jacobianDualMatrix -- computes the Jacobian dual matrix

Synopsis

• Usage:

  M = jacobianDualMatrix(p)

  M = jacobianDualMatrix(phi)

• Inputs:
  • phi, an instance of the type RationalMapping, a rational map between projective varieties
  • p, a ring map, ring map corresponding to a rational map between projective varieties
• Optional inputs:
  • Strategy => a symbol, default value ReesStrategy, choose the strategy to use: ReesStrategy or SaturationStrategy
  • AssumeDominant => a Boolean value, default value false, whether to assume the provided rational map of projective varieties is dominant, if set to true it can speed up computation
  • QuickRank => a Boolean value, default value true, whether to compute rank via the package FastMinors
• Outputs:
  • M, a matrix, a matrix $M$ over the coordinate ring of the image, the kernel of $M$ describes the syzygies of the inverse map, if it exists.

Description

i1 : R=QQ[x,y];

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

i3 : Pi = map(R, S, {x^3, x^2*y, x*y^2, y^3});

o3 : RingMap R <-- S

i4 : jacobianDualMatrix(Pi, Strategy=>SaturationStrategy)

o4 = | -d -c -b 0 0 0 |
     | c  b  a  0 0 0 |

            /               S               \2     /               S               \6
o4 : Matrix |-------------------------------|  <-- |-------------------------------|
            |  2                    2       |      |  2                    2       |
            \(c  - b*d, b*c - a*d, b  - a*c)/      \(c  - b*d, b*c - a*d, b  - a*c)/

See also

• HybridStrategy -- A strategy for determining whether a map is birational and computing its inverse
• SimisStrategy -- a strategy for determining whether a map is birational and computing its inverse
• ReesStrategy -- a strategy for determining whether a map is birational and computing its inverse