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Here I collect some general adivices from other mathematicians to students, especially for those who want to pursue a PhD degree in mathematics. We should learn and try to do. Some of them follows from Prof. Jie Liu’s homepage.
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Here we will introduce some interesting papers and notes here. I may or may not read them.
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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 $\mathrm{char}(k)=0$.
[Kol13] János Kollár. Singularities of the Minimal Model Program. Cambridge University Press, 2013. ↩
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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。
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Here we will introduce some well-known results about the automorphisms of algebraic curves over an algebraic closed field $k$ and their relations between the moduli space of curves.
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This is an interesting problem is that for any finite covering $f:\mathbb{P}^1\to \mathbb{P}^1$, how to compute $f_* \mathscr{O}(m)$?
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Here we give some detailed construction about the projections from the subspaces $\mathbb{P}(V/W)$ of projective spaces $\mathbb{P}(V)\backslash \mathbb{P}(V/W)\to\mathbb{P}(W)$. We follows notes Math 732: Topics in Algebraic Geometry II and Math 732. Rationality of algebraic varieties.
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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.
[FulIT2nd] William Fulton. Intersection Theory, 2nd. Springer New York, NY. 1998. ↩
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This blog aim to give some remarks and complete some details in this book (Kollar and Mori’s Birational Geometry of Algebraic Varieties). I will read first five chapters of this book. This is the third blog which about chapter 4 and chapter 5.
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This is my plan of the reading program of the intersection theory and Chow motives.
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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.
[FulIT2nd] William Fulton. Intersection Theory, 2nd. Springer New York, NY. 1998. ↩
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This blog aim to give some remarks and complete some details in this book (Kollar and Mori’s Birational Geometry of Algebraic Varieties). I will read first five chapters of this book. This is the second blog which about chapter 3.
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This blog aim to give some remarks and complete some details in this book (Kollar and Mori’s Birational Geometry of Algebraic Varieties). I will read first five chapters of this book. This is the first blog from chapter 1 to chapter 2.
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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.
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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.
[FulIT2nd] William Fulton. Intersection Theory, 2nd. Springer New York, NY. 1998. ↩
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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.
[FulIT2nd] William Fulton. Intersection Theory, 2nd. Springer New York, NY. 1998. ↩
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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.
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山穷水复疑无路，柳暗花明又一村。——记我的2022
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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 $\geq i-\dim X+\dim a(X)$.
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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.