[Paper Review] Algebraic structures of groups of birational transformations

Jérémy Blanc|arXiv (Cornell University)|Sep 1, 2015

Algebraic Geometry and Number Theory7 references3 citations

TL;DR

This paper investigates algebraic structures on the group of birational transformations of algebraic varieties, focusing on representability of functors parameterizing families of birational maps. It introduces a flatness condition to construct a scheme structure on Bir(X), showing that while the full Bir(X) is not representable, the flat subfunctor yields a well-behaved scheme structure on P^n for n ≥ 2, though multiplication is not continuous, and the flat and degree-based functors differ for n ≥ 3.

ABSTRACT

A priori, the set of birational transformations of an algebraic variety is just a group. We survey the possible algebraic structures that we may add to it, using in particular parametrised family of birational transformations.

Motivation & Objective

• To understand what algebraic structures can be endowed on the group Bir(X) of birational transformations of an algebraic variety X.
• To analyze the representability of the functor Bir_X(A) parameterizing A-families of birational transformations.
• To compare two key subfunctors: Bir_X^flat (flat families) and Bir_X^deg (degree-bounded families), especially on P^n.
• To determine whether the scheme structure from flat families yields a well-behaved topology and algebraic group-like properties on Bir(X).

Proposed method

• Uses the functorial approach of Demazure and Serre to define morphisms A → Bir(X) via families of birational maps over a base scheme A.
• Introduces the subfunctor Bir_X^flat(A) consisting of families that are flat over A, which is representable by a scheme.
• Applies the graph construction to analyze birational maps and their flatness, particularly for maps on P^n.
• Uses explicit examples (e.g., affine families on P^2 and A^n) to demonstrate unboundedness and non-representability of the full Bir_X functor.
• Compares Bir_X^flat and Bir_X^deg via Hilbert polynomials and degree behavior, especially in P^n.
• Employs the Segre embedding to compute the Hilbert polynomial of the graph of a birational map in P^8.

Experimental results

Research questions

• RQ1 Can the group Bir(X) of birational transformations be given a scheme structure via representable functors of families of transformations?
• RQ2 Is the subfunctor of flat families of birational maps representable, and does it yield a well-behaved scheme structure on Bir(P^n)?
• RQ3 How do the flat and degree-based subfunctors Bir_X^flat and Bir_X^deg compare on P^n, especially for n ≥ 3?
• RQ4 Is the multiplication map on Bir(P^n) continuous or even a morphism under the flat scheme structure?
• RQ5 Does the flatness condition correspond to families with fixed multidegree on connected components?

Key findings

• The full functor Bir_X is not representable by a scheme or ind-scheme, even for X = P^2, due to unbounded families of birational maps.
• The subfunctor Bir_X^flat of flat families is representable by a scheme, providing a well-defined scheme structure on Bir(X) for any X.
• For P^n with n ≥ 2, the set Bir(P^n)_d of maps of degree d is open in the flat scheme structure.
• The multiplication map on Bir(P^n) is not continuous under the flat scheme topology, as shown by a morphism ν' whose image jumps in degree.
• For n ≥ 3, Bir_X^flat and Bir_X^deg are not equal, as demonstrated by families where the inverse has non-constant degree despite constant degree of the map.
• On P^2, it is conjectured that Bir_X^flat = Bir_X^deg, due to the Hilbert polynomial of the graph depending only on degree d.