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Cremona :: exceptionalLocus

exceptionalLocus -- exceptional locus of a birational map

Synopsis

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(RationalMap).

Below, we compute the exceptional locus of the map defined by the linear system of quadrics through the quintic rational normal curve in 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 : psi = inverseMap phi;

o3 : RationalMap (quadratic rational map from 5-dimensional subvariety of PP^9 to PP^5)
i4 : -- a fast probabilistic test
     assert last(p = point source phi, psi phi p == p)
i5 : forceInverseMap(phi,psi)
i6 : E = exceptionalLocus phi;

                ZZ
o6 : Ideal of ------[x , x , x , x , x , x ]
              100003  0   1   2   3   4   5
i7 : E == phi^* ideal psi

o7 = true
i8 : 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

Ways to use exceptionalLocus :