[Paper Review] 2 MORI CONTRACTIONS OF MAXIMAL LENGTH

Carla Novelli|arXiv (Cornell University)|Aug 17, 2016

Algebraic Geometry and Number Theory25 references20 citations

TL;DR

This paper establishes a relative version of the Cho-Miyaoka-Shepherd-Barron theorem for Mori fibre spaces of maximal length, proving that such fibrations are birational to projective bundles. It shows that when the length of an extremal ray is maximal (l(R) = d+1), the fibration is either a projective bundle or becomes one after a birational modification, generalizing classical results on Fano manifolds to singular and non-equidimensional fibrations.

ABSTRACT

International audience

Motivation & Objective

• To generalize the Cho-Miyaoka-Shepherd-Barron theorem to relative, non-equidimensional fibrations in the context of Mori fibre spaces.
• To investigate the structure of degenerations of projective spaces in fibrations with maximal extremal ray length.
• To prove that fibrations of maximal length are birational to projective bundles, even when not equidimensional.
• To provide a precise birational description of the normalization of exceptional divisors in divisorial contractions of maximal length.
• To offer evidence supporting the conjecture that fibrations of length l(R) = n−m+1 are equidimensional under dimension constraints.

Proposed method

• Use the Ionescu–Wiśniewski inequality to analyze the dimension of fibers and exceptional loci in Mori contractions.
• Apply the bend-and-break technique and deformation theory of rational curves to verify the length condition l(R) = d+1.
• Construct a relative version of the Kobayashi–Ochiai theorem using numerical conditions instead of ample Cartier divisors.
• Employ resolution of singularities and the rigidity lemma to construct a birational model where the fibration becomes equidimensional.
• Use Leray spectral sequences and cohomological arguments to prove the existence of a relatively ample Cartier divisor inducing a projective bundle structure.
• Apply local systems and Hodge-theoretic tools to show that the fibration is locally a scroll under mild topological assumptions on the base.

Experimental results

Research questions

• RQ1 Under what conditions is a Mori fibration of maximal length birational to a projective bundle?
• RQ2 Can the condition of equidimensionality be relaxed in the relative Cho–Miyaoka–Shepherd-Barron theorem?
• RQ3 What is the structure of the normalization of the exceptional divisor in a divisorial contraction of maximal length?
• RQ4 Does a fibration of length l(R) = n−m+1 over a base of dimension m necessarily have equidimensional fibers?
• RQ5 To what extent can the existence of a relative hyperplane divisor be replaced by a numerical condition in the relative Kobayashi–Ochiai theorem?

Key findings

• A Mori fibration of maximal length l(R) = d+1 is birational to a projective bundle, even if not equidimensional.
• If the fibration is equidimensional, it is globally a projective bundle, and the anticanonical divisor induces a global hyperplane class.
• For non-equidimensional fibrations, there exists a birational model where the fibration becomes a projective bundle after desingularization of the base and total space.
• The normalization of the exceptional divisor in a divisorial contraction of maximal length is a projective bundle in codimension one.
• The fibration is locally a scroll over a Zariski open neighborhood of the equidimensional locus when the third cohomology of the base is torsion-free.
• The result supports the conjecture that fibrations of length l(R) = n−m+1 are equidimensional when n ≥ 2m−1.