It focuses on finding where a birational map is undefined, checking whether a map is a closed embedding, checking birationality and computing inverse maps

- A. V. Dória, S. H. Hassanzadeh, A. Simis,
*A characteristic free criterion of birationality*, Advances in Mathematics, Volume 230, Issue 1, 1 May 2012, Pages 390-413. - A. Simis,
*Cremona Transformations and some Related Algebras*, Journal of Algebra, Volume 280, Issue 1, 1 October 2004, Pages 162–179

`isBirational`(missing documentation) gives a probabilisitc answer to the question of whether a map between varieties is birational. Furthermore, if the source is projective space, then`degreeOfRationalMap`(missing documentation) with`MathMode=>true`can give a deterministic answer. In some cases, the speed of the latter is comparable with isBirationalMap with`AssumeDominant=>true.``inverseMap`(missing documentation) gives a fast computation of the inverse of a birational map if the source is projective space*and*the map has maximal linear rank. In some cases, even if the map has maximal linear rank, our function inverseOfMap appears to be competitive however. If you pass inverseMap a map not from projective space, then it calls a modified version`invertBirationalMap`(missing documentation) copied from`Parametrization`(missing documentation).

- C.J. Bott <cjamesbott@gmail.com>

- Functions and commands
- baseLocusOfMap -- Computes base locus of a map from a projective variety to projective space
- dimImage -- Computes dimension of image of rational map of projective varieties
- idealOfImageOfMap -- Finds defining equations for the image of a rational map between varieties or schemes
- inverseOfMap -- Computes the inverse map of a given birational map between projective varieties. Returns an error if the map is not birational onto its image.
- isBirationalMap -- Checks if a map between projective varieties is birational.
- isBirationalOntoImage -- Checks if a map between projective varieties is birational onto its image.
- isEmbedding -- Checks whether a map of projective varieties is a closed embedding.
- isRegularMap -- Checks whether a map to projective space is regular
- isSameMap -- Checks whether two maps to projective space are really the same
- jacobianDualMatrix -- Computes the Jacobian Dual Matrix, a matrix whose kernel describing the syzygies of the inverse map.
- mapOntoImage -- Given a map of rings, correspoing to X mapping to Y, this returns the map of rings corresponding to X mapping to f(X).
- relationType -- Given an ideal in a ring this computes the maximum degree, of the new variables, of the minimal generators of the defining ideal of the associated Rees algebra.
- sourceInversionFactor -- Computes the the common factor among the the components of the composition of the inverse map and the original map.

- Symbols
- AssumeDominant -- If true, certain functions assume that the map from X to Y is dominant.
- CheckBirational -- If true, functions will check birationality.
- HybridLimit -- An option to control HybridStrategy
- HybridStrategy -- A strategy for inverseOfMap, isBirationalMap and isEmbedding.
- MinorsCount -- An option controlling the behavior of isBirational and inverseOfMap (and other functions which call those).
- ReesStrategy -- A strategy for inverseOfMap, isBirationalMap, relationType and is Embedding.
- SaturateOutput -- If false, certain functions will not saturate their output.
- SaturationStrategy -- A strategy for inverseOfMap, isBirationalMap, relationType and is Embedding.
- SimisStrategy -- A strategy for inverseOfMap, isBirationalMap and isEmbedding.