[Paper Review] Symplectic birational transformations of the plane

Jérémy Blanc|Osaka City University (Osaka City University)|Dec 3, 2010

Advanced Algebra and Geometry2 references3 citations

TL;DR

This paper proves two conjectures by A. Usnich regarding the group of symplectic birational transformations of the complex plane. It establishes that the group is generated by SL(2,ℤ), the torus (ℂ*)², and a special map P of order 5, and provides a complete presentation for the subgroup H generated by SL(2,ℤ) and P, confirming the conjectured relations including P⁵ = 1 and PCP = I.

ABSTRACT

We study the group of symplectic birational transformations of the plane. It is proved that this group is generated by $\mathrm{SL}(2,\mathbb{Z})$, the torus and a special map of order $5$, as it was conjectured by A. Usnich. Then we consider a special subgroup $H$, of finite type, defined over any field which admits a surjective morphism to the Thompson group of piecewise linear automorphisms of $\mathbb{Z}^2$. We prove that the presentation for this group conjectured by Usnich is correct.

Motivation & Objective

• To prove that the group of symplectic birational transformations of the complex plane is generated by SL(2,ℤ), the torus (ℂ*)², and a specific map P of order 5, as conjectured by Usnich.
• To verify the conjectured presentation of the subgroup H generated by SL(2,ℤ) and P, including the relations I⁴ = C³ = [C, I²] = P⁵ = 1 and PCP = I.
• To clarify the geometric structure of symplectic birational maps by analyzing base-points and their behavior under blow-ups and the symplectic form ω₀ = dx∧dy/(xy).
• method
• research_questions
• key_findings

Proposed method

• The paper uses geometric techniques in algebraic geometry, particularly the study of blow-ups and the behavior of the symplectic form ω₀ = dx∧dy/(xy) under birational transformations.
• It applies the concept of base-points and their multiplicities on surfaces obtained by successive blow-ups to analyze the structure of rational maps.
• The authors use the classification of normal cubic forms and the divisor theory of differential forms to characterize symplectic maps via their pole and zero loci.
• A key method involves analyzing the degree growth of iterated maps and the configuration of base-points in the standard triangle defined by X+Y+Z=0.
• The proof relies on case analysis based on the intersection pattern of base-point sets S and T of consecutive maps, distinguishing cases r=3 and r=4 based on the number of common points.
• It applies a combinatorial argument to find quadratic maps Q with prescribed base-points satisfying multiplicity and incidence conditions, using the fact that three base-points must not be collinear and must lie on the three lines of the triangle X·Y·Z=0.

Experimental results

Research questions

• RQ1 Does the group of symplectic birational transformations of ℂ² admit a finite generating set consisting of SL(2,ℤ), the torus (ℂ*)², and a map P of order 5, as conjectured by Usnich?
• RQ2 Is the subgroup H generated by SL(2,ℤ) and P presented by the relations I⁴ = C³ = [C, I²] = P⁵ = 1 and PCP = I, as conjectured?
• RQ3 How do the base-points of symplectic maps relate to the poles of the differential form ω₀ = dx∧dy/(xy), and what geometric constraints do they satisfy?

Key findings

• The group Symp of symplectic birational transformations of ℂ² is generated by SL(2,ℤ), the torus (ℂ*)², and the map P of order 5, confirming Usnich's conjecture.
• The subgroup H generated by SL(2,ℤ) and P admits the presentation ⟨I, C, P | I⁴ = C³ = [C, I²] = P⁵ = 1, PCP = I⟩, which is proven to be correct.
• All non-toric base-points of elements in Symp arise from the map P, although complex relations exist among the generators.
• The proof shows that any quadratic map Q with three base-points satisfying multiplicity and incidence conditions must have its base-points lying on the three lines of the standard triangle X·Y·Z=0.
• In the case r=3 (one common base-point), at least one of the two candidate triplets of base-points satisfies both the multiplicity and incidence conditions.
• In the case r=4 (no common base-points), at least one of three candidate triplets satisfies the required conditions, and the impossibility of three non-collinear points lying on a single line ensures the existence of a valid Q.