exceptionalLocus -- exceptional locus of a birational map

Synopsis

• Usage:

  exceptionalLocus phi

• Inputs:
  • phi, a rational map, a birational map $X\dashrightarrow Y$
• Optional inputs:
  • Certify => ..., default value false, whether to ensure correctness of output
• Outputs:
  • an ideal, the ideal defining the closure in X of the locus where phi is not a local isomorphism

Description

This method simply calculates the inverse image of the base locus of the inverse map, which in turn is determined through the method inverse.

Below, we compute the exceptional locus of the map defined by the linear system of quadrics through the quintic rational normal curve in $\mathbb{P}^5$.

i1 : P5 := ZZ/100003[x_0..x_5];

i2 : phi = rationalMap(minors(2,matrix{{x_0,x_1,x_2,x_3,x_4},{x_1,x_2,x_3,x_4,x_5}}),Dominant=>2);

o2 : RationalMap (quadratic rational map from PP^5 to 5-dimensional subvariety of PP^9)

i3 : E = exceptionalLocus phi;

                ZZ
o3 : Ideal of ------[x ..x ]
              100003  0   5

i4 : assert(E == phi^* ideal phi^-1)

i5 : assert(E == minors(3,matrix{{x_0,x_1,x_2,x_3},{x_1,x_2,x_3,x_4},{x_2,x_3,x_4,x_5}}))

See also

• ideal(RationalMap) -- base locus of a rational map
• inverseMap(RationalMap) -- inverse of a birational map
• RationalMap ^** Ideal -- inverse image via a rational map
• isIsomorphism(RationalMap) -- whether a birational map is an isomorphism
• forceInverseMap -- declare that two rational maps are one the inverse of the other