Addendum to "The Noether inequality for algebraic threefolds"

TL;DR

This paper refines previous results on the volume bounds of algebraic threefolds of general type, providing sharper inequalities for certain geometric genus ranges.

Contribution

It improves the known volume inequalities for projective threefolds of general type with specific geometric genus conditions.

Findings

Established refined volume bounds for p_g(X) ≤ 4.

Established refined volume bounds for p_g(X) ≥ 11.

Extended the Noether inequality to a broader class of threefolds.

Abstract

In this short note, we give a refinement of our previous work (arXiv:1803.05553) stating that for a projective $3$-fold $X$ of general type with either $p_g(X)\leq 4$ or $p_g(X)\geq 11$, $$\text{vol}(X)\geq \frac{4}{3}p_g(X)-{\frac{10}{3}}.$$