby Andres Fernandez Herrero (Cornell University) as part of UCSB Seminar on G eometry and Arithmetic\n\n\nAbstract\nModuli spaces arise as a geometric w ay of classifying objects of interest in algebraic geometry. For example\, there exists a quasiprojective moduli space that parametrizes stable vect or bundles on a smooth projective curve C. In order to further understand the geometry of this space\, Mumford constructed a compactification by add ing a boundary parametrizing semistable vector bundles. If the smooth curv e C is replaced by a higher dimensional projective variety X\, then one ca n compactify the moduli problem by allowing vector bundles to degenerate t o an object known as a "torsion-free sheaf". Gieseker and Maruyama constru cted moduli spaces of semistable torsion-free sheaves on such a variety X. More generally\, Simpson proved the existence of moduli spaces of semista ble pure sheaves supported on smaller subvarieties of X. All of these cons tructions use geometric invariant theory (GIT).\n\nFor a projective variet y X\, the moduli problem of coherent sheaves on X is naturally parametrize d by an algebraic stack M\, which is a geometric object that naturally enc odes the notion of families of sheaves. In this talk I will explain a GIT- free construction of the moduli space of Gieseker semistable pure sheaves which is intrinsic to the moduli stack M. This approach also yields a Hard er-Narasimhan stratification of the unstable locus of the stack. Our main technical tools are the theory of Theta-stability introduced by Halpern-Le istner\, and some recent techniques developed by Alper\, Halpern-Leistner and Heinloth. In order to apply these results\, one needs to prove some mo notonicity conditions for a polynomial numerical invariant on the stack. W e show monotonicity by defining a higher dimensional analogue of the affin e grassmannian for pure sheaves. I will also explain some applications of these ideas to other moduli problems. This talk is based on joint work wit h Daniel Halpern-Leistner and Trevor Jones\, as well as work with Tomas Go mez and Alfonso Zamora.\n LOCATION:https://researchseminars.org/talk/UCSBsga/32/ END:VEVENT END:VCALENDAR