LIU Xiaolong

LIU Xiaolong

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

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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.

Generic vanishing theorem I - 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 $\geq i-\dim X+\dim a(X)$.

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

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.

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. ↩

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. ↩

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 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. ↩

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. ↩

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。

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. ↩

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notes

Informal Notes

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Here consist ofsome informal notes about math and more things.

[2024/08/30] Lecture note about the symplectic resolutions, symplectic duality and coulomb branches after reading the survey paper Kamnitzer22. Here is my notes: Informal-Symp-dual, updated at 2024-09-02.

[2022/01/01] Here including some naive notes during undergraduate. See: Other Notes.

Modern Theory of Moduli Spaces, Stability and Its related topics (finished)

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We will first introduce the theory of good moduli space of Prof. Jarod Alper and his cooperators, more precisely, the first 6 sections in their paper1 (the more earier version since the results in the paper are too general which we will not use).

Then we will introduce a very general moduli space in the 7th section in paper1, the good moduli space of objects in abelian categories which will give our a very general idea to construct the moduli space of semistable sheaves and Bridgeland stable complexes.

Next we will see the detailed theory of stability and moduli theory of stable sheaves.

Fianlly We will introduce the general theory of Bridgeland stablility on the triangulated category and construct the good moduli space of Bridgeland semistable objects in heart. We will also prove the Wall-crossing structure and give some examples. Moreover we will construct the Bridgeland stablility on the surface in detail and give the second tilt about threefolds with some conjectures.

Here is my notes about this part: ModernModuliTheory, finished at 2023-10-14.

  1. Jarod Alper, Daniel Halpern-Leistner, and Jochen Heinloth. Existence of moduli spaces for algebraic stacks. Invent. Math., pages 1–90, 2023. ↩ ↩2

Varieties of Minimal Rational Tangents on the Fano Varieties (finished)

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The theory of Varieties of Minimal Rational Tangents (VMRT) motivated from the proof of Hartshorne conjecture due to S. Mori. Then developed by Ngaiming Mok and Junmuk Hwang in 80s to 00s. Now this is a important area to discover the geometric properties of Fano varieties of Picard number $1$. Here is our main topics:

(a) First we introduce the basic theory of rational curves, including Hilbert schemes of morphisms, the Chow scheme of rational components $\mathrm{RatCurves}^n_d(X)$ and their useful properties. Then we will give some applications, such as Bend-and-Break, boundedness of smooth Fano varieties and the proof of Hartshorne’s conjecture using our theory.

(b) Then we will introduce some more general facts about Fano manifolds. Moreover we will also introduce some special Fano manifolds such as Gushel-Mukai varieties, rational homogeneous varieties, Hermitian symmetric space and Del-Pezzo manifolds.

(c) Then we will construct the theory of VMRT over smooth uniruled varieties and mainly focused on the Fano varieties of Picard number $1$. We will dicuss the properties of distributions $\mathcal{D}\subset T_X$ and Cartan-Fubini type extension theorem.

(d) Then we will discuss some basic applications of VMRT, such as stability of the tangent bundles and rigidity of generically finite morphisms.

(e) We will consider the VMRT of rational homogeneous varieties which is very important We will consider the properties of minimal sections and dual VMRT.

(g) We will talk about Campana-Peternell conjecture and prove the case $T_X$ is nef, big and $1$-ample.

(h) Consider the most recent work of the conjecture of Fano manifolds with non-isomorphic surjective endomorphism and the most recent work of the conjecture of Fano manifolds with big automorphism group.

(i) Remmert-Van de Ven/Lazarsfeld problem which is also fundamental.

(j) We will again talk about Campana-Peternell conjecture and prove the case $G/B$ and some special cases, such as lower dimension case and large Picard number case.

Here is my notes: VMRT-BasicTheory, finished at 2024-03-25.

Dualizing Complexes Using Derived Categories (finished)

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This is the notes about dualizing complexes in commutative algebra which is the reading report of the course “advanced commutative algebra” in AMSS, CAS. This is a very interesting course about Cohen-Macaulay rings and local cohomology.

Here is my notes: DualizingComplexes, finished at 2024-05-20.

Note for the Virtual Fundamental Class (finished)

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This is the basic theory of virtual fundamental class which is the fundation of the enumerative geometry, including the construction of virtual fundamental class and the localization formula over it.

Here is my notes: Virtual-Fundamental-Class, finished at 2024-07-25. More general and relative version of intrisic normal cone and virtual pullback, we refer 虚拟基本类 (only for chinese) for an introduction.

Here is my note about DAG: DAG which we might use in the enumerative geometry, finished at 2024-06-19.

Here is my note about Gromov-Witten invariants and topological recursion relations: TRR-GW, (has been abandoned for now) 2024-09-30.

Notes on the Geometry of Hypertoric Varieties (finished)

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This is the basic theory of hypertoric varieties (or toric hyperkähler varieties) and its geometry, including their symplectic resolutions via hyperplane arrangements and consider their cores. We also consider their universal Poisson structure and wall-crossing structures via the family-version of Mukai-flops. This is the seminar notes in MCM algebraic-geometry seminar, 03/12/2024.

Here is my notes: Hypertoric_Varieties, finished at 2024-12-02.

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Teaching experience 2

Workshop, University 1, Department, 2015

This is a description of a teaching experience. You can use markdown like any other post.