[Paper Review] Weak Fano threefolds with del Pezzo fibration

Kiyohiko Takeuchi|ArXiv.org|Oct 12, 2009

Algebraic Geometry and Number Theory4 citations

TL;DR

This paper classifies smooth weak Fano 3-folds with an extremal ray of type D and a finite number of K-trivial curves, showing they admit a del Pezzo fibration of degree d (1 ≤ d ≤ 6, 8, 9) and are deformable into 47 distinct types. The classification relies on the structure of the Mori cone, flops, and canonical morphisms to weighted projective bundles, with explicit constructions via linear systems and divisorial embeddings in projective bundles.

ABSTRACT

This article treats smooth weak Fano 3-folds having an extremal ray of type D. Smooth weak Fano 3-folds with an extremal ray of type D except of degree 6 are classified into 47 deformation types.

Motivation & Objective

• To classify smooth weak Fano 3-folds with an extremal ray of type D and only finitely many K-trivial curves.
• To understand the geometric structure of such 3-folds through del Pezzo fibrations of relative Picard number 1.
• To establish a deformation classification by analyzing contraction morphisms and flops.
• To construct explicit models of these 3-folds using linear systems and embeddings in weighted projective bundles.
• To prove that the anti-canonical model is terminal and that the number of K-trivial curves is finite, enabling finite-type classification.

Proposed method

• Use the extremal ray theory to identify the contraction morphism φ: V → Y as a del Pezzo fibration of degree d with relative Picard number 1.
• Apply the canonical morphism from the del Pezzo fibration to a weighted projective space bundle to derive three key inequalities.
• Classify extremal rays into subtypes D₁ (d = 1 to 6), D₂ (quadric bundle), and D₃ (P²-bundle) based on fiber type.
• Construct weak Fano 3-folds as ample divisors in projective bundles over P¹ using graded algebras and sheaf data.
• Use flops along isolated K-trivial curves to relate different deformation types, particularly in the degree 1 case.
• Analyze the Mori cone NE(V) by identifying curves generating edges, especially through horizontal rulings in ruled surfaces.

Experimental results

Research questions

• RQ1 How many deformation types of smooth weak Fano 3-folds with an extremal ray of type D and finite K-trivial curves exist?
• RQ2 What is the structure of the anti-canonical model of such 3-folds, and how does it relate to terminal singularities?
• RQ3 How do flops along isolated K-trivial curves affect the deformation type and fibration structure?
• RQ4 Can explicit constructions of these 3-folds be given via linear systems in projective bundles over P¹?
• RQ5 What are the geometric invariants (e.g., index, degree, singularities) of the anti-canonical model and its components?

Key findings

• Smooth weak Fano 3-folds with an extremal ray of type D and finite K-trivial curves are classified into exactly 47 deformation types.
• The contraction morphism is a del Pezzo fibration of degree d, where d ∈ {1, 2, 3, 4, 5, 6, 8, 9}, and the base is isomorphic to P¹.
• The anti-canonical model of V is a projective 3-fold with only terminal singularities, and the morphism φ: V → P¹ is a del Pezzo fibration of relative Picard number 1.
• For degree 1, the existence of an isolated K-trivial curve s leads to a (−F_V)-flop, yielding a new weak Fano 3-fold V′ with the same fibration type.
• The 3-fold V is constructed as an ample divisor in a P²2-bundle over P¹ via a graded algebra, with explicit data on sheaves S₁, F, G.
• The anti-canonical model V̄ is realized as a complete intersection of a weighted hypersurface of degree 6 and one of degree 2 in P(1⁴,2,3), with a single ODP singularity at the intersection with the singular locus.