Generic vanishing theorem I - Hacon's approach

01 Dec 2022

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

15 Nov 2022

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.

Some Books and References in Learning Algebraic Geometry

14 Nov 2022

In this blog, we will give a long list of books and notes about many areas in algebraic geometry.

Some Toy Versions of Big Conjectures in Mathematics

12 Nov 2022

Here we use the basic algebraic geometry to prove some toy versions of big conjectures in math, such as Fermat’s conjecture for polynomials and $ABC$-conjecture for polynomials.

Some of the rules of typing blogs.

07 Nov 2019

Some basic rules and codes in this Jekyll and HTML.