[Paper Review] On covering and quasi-unsplit families of rational curves

Bonavero, L., Casagrande, C.+1 more|ArXiv.org|Apr 4, 2005

Algebraic Geometry and Number Theory6 references13 citations

TL;DR

This paper establishes that on normal, Q-factorial projective varieties of dimension at most 4, every covering and quasi-unsplit family of rational curves generates a geometric extremal ray in the Mori cone. The authors prove that such families define extremal contractions and, under mild singularities or toric conditions, yield geometric quotients via Mori theory, resolving a key question on the extremality of quasi-unsplit families.

ABSTRACT

We study extremality properties of covering families of rational curves on projective varieties. Among others, we show that on a normal and Q-factorial projective variety of dimension at most 4, every covering and quasi-unsplit family of rational curves generates a geometric extremal ray of the Mori cone.

Motivation & Objective

• To determine whether covering and quasi-unsplit families of rational curves on projective varieties generate geometric extremal rays in the Mori cone.
• To investigate the existence and structure of geometric quotients for such families.
• To establish conditions under which the Mori contraction associated to a family is a geometric quotient.
• To extend known results on extremal rays to varieties with canonical singularities and toric varieties.
• To resolve the question of whether quasi-unsplitness and covering imply extremality in low-dimensional and toric settings.

Proposed method

• Uses the theory of covering families of rational curves and their associated equivalence relations via rational maps to $X \dasharrow Y$.
• Applies Campana's theory of $V$-equivalence classes to associate a rational map to any covering family $V$.
• Employs the dimension $f_V$ of general $V$-equivalence classes as a key invariant to bound the codimension of fibers.
• Utilizes the Mori cone $\overline{\rm NE}(X)$ and the notion of geometric extremal rays to analyze extremality.
• Applies toric geometry techniques, including fans and primitive generators, to prove results in the toric case.
• Uses induction on dimension and projection to invariant divisors to verify the combinatorial condition (4) for the existence of geometric quotients.

Experimental results

Research questions

• RQ1 Does every covering and quasi-unsplit family of rational curves on a Q-factorial projective variety of dimension ≤4 generate a geometric extremal ray in the Mori cone?
• RQ2 Under what conditions does the Mori contraction associated to such a family yield a geometric quotient?
• RQ3 Can the existence of a geometric quotient be guaranteed for quasi-unsplit families in the toric setting?
• RQ4 Is the dimension of the $V$-equivalence class a sufficient invariant to ensure extremality of the ray $\mathbb{R}_{\geq 0}[V]$?
• RQ5 What is the relationship between the numerical class $[V]$ and the existence of a flat, equivariant morphism $q^\prime: X \to Y^\prime$ contracting curves in $V$?

Key findings

• For any normal, Q-factorial projective variety $X$ of dimension $n \leq 4$, every covering and quasi-unsplit family $V$ of rational curves generates a geometric extremal ray in $\overline{\rm NE}(X)$.
• If $f_V \geq n-3$, where $f_V$ is the dimension of a general $V$-equivalence class, then $\mathbb{R}_{\geq 0}[V]$ is a geometric extremal ray.
• When $X$ has canonical singularities and $f_V \geq n-3$, the Mori contraction $\mathrm{cont}_{[V]}: X \to Y^\prime$ is the geometric quotient for $V$, and is equidimensional if $f_V \geq n-2$.
• In the toric case, any covering and quasi-unsplit family $V$ generates a geometric extremal ray, and the Mori contraction $\mathrm{cont}_{[V]}$ is a geometric quotient.
• The existence of a geometric quotient is equivalent to a combinatorial condition on the fan of $X$, which is verified via induction on dimension and projection to invariant divisors.
• For a variety $X$ covered by lines, if $X$ is toric or has canonical singularities and $\dim X \leq 4$, there exists a morphism $q^\prime: X \to Y^\prime$ with $\rho_{Y^\prime} = \rho_X - 1$ contracting all lines in $V$.