[Paper Review] Almost del Pezzo manifolds

Priska Jahnke, Thomas Peternell|ArXiv.org|Dec 18, 2006

Algebraic Geometry and Number Theory11 references4 citations

TL;DR

This paper classifies almost del Pezzo manifolds—smooth projective manifolds with nef and big anticanonical bundle—by analyzing their Mori contractions and anticanonical morphisms to singular Gorenstein Fano varieties. The key contribution is a complete classification in all dimensions, showing that such manifolds arise as blow-ups of points, projectivizations of rank-2 bundles, or quadric bundles over P^1, with explicit geometric realizations and degree constraints.

ABSTRACT

We classify almost del Pezzo manifolds in arbitrary dimension n, i.e., projective manifolds X with big and nef anticanonical bundle -K_X, such that -K_X is divisible by n-1.

Motivation & Objective

• To classify smooth projective manifolds with nef and big anticanonical bundle, termed 'almost del Pezzo' manifolds, extending the classical classification of del Pezzo manifolds.
• To understand the structure of these manifolds through their anticanonical morphisms to singular Gorenstein Fano varieties with canonical singularities.
• To determine when such manifolds arise as blow-ups of smooth points or as projectivizations of rank-2 vector bundles over surfaces.
• To classify almost del Pezzo manifolds in dimensions ≥4 using fibrations and blow-ups, building on lower-dimensional results.
• To resolve the existence question for certain singular anticanonical models by analyzing singularities and factorizations of crepant resolutions.

Proposed method

• Apply Mori theory to classify extremal contractions of threefolds with Picard number two, identifying them as quadric fibrations, P^1-bundles, or blow-ups of smooth points.
• Use the anticanonical morphism ψ:X→X′ to map almost del Pezzo manifolds to singular Gorenstein Fano threefolds X′ with ρ(X′)=1 and −K_X′=(n−1)H′.
• Analyze the birational geometry of X′ via Fujita’s classification of Gorenstein Fano n-folds, especially for n≥4.
• Employ the Riemann–Roch theorem and vanishing theorems to compute h⁰(X,H)=d+n−1, where d=Hⁿ is the degree of the manifold.
• Use flops and crepant resolutions to reduce the classification to known cases, particularly ruling out non-terminal or non-Q-factorial models.
• Construct explicit examples via projectivization of rank-2 bundles over P², F₂, or P¹×P¹, and via hyperplane sections of Grassmannians or Veronese cones.

Experimental results

Research questions

• RQ1 Which smooth projective manifolds with nef and big anticanonical bundle can be classified as almost del Pezzo in arbitrary dimension?
• RQ2 What are the possible Mori contractions for threefolds with Picard number two and non-Fano anticanonical bundle?
• RQ3 When does the anticanonical morphism ψ:X→X′ of an almost del Pezzo manifold factor through a small or divisorial contraction, and what are the implications for the singularities of X′?
• RQ4 Under what conditions does the blow-up of a smooth point in an almost del Pezzo manifold remain almost del Pezzo?
• RQ5 Can singular anticanonical models X′ of almost del Pezzo manifolds be realized as hyperplane sections of smooth del Pezzo varieties, and when are they non-Q-factorial or non-terminal?

Key findings

• For threefolds with ρ(X)=2, the unique extremal contraction is either a quadric fibration, a P¹-bundle, or the blow-up of a smooth point; the anticanonical morphism is small if and only if X′ is singular and non-Q-factorial.
• In dimension 3 with ρ(X)≥3, after finitely many flops, X is either a blow-up of a point in a threefold classified in the ρ=2 case or a projectivization of a rank-2 bundle over P², F₂, or P¹×P¹.
• For n≥4, any Mori contraction of an almost del Pezzo manifold is a P^{n−2}-bundle over a smooth surface, a quadric bundle over P¹, or a blow-up of a smooth point in another almost del Pezzo manifold.
• The anticanonical model X′ of an almost del Pezzo manifold of dimension 5 with H⁵=5 is a singular hyperplane section of the Grassmannian G(1,4), i.e., a del Pezzo 5-fold of degree 5.
• For dimension 4 with H⁵=4, X′ is a hyperplane section of the 5-fold in the previous case, and such manifolds exist.
• The case where X′ is a cone over a singular Fano threefold with canonical non-terminal singularities leads to a contradiction if X is assumed to exist, ruling out such models as anticanonical images.