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Giancarlo Urzua, N-resolutions

February 27, 2023 @ 1:30 pm - 2:20 pm KST

B236-1, IBS Korea, Republic of


Giancarlo Urzua
UC Chille

(This is a part of Seminars on Algebraic Surfaces and Related Topics.)

I will introduce N-resolutions, which are the negative analog of the Kollár–Shepherd-Barron (1988) P-resolutions of a 2-dimensional cyclic quotient singularity. (We instead work with the corresponding M-resolutions of Benkhe-Christophersen (1994).) I will start by describing an algorithm to find all of them based on the explicit algorithm for P-resolutions in Park-Park-Shin-Urzúa (2018) (that geometrically recovers Christophersen-Stevens’ zero continued fractions correspondence (1991)), which in turn is based on the explicit MMP described by Hacking-Tevelev-Urzúa (HTU 2017). I will also describe another way to find N-resolutions via antiflips (HTU 2017) starting with an M-resolution, showing an action of the braid group on all its associated Wahl resolutions. This will bring us to Hacking exceptional collections (2013-2016) on surfaces that are Q-Gorenstein smoothings of particular singular surfaces, where Karmazyn-Kuznetsov-Shinder (2022) have described their derived categories via derived categories of the Kalck-Karmazyn algebras (2017). This can be put together through Kawamata’s bundles (2018-2022), and I will describe our main theorem on semi-orthogonal decompositions defined by these M- and N-resolutions. I will end with applications to all simply-connected Dolgachev surfaces. I will mention open problems. This is on the joint recent work with Jenia Tevelev. This computer program finds all M- and N-resolutions.


February 27, 2023
1:30 pm - 2:20 pm KST
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IBS Korea, Republic of


Yongnam Lee
IBS 복소기하학연구단 Center for Complex Geometry
기초과학연구원 복소기하학연구단
대전 유성구 엑스포로 55 (우) 34126
IBS Center for Complex Geometry
Institute for Basic Science (IBS)
55 Expo-ro Yuseong-gu Daejeon 34126 South Korea
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