On moduli spaces of canonical threefolds with small genera and minimal volumes

Abstract: 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 $\mathcal{M}<em>{\frac{1}{3}, 2}$, parametrizing all canonical $3$-folds with canonical volume $\frac{1}{3}$ and geometric genus $2$, is an irreducible unirational variety of dimension $189$. Parallel studies show that $\mathcal{M}</em>{1, 3}$ is irreducible unirational as well and is of dimension $236$, and that $\mathcal{M}_{2, 4}$ is irreducible unirational and is of dimension $270$. As being conceived, every member in these 3 families is simply connected. Additionally, our method yields $\textrm{Vol}\geq \frac{4}{3}p_g-\frac{10}{3}$ for $3$-folds of general type with $5\leq p_g\leq 10$, which completely solves all remaining cases of the Noether inequality of $3$-folds.