Moduli spaces of threefolds on the Noether line

TL;DR

The paper classifies canonical threefolds on the refined Noether line with $p_g\ge5$ by establishing an explicit stratification of their moduli into unirational families arising from simple fibrations in $(1,2)$-surfaces. Central to the method is the reduction to a relative canonical model and realizing these threefolds as hypersurfaces in toric fourfolds, parameterized by triples $(d,N,d_0)$, with $N\in\{0,1,2\}$. The authors compute dimensions for all strata, bound and describe the number of irreducible components, and prove linear growth of components with $p_g$, revealing new phenomena that differ from surface cases. They also prove the existence of simple fibrations for large $p_g$ and analyze the canonical divisor, singularities, and nef/ample behavior, including explicit toric realizations for $N\le4$ and a detailed account of the moduli stratification and its limits. The work provides a Horikawa-style, higher-dimensional geography for varieties of general type and highlights the rich, stratified structure of the corresponding moduli spaces along the refined Noether line, with significant implications for understanding the geometry and deformations of canonical threefolds.

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.

Figures (1)

• Figure 1: Dimension of modular families for $d=24$, $N=0$

Theorems & Definitions (81)

• Theorem 1.1
• Theorem 1.2
• Conjecture 1.3
• Theorem 1.4: See Corollary \ref{['cor: simple fibration']}
• Theorem 1.5: See Theorem \ref{['thm: X is hypersurface']}
• Theorem 2.1: Noether inequality
• Theorem 2.2: Refined Noether inequality
• proof
• Definition 2.3
• Theorem 2.4