[Paper Review] Equivalent Birational Embeddings III: cones

Massimiliano Mella|arXiv (Cornell University)|Jul 30, 2014

Algebraic Geometry and Number Theory9 references3 citations

TL;DR

This paper establishes that two irreducible cones in projective space are Cremona equivalent if and only if their general hyperplane sections are birational. The key innovation is a log resolution technique that reduces the Cremona equivalence problem of cones to lower-dimensional subvarieties, enabling the lifting of Cremona maps from codimension-2 subvarieties back to the cones. The result resolves a long-standing question about rational cones in ℙ³, showing they are all Cremona equivalent to a plane.

ABSTRACT

Two divisors in $\mathbb P^n$ are said to be Cremona equivalent if there is a Cremona modification sending one to the other. In this paper I study irreducible cones in $\mathbb P^n$ and prove that two cones are Cremona equivalent if their general hyperplane sections are birational. In particular I produce examples of cones in $\mathbb P^3$ Cremona equivalent to a plane whose plane section is not Cremona equivalent to a line in $\mathbb P^2$.

Motivation & Objective

• To determine when two irreducible cones in ℙⁿ are Cremona equivalent.
• To address the difficulty of Cremona equivalence for rational surfaces in ℙ³, especially cones.
• To extend the Cremona equivalence criterion from plane curves to higher-dimensional cones.
• To clarify the role of the sup-threshold invariant in the context of cone singularities.
• To provide a geometric criterion for Cremona equivalence of rational cones in ℙ³ using their plane sections.

Proposed method

• Develops a specialized log resolution of the pair (ℙⁿ, S) for a cone S, enabling blow-down of the strict transform of S to a codimension-2 subvariety.
• Reduces the Cremona equivalence problem of cones to the equivalence of their general hyperplane sections via a birational transformation strategy.
• Applies the main result from [MP] on Cremona equivalence of codimension-2 subvarieties to lift Cremona maps from the base to the cone.
• Uses the ∗-Minimal Model Program (MMP) to analyze the birational geometry of pairs (T, S_T) associated with cones.
• Constructs birational maps that contract the cone to a curve, allowing translation into the language of scroll and conic bundle structures.
• Employs elementary transformations and birational maps to preserve the Cremona equivalence class under reduction to lower-dimensional cases.

Experimental results

Research questions

• RQ1 When are two cones in ℙⁿ Cremona equivalent?
• RQ2 Can Cremona equivalence of cones be determined solely by the birational type of their general hyperplane sections?
• RQ3 Does every rational cone in ℙ³ admit a Cremona transformation to a plane?
• RQ4 What is the role of the sup-threshold invariant in the birational geometry of cones?
• RQ5 Under what conditions does a conic bundle structure on a 3-fold model ensure positive sup-threshold for the original cone?

Key findings

• Two cones in ℙⁿ are Cremona equivalent if and only if their general hyperplane sections are birational.
• Every rational cone in ℙ³ is Cremona equivalent to a plane, as shown by the geometric genus of its plane section being a birational invariant.
• The sup-threshold of any rational cone in ℙ³ is positive, confirming a key condition in the ∗-MMP framework.
• The Cremona equivalence of cones can be reduced to Cremona equivalence of codimension-2 subvarieties via a carefully constructed log resolution.
• For rational surfaces in ℙ³ with a ∗-minimal model as a conic bundle, the sup-threshold is positive if the surface is the pullback of a curve under a conic bundle with a section.
• The method successfully lifts Cremona maps from the base curve of a cone to the cone itself, even when the base is not Cremona equivalent to a line.