The Noether inequality for threefolds and three moduli spaces with minimal volumes

We establish the Noether inequality \[\textrm{Vol}(X)\geq \frac{4}{3}p_g(X)-\frac{10}{3}\] for all projective $3$-folds $X$ of general type with geometric genus $5\leq p_g(X)\leq 10$ where $\textrm{Vol}(X)$ is the canonical volume. This result resolves all remaining cases of the Noether inequality for $3$-folds. We further investigate the moduli spaces of canonical $3$-folds with small genera and minimal volumes. For a $3$-fold of general type with geometric genus $2$ and with minimal canonical volume $\frac{1}{3}$, we prove that its canonical model is 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 unirational variety of dimension $189$. Parallel studies show that $\mathcal{M}_{1, 3}$ is irreducible, unirational, and $236$-dimensional, and that $\mathcal{M}_{2, 4}$ is irreducible, unirational, and $270$-dimensional. As being conceived, every member in these 3 families is simply-connected.