Haiden-Katzarkov-Kontsevich-Pandit

Papers Semistability, modular lattices, and iterated logarithms Iterated logarithms and gradient flows Talks Katzarkov, Towards a categorical Donaldson-Uhlenbeck-Yau correspondence, November 2015, slides, video Kontsevich, Kahler stability, November 2015, slides, video Pandit, Categorical Kahler geometry, Jan 2017, video Haiden, Refined Harder-Narasimhan filtrations in modular lattices and iterated logarithms, Jan 2017, slides, video Haiden, Categorical Kahler Geometry, Jan 2017 video 1, video 2 Haiden, Semistability and iterated logarithms, Dec 15 2017, video Pandit, From homotopical mathematics to emergent geometry, Jan 24 2018, slides Pandit, Categorical Kahler geometry, June 2018, slides, video Pandit, Gradient flows, iterated logarithms and semistability, June 2018, slides, video Pandit, Towards a categorigal Donaldson-Uhlenbeck-Yau correspondence, July 2018, video Pandit, Categorical Kahler geometry, April 2022, video

Schemes from varieties

In scheme-theoretic algebraic geometry, we can define a variety as an reduced1 separated scheme which is finite type over a field $k$. In this post, we’ll start off with the notions of affine and projective varieties from classical algebraic geometry, and then generalise these incrementally, to end up at the definition of a scheme. First of all, recall that we have a bijection between affine varieties and finitely generated reduced $k$ algebras, given by sending an affine variety $X$ to it’s ring of regular functions $k[X]$.

Functor of points

One idea which has been coming up a lot in the SMSTC course on Algebraic Geometry, lectured by Clark Barwick, has been the idea of the functor of points. Here, we’ll try and motivate that definition with some examples. Throughout, we work over an algebraically field $k$, and so schemes and morphisms are relative to $k$. Let $X$ be a scheme. Associated to this, we have a functor $h_X : \mathrm{Sch}/k^{\mathrm{op}} \to \mathrm{Sets}$, given by $$h_X(Y) = \mathrm{Mor}(Y, X)$$ and this is called the functor of points.