On generic finiteness of pluricanonical maps of threefolds of general type

Abstract: We prove that $|6K_X|$ defines a generically finite map for all minimal 3-folds $X$ of general type with $P_2(X)\geq 2$, which is optimal. We also prove that $|nK_X|$ defines a generically finite map for all minimal 3-folds $X$ of general type when $n\geq 38$. The essential technical ingredients of this paper are a new generic finiteness criterion for surfaces and an effective comparison inequality under a special resolution.

• The paper proves that |6Kₓ| is generically finite for minimal threefolds with P₂(X) ≥ 2, establishing sharp effective bounds.
• It introduces a new generic finiteness criterion for surfaces and refines Reid basket calculus to compare pluricanonical restrictions.
• The results impact moduli theory and the minimal model program by setting precise benchmarks for the behavior of pluricanonical maps.

Generic Finiteness of Pluricanonical Maps on Threefolds of General Type

Introduction and Motivation

The paper "On generic finiteness of pluricanonical maps of threefolds of general type" (2607.02066) addresses the longstanding problem of determining effective bounds for the pluricanonical systems $|nK_X|$ on minimal threefolds $X$ of general type such that the associated rational map is generically finite. This topic is central to the birational classification theory of higher-dimensional algebraic varieties and the construction of their moduli spaces. Existing literature provides effective birationality results and partial generic finiteness results, but optimal bounds, especially in low plurigenera and for minimal threefolds with $P_2(X) \geq 2$, have been elusive.

Main Results

The author proves three central theorems, each constituting an optimal or near-optimal effective bound for the generic finiteness of pluricanonical maps:

• For all minimal threefolds $X$ of general type with $P_2(X) \geq 2$, $|6K_X|$ defines a generically finite map. The paper demonstrates this bound is sharp using explicit examples (e.g., certain weighted complete intersections with $P_2=2$ for which $|5K_X|$ fails to be generically finite).
• For all minimal threefolds $X$ of general type and all $n \geq 38$, $X$0 defines a generically finite map.
• For all minimal threefolds $X$1 of general type with $X$2, $X$3 defines a generically finite map for $X$4.

Furthermore, for each result, optimality is attested by either construction or citation of explicit threefolds for which the bound is attained.

Technical Innovations

New Generic Finiteness Criterion for Surfaces

The core technical development underpinning the threefold results is a new criterion for when a linear system of the form $X$5 on a smooth projective surface $X$6 of general type, where $X$7 is the minimal model and $X$8 is effective, defines a generically finite map. This result overcomes prior obstructions given by $X$9-surfaces (those with $P_2(X) \geq 2$0, $P_2(X) \geq 2$1), where Bombieri’s and Xiao’s earlier theorems do not apply sharply at the needed level. The analysis hinges on intersection-theoretic inequalities and careful use of surface classification.

Effective Comparison Inequalities and Modified Reid Basket Calculus

The study leverages and extends the formalism of baskets of singularities (Reid’s basket), which encodes cyclic quotient singularities arising on minimal models. A key insight is the introduction of an effective comparison inequality under a specialized resolution (Chen resolution), allowing the author to compare pluricanonical restrictions to minimal surface components effectively (with actual divisorial inequalities $P_2(X) \geq 2$2 rather than $P_2(X) \geq 2$3-linear equivalence).

The construction of truncated or "packed" baskets and the level $P_2(X) \geq 2$4 truncation, along with programmatic computation, enables exhaustive checking of possible baskets for threefolds with large pluricanonical section index. For each possible basket, the pluricanonical dimensions and invariants are checked against the non-vanishing, semigroup, and volume inequalities.

Non-Vanishing and Semigroup Criteria

By establishing improved non-vanishing results (notably, that $P_2(X) \geq 2$5 for $P_2(X) \geq 2$6 when $P_2(X) \geq 2$7) and by relating these to the properties of the basket, sharp lower bounds for the canonical volume are acquired as a function of specified plurigenera. Furthermore, analysis of the semigroups formed by the plurigenera allows recursive propagation of non-vanishing, contributing directly to the effectiveness of the main theorems.

Handling of Threefolds with Large Pluricanonical Section Index

A careful case analysis for threefolds with large $P_2(X) \geq 2$8 (the minimal $P_2(X) \geq 2$9 with $X$0) entails enumerative and computational techniques. For all possible baskets consistent with the known invariants, the requisite pluricanonical sections are shown to exist.

Optimality and Explicit Examples

The examples demonstrate the sharpness of the results. For instance, the cited threefold $X$1 in weighted projective space with $X$2 lacks a generically finite map for $X$3 with $X$4, but $X$5 achieves generic finiteness. Similar optimality appears for the general bounds in $X$6.

Implications and Future Directions

From the perspective of moduli theory, these results provide effective criteria for when the natural pluricanonical maps of a family of threefolds yield generically finite morphisms, directly impacting the construction of moduli spaces and the minimal model program in dimension three. The established bound $X$7 for arbitrary minimal threefolds with no generic vanishing conditions provides a definitive benchmark for further study; the author explicitly poses the question of whether this lower bound can be further reduced, especially in comparison to the (much larger) known birationality threshold $X$8.

Methodologically, the new generic finiteness criterion for surfaces and the refined handling of basket invariants have potential applicability for broader classes of higher-dimensional varieties, and may enable new progress on effective birationality and related problems.

Conclusion

This paper achieves a set of sharp, effective bounds for the generic finiteness of pluricanonical maps on minimal threefolds of general type, overcoming previous barriers associated with low plurigenera and exceptional surface fibers. The technical contributions—most notably a new generic finiteness criterion for surfaces of general type and enhanced basket calculus—yield both theoretical advances and concrete tools for further work in high-dimensional birational geometry. The results reduce the gap between known generic finiteness and birationality thresholds, and establish optimal bounds in several cases, providing both answers and new questions for the birational classification of threefolds.