Seminar| Institute of Mathematical Sciences
Time: Wednesday, April 9th, 2025,16:00-17:00
Location: IMS, RS408
Speaker:Yong Hu (胡勇), Shanghai Jiaotong University
Abstract:In this talk, we prove that the canonical model of a $3$-fold of general type with geometric genus $2$ and with minimal canonical volume $\frac{1}{3}$ must be a hypersurface of degree $16$ in $\mathbb{P}(1,1,2,3,8)$, which gives an explicit description of its canonical ring. This implies that the coarse moduli space $\mathcal{M}_{\frac{1}{3}, 2}$, parametrizing all canonical $3$-folds with canonical volume $\frac{1}{3}$ and geometric genus $2$, is an irreducible variety of dimension $189$. Parallel studies show that $\mathcal{M}_{1, 3}$ is irreducible as well and is of dimension $236$, and that $\mathcal{M}_{2, 4}$ is irreducible and is of dimension $270$. As being conceived, every member in these 3 families is simply connected. Additionally, our method yields the expected Noether inequality $\textrm{Vol}\ge\frac{4}{3}p_g-\frac{10}{3}$ for $3$-folds of general type with $5\leq p_g\leq 10$, which completely solves all remaining cases of the Noether inequality for $3$-folds. This is a joint work with Meng Chen and Chen Jiang.