Algebraic threefolds of general type with small volume

It is known that the optimal Noether inequality \({\text {vol} }(X) \ge \frac{4}{3}p_g(X) - \frac{10}{3}\) holds for every 3-fold X of general type with \(p_g(X) \ge 11\). In this paper, we give a complete classification of 3-folds X of general type with \(p_g(X) \ge 11\) satisfying the above equality by giving the explicit structure of a relative canonical model of X. This model coincides with the canonical model of X when \(p_g(X) \ge 23\). We also establish the second and third optimal Noether inequalities for 3-folds X of general type with \(p_g(X) \ge 11\). A novel phenomenon shows that there is a one-to-one correspondence between the three Noether inequalities and three possible residues of \(p_g(X)\) modulo 3. These results answer two open questions by Chen et al. (Duke Math J 169(9):1603–1164, 2020), and in dimension three an open question by Chen and Lai (Int J Math 31(1):2050005, 2020).