LIU Xiaolong

LIU Xiaolong

I’m a second-year graduate student in the area of the algebraic geometry.

Blogs

Basic Theory of the Generalized Donaldson-Thomas Invariants of Joyce-Song

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This is a basic and quick introduction of the theory of generalized Donaldson-Thomas invariants of strict Calabi-Yau $3$-folds due to Joyce-Song in [JS11]1. We work over $\mathbb C$ to defined consider the generalized DT-invariants. But most of the theory before that we will work over any algebraically closed field $\mathbb K$ of characteristic zero.

  1. [JS11] Dominic Joyce and Yinan Song. A theory of generalized Donaldson-Thomas invariants. Amer. Math. Soc. 2011. See also ArXiv:0810.5645. ↩

A Brief Introduction to the Virtual Intersection Theory

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This is a basic and quick introduction of the virtual intersection theory introduced by [BF97]1 and [Man12]2 and give some more properties of these theories. This generalize the constructions and results in my notes.

  1. [BF97] K. Behrend and B. Fantechi. The intrinsic normal cone. Invent. Math., 128:45–88, 1997. ↩

  2. [Man12] Cristina Manolache. Virtual pull-backs. J. Algebraic Geom., 21:201–245, 2012. ↩

About Affine Cones associated to ample line bundles

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Here we will introduce some well-known results about the cones which is the simplest examples of terminal, canonical, etc. singularities. Here we will follow the book [Kol13]1. We just consider $\mathrm{char}(k)=0$.

  1. [Kol13] János Kollár. Singularities of the Minimal Model Program. Cambridge University Press, 2013. ↩

魏晋玄学，帖学和兰亭序

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魏晋南北朝时期是书法全面转向艺术的重要时期，本文是我将这段时间对南派帖学和兰亭的所学所想写的一些综述，总结和读后感，而北派碑学则不做过多解释。我本人则不太喜欢兰亭本身的书法（对内容还是喜欢的），这也就导致我想要探究为何兰亭想去晋人那么远。

Some Gaps and Examples in Intersection Theory by Fulton IV (The End)

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This is one of a series of blogs aiming to complete some details of the examples in this book (Intersection Theory, 2nd edition by William Fulton1) and give some comments. This blog we consider chapter 14 to chapter 15.

  1. [FulIT2nd] William Fulton. Intersection Theory, 2nd. Springer New York, NY. 1998. ↩

Some Gaps and Examples in Intersection Theory by Fulton III

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This is one of a series of blogs aiming to complete some details of the examples in this book (Intersection Theory, 2nd edition by William Fulton1) and give some comments. This blog we consider chapter 10 to chapter 13.

  1. [FulIT2nd] William Fulton. Intersection Theory, 2nd. Springer New York, NY. 1998. ↩

Birational Geometry Reading Seminar

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This is my plan of the reading program of birational geometry for the beginner of this area! Aiming to read the basic aspect in the birational geometry, both lower dimensional ($\dim X=2$) and higher dimensional ($\dim X\geq 3$) in algebraic geometry.

Some Gaps and Examples in Intersection Theory by Fulton II

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This is one of a series of blogs aiming to complete some details of the examples in this book (Intersection Theory, 2nd edition by William Fulton1) and give some comments. This blog we consider chapter 7 to chapter 9.

  1. [FulIT2nd] William Fulton. Intersection Theory, 2nd. Springer New York, NY. 1998. ↩

Some Gaps and Examples in Intersection Theory by Fulton I

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This is one of a series of blogs aiming to complete some details of the examples in this book (Intersection Theory, 2nd edition by William Fulton1) and give some comments. This blog we consider chapter 1 to chapter 6.

  1. [FulIT2nd] William Fulton. Intersection Theory, 2nd. Springer New York, NY. 1998. ↩

A Quick Tour of Géométrie algébrique et géométrie analytique

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In this blog, we will introduce some basic fact about GAGA-principle. Actually I only vaguely knew that this is a correspondence between analytic geometry and algebraic geometry over $\mathbb{C}$ before. So as we may use GAGA frequently, we will summarize in this blog to facilitate learning and use.

我的2022

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山穷水复疑无路，柳暗花明又一村。——记我的2022

Generic vanishing theorem - Hacon’s approach

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This is the first blog aim to introduce the generic vanishing theorem, follows Hacon’s proof. He use the Fourier-Mukai transform and something about Abelian varieties to solve this problem:

Main Theorem. Let $X$ be a smooth projective variety over $\mathbb{C}$. Let $a:X\to\mathrm{Alb}(X)$ be the Albanese map. Let \[S^i(\omega_X)=\{M\in A^t(\mathbb{C}): H^i(X,\omega_X\otimes M)\geq 0\},\] then $S^i(\omega_X)\subset A^t$ is closed of codimension $\geq i-\dim X+\dim a(X)$.

Reading program on moduli space of curves

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This is my plan of the reading program of the moduli space of curves, aiming to discover the geometrical properties of stacks $\mathscr{M}_{g,n}$ and $\overline{\mathscr{M}}_{g,n}$ and their coarse moduli spaces.