This defines the locus where a given map of projective varieties is not defined. If the option SaturateOutput is set to false, the output will not be saturated. The default value is true. Consider the following rational map from P2 to P1
i1 : R = QQ[x,y,z]; |
i2 : S = QQ[a,b]; |
i3 : f = map(R, S, {x,y}); o3 : RingMap R <--- S |
i4 : baseLocusOfMap(f) o4 = ideal (y, x) o4 : Ideal of R |
Observe it is not defined at the point [0:0:1], which is exactly what one expects. However, we can restrict the map to a curve on P2 and then it will be defined everywhere.
i5 : R=QQ[x,y,z]/(y^2*z-x*(x-z)*(x+z)); |
i6 : S=QQ[a,b]; |
i7 : f=map(R,S,{x,y}); o7 : RingMap R <--- S |
i8 : baseLocusOfMap(f) o8 = ideal 1 o8 : Ideal of R |
Let us next consider the quadratic Cremona transformation.
i9 : R=QQ[x,y,z]; |
i10 : S=QQ[a,b,c]; |
i11 : f=map(R,S,{y*z,x*z,x*y}); o11 : RingMap R <--- S |
i12 : J=baseLocusOfMap(f) o12 = ideal (y*z, x*z, x*y) o12 : Ideal of R |
i13 : minimalPrimes J o13 = {ideal (y, x), ideal (z, x), ideal (z, y)} o13 : List |
The base locus is exactly the three points one expects.