quadroQuadricCremonaTransformation -- quadro-quadric Cremona transformations

Synopsis

Usage:

quadroQuadricCremonaTransformation(n,i)

quadroQuadricCremonaTransformation(n,i,K)
Inputs:
- n, an integer, the dimension of the projective space
- i, an integer, the i-th case in the classification for P^n (for instance, if n=5 then 1<=i<=39)
- K, a ring, the ground field (optional, the default value is QQ)
Outputs:
- a rational map, an example of quadro-quadric Cremona transformation over K, according to the classifications given in the paper Quadro-quadric Cremona transformations in low dimensions via the JC-correspondence, by Pirio and Russo.

Description

i1 : quadroQuadricCremonaTransformation(5,23)

o1 = -- rational map --
     source: Proj(QQ[x, y, z, t, u, v])
     target: Proj(QQ[x, y, z, t, u, v])
     defining forms: {
                      x*y,
                      
                       2
                      x ,
                      
                               2    2
                      - y*z + t  + u ,
                      
                      -x*t,
                      
                      -x*u,
                      
                      -x*v
                     }

o1 : RationalMap (Cremona transformation of PP^5 of type (2,2))

i2 : describe oo

o2 = rational map defined by forms of degree 2
     source variety: PP^5
     target variety: PP^5
     dominance: true
     birationality: true
     projective degrees: {1, 2, 2, 2, 2, 1}
     number of minimal representatives: 1
     dimension base locus: 3
     degree base locus: 2
     coefficient ring: QQ

In addition, the four pairs (n,i)=(5,1),(8,1),(14,1),(26,1) correspond to the four examples of special quadro-quadric Cremona transformations:

i3 : describe quadroQuadricCremonaTransformation(5,1)

o3 = 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: QQ

i4 : describe quadroQuadricCremonaTransformation(8,1)

o4 = rational map defined by forms of degree 2
     source variety: PP^8
     target variety: PP^8
     dominance: true
     birationality: true
     number of minimal representatives: 1
     dimension base locus: 4
     degree base locus: 6
     coefficient ring: QQ

i5 : describe quadroQuadricCremonaTransformation(14,1)

o5 = rational map defined by forms of degree 2
     source variety: PP^14
     target variety: PP^14
     dominance: true
     birationality: true
     number of minimal representatives: 1
     dimension base locus: 8
     degree base locus: 14
     coefficient ring: QQ

i6 : describe quadroQuadricCremonaTransformation(26,1)

o6 = rational map defined by forms of degree 2
     source variety: PP^26
     target variety: PP^26
     dominance: true
     birationality: true
     number of minimal representatives: 1
     dimension base locus: 16
     degree base locus: 78
     coefficient ring: QQ

Ways to use quadroQuadricCremonaTransformation:

quadroQuadricCremonaTransformation(Ring,ZZ,ZZ)
quadroQuadricCremonaTransformation(ZZ,ZZ)
quadroQuadricCremonaTransformation(ZZ,ZZ,Ring)

For the programmer

The object quadroQuadricCremonaTransformation is a method function.

quadroQuadricCremonaTransformation -- quadro-quadric Cremona transformations

Synopsis

Description

See also

Ways to use quadroQuadricCremonaTransformation:

For the programmer