isBirationalMap -- Checks if a map between projective varieties is birational.

Synopsis

• Usage:

  val = isBirationalMap(a,b,f)

  val = isBirationalMap(R,S,f)

  val = isBirationalMap(Pi)

• Inputs:
  • a, an ideal, defining equations for X
  • b, an ideal, defining equations for Y
  • f, a basic list, A list of where to send the variables in the ring of b, to in the ring of a.
  • R, a ring, the homogeneous coordinate ring of X
  • S, a ring, the homogeneous coordinate ring of Y
  • Pi, a ring map, A ring map S to R corresponding to X mapping to Y
• Optional inputs:
  • AssumeDominant => ..., default value false, If true, certain functions assume that the map from X to Y is dominant.
  • HybridLimit => ..., default value 15, An option to control HybridStrategy
  • MinorsCount => ..., default value null, An option controlling the behavior of isBirational and inverseOfMap (and other functions which call those).
  • QuickRank => ..., default value true, An option for computing how rank is computed
  • Strategy => ..., default value HybridStrategy, Determines the desired Strategy in each function.
  • Verbose => ..., default value true, generate informative output
• Outputs:
  • val, a Boolean value, true if the map is birational, false if otherwise

Description

i1 : R=QQ[x,y,z];

i2 : S=QQ[a,b,c];

i3 : Pi = map(R, S, {x*y, x*z, y*z});

o3 : RingMap R <--- S

i4 : isBirationalMap(Pi, Verbose=>false, Strategy=>SimisStrategy )

o4 = true

i5 : R=QQ[x,y,z]/(x^3+y^3-z^3);

i6 : S=QQ[s,t];

i7 : Pi = map(R, S, {x, y-z});

o7 : RingMap R <--- S

i8 : isBirationalMap(Pi, Verbose=>false)

o8 = false

i9 : R = ZZ/5[x,y,z]/(x^3+y^3-z^3);

i10 : S = ZZ/5[a,b,c]/(a^3+b^3-b^3);

i11 : h = map(R, S, {x^5, y^5, z^5});

o11 : RingMap R <--- S

i12 : isBirationalMap(h, Strategy=>SaturationStrategy)
isBirationalMap: About to find the image of the map.  If you know the image, 
        you may want to use the AssumeDominant option if this is slow.
isBirationalMap: Found the image of the map.
Starting isBirationalOntoImage
isBirationalOntoImageRees:  About to compute the Jacobian Dual Matrix,
if it is slow, run again and  set Strategy=>HybridStrategy or SimisStrategy.
isBirationalOntoImageRees: computed Jacobian Dual Matrix- barJD
Jacobain dual matrix has  3 columns  and   0 rows.
isBirationalOntoImageRees: is computing the rank of the  Jacobian Dual Matrix- barJD

o12 = false

Caveat

See also

• isBirationalOntoImage -- Checks if a map between projective varieties is birational onto its image.