Moduli spaces of threefolds on the Noether line

TL;DR

This paper classifies moduli spaces of canonical threefolds on the Noether line with prescribed geometric genus, revealing a linear growth in irreducible components and relating threefolds to fibrations in (1,2)-surfaces.

Contribution

It provides an explicit stratification, dimension computation, and irreducible component estimation for these moduli spaces, offering a complete classification and new insights into their structure.

Findings

Number of irreducible components grows linearly with p_g.

Moduli space of surfaces on the Noether line has at most two components.

Relation between threefolds and fibrations in (1,2)-surfaces.

Abstract

In this paper, we study the moduli spaces of canonical threefolds with any prescribed geometric genus $p_g \ge 5$ which have the smallest possible canonical volume. This minimal volume is equal to the smallest half-integer that is larger than or equal to $\frac43 p_g -\frac{10}3$, and the threefolds in question are said to lie on the (refined) Noether line. For every such moduli space, we establish an explicit stratification, compute the dimension of all strata, and estimate the number of its irreducible components. Thus it yields a complete classification of threefolds on the (refined) Noether line. A new and unexpected phenomenon is that the number of irreducible components of the moduli space grows linearly with $p_g$, while the moduli space of canonical surfaces on the Noether line with any prescribed geometric genus has at most two irreducible components. The key idea in the proof is to relate these canonical threefolds $X$ to simple fibrations in $(1, 2)$-surfaces. In turn, this depends on the observation that a general member in $|K_X|$ is a canonical surface on the Noether line.