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

We prove that the canonical model of a $3$-fold of general type with geometric genus $2$ and with minimal canonical volume 1/3 must be a hypersurface of degree $16$ in P(1,1,2,3,8), which gives an explicit description of its canonical ring. This implies that the coarse moduli space M1/3, 2, parametrizing all canonical $3$-folds with canonical volume 1/3 and geometric genus $2$, is an irreducible variety of dimension $189$. Parallel studies show that M1, 3 is irreducible as well and is of dimension $236$, and that M2, 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 Vol>4/3pg-10/3 for $3$-folds of general type with pg=5.