[Paper Review] Birational rigidity is not an open property

Ivan Cheltsov, Михаил Михайлович Гриненко|ArXiv.org|Dec 6, 2006

Organometallic Complex Synthesis and Catalysis4 references19 citations

TL;DR

This paper constructs a counterexample to the conjecture that birational rigidity is an open property in moduli: a Fano threefold defined as a complete intersection of a quadric and a cubic in ℙ⁵ with an ordinary double point is birationally rigid, but small deformations of it yield non-rigid Fano varieties. The result shows that rigidity is not preserved under deformation, disproving a long-standing conjecture in birational geometry.

ABSTRACT

We construct an example of the birationally rigid complete intersection of a quadric and a cubic in $\PA^5$ with an ordinary double point, which under a small deformation gives a non-rigid Fano variety. Thus we show that birational rigidity is not open in moduli.

Motivation & Objective

• To investigate whether birational rigidity is an open property in moduli, as conjectured in prior work.
• To construct explicit examples of Fano threefolds where rigidity fails under small deformations.
• To demonstrate that a singular complete intersection of a quadric and a cubic in ℙ⁵ can be rigid, yet become non-rigid upon deformation.
• To extend the counterexample to del Pezzo fibrations, showing rigidity is not open in their moduli as well.
• To provide a counterexample to the expectation that small deformations preserve birational rigidity in Fano and del Pezzo fibrations.

Proposed method

• Construct a singular complete intersection X ⊂ ℙ⁴ defined by ht − q₁q₂ = 0, where h, t are linear, q₁, q₂ quadratic forms.
• Use two unprojections to construct two non-singular Fano threefolds V₁ and V₂ in ℙ⁵ via the equations y₅h = q₁, y₅q₂ = t and y₅h = q₂, y₅q₁ = t.
• Show that V₁ and V₂ are birational to X and to each other via a flop ψ: V₁ ⇢ V₂, which is not square.
• Analyze the linear systems on V₁ and V₂ using the pushforward of divisors under ψ, deriving degree bounds via Lemma 7.2.
• Apply the method of maximal singularities and birational automorphisms (from Lemma 7.1) to show that any birational map to a Mori fibration must be square, proving rigidity.
• Use the fact that V₁ and V₂ are not isomorphic in general, but become isomorphic under symmetric conditions, to construct the deformation family.

Experimental results

Research questions

• RQ1 Is birational rigidity an open property in moduli for Fano threefolds?
• RQ2 Can a birationally rigid Fano threefold degenerate to a non-rigid one under small deformation?
• RQ3 Does the failure of rigidity under deformation occur in the context of del Pezzo fibrations as well?
• RQ4 Can unprojection and flop techniques be used to construct explicit counterexamples to the openness of rigidity?
• RQ5 Under what symmetric conditions do non-isomorphic Fano fibrations become isomorphic, leading to rigidity?

Key findings

• A Fano threefold X ⊂ ℙ⁵ defined as a complete intersection of a quadric and a cubic with an ordinary double point is birationally rigid.
• Small deformations of X yield non-rigid Fano threefolds, demonstrating that birational rigidity is not an open property in moduli.
• The varieties V₁ and V₂ constructed via unprojection are non-isomorphic in general, but become isomorphic when the section Q_S is symmetric under an involution, leading to rigidity.
• The flop ψ: V₁ ⇢ V₂ is not square, and the birational map between V₁ and V₂ does not preserve the fibration structure, which is key to the non-rigidity of the general deformation.
• The counterexample extends to del Pezzo fibrations: symmetric choices of branch divisors yield rigid fibrations, but generic choices break symmetry and yield non-rigid fibrations.
• The proof relies on analyzing linear systems via pushforwards and using the method of maximal singularities to show that any birational map to a Mori fibration must be square, thus proving rigidity in the symmetric case.