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# specialCremonaTransformation -- special Cremona transformations whose base locus has dimension at most three

- Usage:
    `specialCremonaTransformation i`
    `specialCremonaTransformation(i,K)`
- Inputs:
  - i, an integer, an integer between 1 and 12
  - K, a ring, the ground field (optional, the default value is QQ)
- Outputs:
  - a rational map, an example of special Cremona transformation over K, according to the classification given in Table 1 of Special cubic Cremona transformations of P6 and P7.

## Description

A Cremona transformation is said to be special if the base locus scheme is smooth and irreducible. To ensure this condition, the field K must be large enough but no check is made.

    i1 : time apply(1..12,i -> describe specialCremonaTransformation(i,ZZ/3331))
     -- used 1.27476s (cpu); 1.07289s (thread); 0s (gc)

    o1 = (rational map defined by forms of degree 3,
          source variety: PP^3
          target variety: PP^3
          dominance: true
          birationality: true
          projective degrees: {1, 3, 3, 1}
          number of minimal representatives: 1
          dimension base locus: 1
          degree base locus: 6
          coefficient ring: ZZ/3331
         ------------------------------------------------------------------------
         rational map defined by forms of degree 2,
         source variety: PP^4
         target variety: PP^4
         dominance: true
         birationality: true
         projective degrees: {1, 2, 4, 3, 1}
         number of minimal representatives: 1
         dimension base locus: 1
         degree base locus: 5
         coefficient ring: ZZ/3331
         ------------------------------------------------------------------------
         rational map defined by forms of degree 3,
         source variety: PP^4
         target variety: PP^4
         dominance: true
         birationality: true
         projective degrees: {1, 3, 4, 2, 1}
         number of minimal representatives: 1
         dimension base locus: 2
         degree base locus: 5
         coefficient ring: ZZ/3331
         ------------------------------------------------------------------------
         rational map defined by forms of degree 4,
         source variety: PP^4
         target variety: PP^4
         dominance: true
         birationality: true
         projective degrees: {1, 4, 6, 4, 1}
         number of minimal representatives: 1
         dimension base locus: 2
         degree base locus: 10
         coefficient ring: ZZ/3331
         ------------------------------------------------------------------------
         rational map defined by forms of degree 2,
         source variety: PP^5
         target variety: PP^5
         dominance: true
         birationality: true
         projective degrees: {1, 2, 4, 4, 2, 1}
         number of minimal representatives: 1
         dimension base locus: 2
         degree base locus: 4
         coefficient ring: ZZ/3331
         ------------------------------------------------------------------------
         rational map defined by forms of degree 2,
         source variety: PP^6
         target variety: PP^6
         dominance: true
         birationality: true
         projective degrees: {1, 2, 4, 8, 9, 4, 1}
         number of minimal representatives: 1
         dimension base locus: 2
         degree base locus: 7
         coefficient ring: ZZ/3331
         ------------------------------------------------------------------------
         rational map defined by forms of degree 2,
         source variety: PP^6
         target variety: PP^6
         dominance: true
         birationality: true
         projective degrees: {1, 2, 4, 8, 8, 4, 1}
         number of minimal representatives: 1
         dimension base locus: 2
         degree base locus: 8
         coefficient ring: ZZ/3331
         ------------------------------------------------------------------------
         rational map defined by forms of degree 5,
         source variety: PP^5
         target variety: PP^5
         dominance: true
         birationality: true
         projective degrees: {1, 5, 10, 10, 5, 1}
         number of minimal representatives: 1
         dimension base locus: 3
         degree base locus: 15
         coefficient ring: ZZ/3331
         ------------------------------------------------------------------------
         rational map defined by forms of degree 2         ,
         source variety: PP^8
         target variety: PP^8
         dominance: true
         birationality: true
         projective degrees: {1, 2, 4, 8, 16, 20, 14, 5, 1}
         number of minimal representatives: 1
         dimension base locus: 3
         degree base locus: 12
         coefficient ring: ZZ/3331
         ------------------------------------------------------------------------
         rational map defined by forms of degree 2         ,
         source variety: PP^8
         target variety: PP^8
         dominance: true
         birationality: true
         projective degrees: {1, 2, 4, 8, 16, 19, 13, 5, 1}
         number of minimal representatives: 1
         dimension base locus: 3
         degree base locus: 13
         coefficient ring: ZZ/3331
         ------------------------------------------------------------------------
         rational map defined by forms of degree 3  ,
         source variety: PP^6
         target variety: PP^6
         dominance: true
         birationality: true
         projective degrees: {1, 3, 9, 13, 11, 5, 1}
         number of minimal representatives: 1
         dimension base locus: 3
         degree base locus: 14
         coefficient ring: ZZ/3331
         ------------------------------------------------------------------------
         rational map defined by forms of degree 3  )
         source variety: PP^6
         target variety: PP^6
         dominance: true
         birationality: true
         projective degrees: {1, 3, 9, 14, 12, 5, 1}
         number of minimal representatives: 1
         dimension base locus: 3
         degree base locus: 13
         coefficient ring: ZZ/3331

    o1 : Sequence

## See also

- quadroQuadricCremonaTransformation -- quadro-quadric Cremona transformations
- specialQuadraticTransformation -- special quadratic transformations whose base locus has dimension three
- specialCubicTransformation -- special cubic transformations whose base locus has dimension at most three

## Ways to use specialCremonaTransformation:

- `specialCremonaTransformation(Ring,ZZ)`
- `specialCremonaTransformation(ZZ)`
- `specialCremonaTransformation(ZZ,Ring)`

## For the programmer

The object specialCremonaTransformation is a method function.

The source of this document is in /build/macaulay2-88fgJW/macaulay2-1.25.11+ds/M2/Macaulay2/packages/Cremona/documentation.m2:897:0.