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 $\geq i-\dim X+\dim a(X)$.
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.
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.
