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On bicanonical maps of threefolds of general type with large volumes

Chen Jiang, Ziqi Liu

TL;DR

The paper proves that a smooth projective 3-fold of general type with canonical volume exceeding 12^6 cannot have a pencil as its bicanonical image, by constructing non-klt centers via boundary divisors and employing effective extension theorems to lift sections. It develops an extension framework for 3-folds fibred over curves and applies it to (1,2) and (2,3)-surfaces, deriving precise behavior for pluricanonical maps and bicanonical maps. A parallel extension theory for curve non-klt centers, together with inversion of adjunction and Nadel vanishing, enables control over two-genus curve centers and ensures nontrivial bicanonical images. Collectively these results yield a robust approach to birational geometry of high-volume 3-folds and yield concrete thresholds for the finite-ness and birationality of lower-degree pluricanonical maps.

Abstract

We prove that for any smooth projective $3$-fold of general type with canonical volume greater than $12^6$, the image of its bicanonical map has dimension at least $2$. We also study pluricanonical maps of $3$-folds of general type with large canonical volume and fibered by $(1,2)$-surfaces or $(2,3)$-surfaces.

On bicanonical maps of threefolds of general type with large volumes

TL;DR

The paper proves that a smooth projective 3-fold of general type with canonical volume exceeding 12^6 cannot have a pencil as its bicanonical image, by constructing non-klt centers via boundary divisors and employing effective extension theorems to lift sections. It develops an extension framework for 3-folds fibred over curves and applies it to (1,2) and (2,3)-surfaces, deriving precise behavior for pluricanonical maps and bicanonical maps. A parallel extension theory for curve non-klt centers, together with inversion of adjunction and Nadel vanishing, enables control over two-genus curve centers and ensures nontrivial bicanonical images. Collectively these results yield a robust approach to birational geometry of high-volume 3-folds and yield concrete thresholds for the finite-ness and birationality of lower-degree pluricanonical maps.

Abstract

We prove that for any smooth projective -fold of general type with canonical volume greater than , the image of its bicanonical map has dimension at least . We also study pluricanonical maps of -folds of general type with large canonical volume and fibered by -surfaces or -surfaces.

Paper Structure

This paper contains 16 sections, 24 theorems, 84 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Conventions
  4. Volumes
  5. Singularities
  6. Multiplier ideal sheaves
  7. Tankeev's principle
  8. Glct for minimal surfaces of general type
  9. An extension theorem for fibrations over curves and applications
  10. An extension theorem for fibrations over curves
  11. An application to fibrations of $(1,2)$-surfaces and $(2,3)$-surfaces
  12. Bicanonical maps of fibrations over curves
  13. An extension theorem for curve non-klt centers
  14. Inversion of adjunction on curves
  15. An extension theorem for curve non-klt centers
  16. ...and 1 more sections

Key Result

Theorem 1.1

Let $X$ be a smooth projective $3$-fold of general type. If $\text{\rm Vol}(X)>12^6$, then $\dim \overline{\Phi_{|2K_X|}(X)}\geq 2$.

Theorems & Definitions (55)

  • Theorem 1.1
  • Theorem 1.2
  • Definition 2.1
  • Lemma 2.2: cf. Jiang-AJM
  • proof
  • Definition 2.3
  • Definition 2.4
  • Proposition 2.5: cf. Amb06
  • proof
  • Definition 2.6: cf. Positivity2
  • ...and 45 more