Deautonomisation by singularity confinement: an algebro-geometric justification

TL;DR

This paper provides an algebro-geometric justification for the singularity confinement criterion used in deautonomizing integrable mappings, showing its equivalence to conditions derived from blow-up regularization in discrete Painleve equations.

Contribution

It demonstrates that singularity confinement at the first opportunity aligns with algebro-geometric conditions from blow-up analysis, validating the standard criterion.

Findings

Confinement at later stages leads to nonintegrability.

Conditions from singularity confinement match those from blow-up analysis.

Parameter conditions are equivalent to linear transformations on the Picard group.

Abstract

The `deautonomisation' of an integrable mapping of the plane consists in treating the free parameters in the mapping as functions of the independent variable, the precise expressions of which are to be determined with the help of a suitable criterion for integrability. Standard practice is to use the singularity confinement criterion and to require that singularities be confined at the very first opportunity. An algebro-geometrical analysis will show that confinement at a later stage invariably leads to a nonintegrable deautonomized system, thus justifying the standard singularity confinement approach. In particular, it will be shown on some selected examples of discrete Painleve equations, how their regularisation through blow-up yields exactly the same conditions on the parameters in the mapping as the singularity confinement criterion. Moreover, for all these examples, it will be shown that the conditions on the parameters are in fact equivalent to a linear transformation on part of the Picard group, obtained from the blow-up.