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

TL;DR

This paper proves a key inequality for threefolds of general type with certain invariants, fully characterizes moduli spaces of minimal-volume threefolds, and describes their geometric properties and dimensions.

Contribution

It establishes the Noether inequality for all relevant threefolds and explicitly describes the moduli spaces of threefolds with minimal volumes and small genera.

Findings

Proves the Noether inequality for 3-folds with 5 ≤ p_g(X) ≤ 10.

Describes the canonical models of 3-folds with minimal volume and genus 2.

Determines the dimension and unirationality of specific moduli spaces.

Abstract

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.