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SUMMARY:Geometric construction of toric NCRs
DESCRIPTION:Algebraic Geometry in String Theory Seminar \nSpeaker: Jesse Huang\, University of Alberta \nTitle: Geometric construction of toric NCRs \nAbstract: The Rouquier dimension of a toric variety is recently shown to be achieved by the Frobenius pushforward of O via coherent-constructible correspondence. From the perspective of noncommutative geometry\, this result leads to a geometric construction of toric NCR of the invariant ring of the Cox ring with respect to a multi-grading which also gives the information about its global dimension. From the perspective of mirror symmetry\, the same construction provides a universal “wall skeleton” capturing VGIT wall-crossings\, which contains a window for each chamber as a full subcategory. From the perspective of commutative algebra\, the same construction indicates the existence of virtual resolutions of the multigraded diagonal bimodule\, which agrees with a recent result of Hanlon-Hicks-Larzarev constructing one such resolution explicitly. In this talk\, I will survey these perspectives. The talk is based on joint works with P. Zhou\, joint works with D. Favero\, and work in progress with D. Favero. \n  \n 
URL:https://cmsa.fas.harvard.edu/event/agst-3724/
LOCATION:CMSA Room G10\, CMSA\, 20 Garden Street\, Cambridge\, MA\, 02138\, United States
CATEGORIES:Algebraic Geometry in String Theory Seminar
ATTACH;FMTTYPE=image/png:https://cmsa.fas.harvard.edu/media/CMSA-Algebraic-Geometry-in-String-Theory-03.07.2024.png
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SUMMARY:The KSBA moduli space of log Calabi-Yau surfaces
DESCRIPTION:Algebraic Geometry in String Theory Seminar \nSpeaker: Pierrick Bousseau\, University of Georgia \nTitle: The KSBA moduli space of log Calabi-Yau surfaces \nAbstract: The KSBA moduli space\, introduced by Kollár–Shepherd-Barron\, and Alexeev\, is a natural generalization of “the moduli space of stable curves” to higher dimensions. It parametrizes stable pairs (X\,B)\, where X is a projective algebraic variety satisfying certain conditions and B is a divisor such that K_X+B is ample. This moduli space is described concretely only in a handful of situations: for instance\, if X is a toric variety and B=D+\epsilon C\, where D is the toric boundary divisor and C is an ample divisor\, it is shown by Alexeev that the KSBA moduli space is a toric variety. Generally\, for a log Calabi-Yau variety (X\,D) consisting of a projective variety X and an anticanonical divisor D\, with B=D+\epsilon C where C is an ample divisor\, it was conjectured by Hacking–Keel–Yu that the KSBA moduli space is still toric (up to passing to a finite cover). In joint work with Alexeev and Argüz\, we prove this conjecture for all log Calabi-Yau surfaces. This uses tools from the minimal model program\, log smooth deformation theory and mirror symmetry. \n 
URL:https://cmsa.fas.harvard.edu/event/agst-32124/
LOCATION:CMSA Room G10\, CMSA\, 20 Garden Street\, Cambridge\, MA\, 02138\, United States
CATEGORIES:Algebraic Geometry in String Theory Seminar
ATTACH;FMTTYPE=image/png:https://cmsa.fas.harvard.edu/media/CMSA-Algebraic-Geometry-in-String-Theory-03.21.2024.png
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