Algebraic threefolds of general type with small volume

TL;DR

This paper classifies algebraic threefolds of general type with small volume, establishing explicit structures for those meeting the optimal Noether inequality and exploring related inequalities and residue phenomena.

Contribution

It provides a complete classification of 3-folds satisfying the optimal Noether inequality with explicit models, and introduces new inequalities and residue-based phenomena.

Findings

Classification of 3-folds with equality in Noether inequality

Explicit structure of relative canonical models

Connection between Noether inequalities and residues modulo 3

Abstract

It is known that the optimal Noether inequality $\mathrm{vol}(X) \ge \frac{4}{3}p_g(X) - \frac{10}{3}$ 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 $p_g(X) \ge 11$. These results answer two open questions raised by J. Chen, M. Chen and C. Jiang, and in dimension three an open question raised by J. Chen and C. Lai. 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 $3$.