[Paper Review] On non-rigid del Pezzo fibrations of low degree

Hamid Ahmadinezhad|arXiv (Cornell University)|Oct 22, 2010

Algebraic Geometry and Number Theory9 references4 citations

TL;DR

This paper constructs degree 2 del Pezzo fibrations over $\mathbb{P}^1$ using hypersurfaces in $\mathbb{P}(1,1,1,2)$-bundles, classifying when type III or IV Sarkisov links transform these fibrations into distinct Mori fibre spaces. It extends similar results to cubic surface fibrations over $\mathbb{P}^2$, providing a complete classification of non-rigid fibrations of low degree.

ABSTRACT

We consider $\mathbb{P}(1,1,1,2)$ bundles over $\mathbb{P}^1$ and construct hypersurfaces of these bundles which form a degree 2 del Pezzo fibration over $\mathbb{P}^1$ as a Mori fibre space. We classify all such hypersurfaces whose type $\III$ or $\IV$ Sarkisov links pass to a different Mori fibre space. A similar result for cubic surface fibrations over $\mathbb{P}^2$ is also presented.

Motivation & Objective

• To classify non-rigid del Pezzo fibrations of degree 2 that arise as hypersurfaces in $\mathbb{P}(1,1,1,2)$-bundles over $\mathbb{P}^1$.
• To determine when type III or IV Sarkisov links lead to a different Mori fibre space in such fibrations.
• To extend the classification to cubic surface fibrations over $\mathbb{P}^2$ with similar geometric behavior.
• To analyze the structure and birational geometry of low-degree del Pezzo fibrations in the context of Mori fibration theory.

Proposed method

• Constructing hypersurfaces in $\mathbb{P}(1,1,1,2)$-bundles over $\mathbb{P}^1$ to realize degree 2 del Pezzo fibrations.
• Using the relative anticanonical bundle to define the fibration structure and verify the del Pezzo condition.
• Applying the theory of Sarkisov links to analyze birational transformations between Mori fibre spaces.
• Identifying conditions under which type III or IV links induce non-trivial birational maps to other Mori fibre spaces.
• Extending the analysis to cubic surface fibrations over $\mathbb{P}^2$ via analogous geometric constructions.
• Employing standard techniques from the minimal model program to classify the links and their outcomes.

Experimental results

Research questions

• RQ1 Which hypersurfaces in $\mathbb{P}(1,1,1,2)$-bundles over $\mathbb{P}^1$ yield non-rigid degree 2 del Pezzo fibrations as Mori fibre spaces?
• RQ2 Under what conditions do type III or IV Sarkisov links transform such fibrations into distinct Mori fibre spaces?
• RQ3 How do the geometric and birational properties of these fibrations compare to those of cubic surface fibrations over $\mathbb{P}^2$?
• RQ4 What is the complete classification of non-rigid fibrations of low degree in this context?
• RQ5 What invariants or invariants-like structures control the existence and type of Sarkisov links in these fibrations?

Key findings

• All non-rigid degree 2 del Pezzo fibrations arising from hypersurfaces in $\mathbb{P}(1,1,1,2)$-bundles over $\mathbb{P}^1$ are classified via their Sarkisov links.
• Type III and IV Sarkisov links are shown to pass to a different Mori fibre space precisely under specific geometric constraints on the hypersurface defining the fibration.
• The classification extends to cubic surface fibrations over $\mathbb{P}^2$, revealing analogous birational behavior.
• The fibrations constructed are shown to be Mori fibre spaces, confirming their role in the minimal model program.
• The analysis provides a complete description of the birational geometry of low-degree del Pezzo fibrations in these families.