Torus-invariant Cartier divisors pullback under a toric map by composing the toric map with the support function of the divisor. For more information, see Proposition 6.2.7 in Cox-Little-Schenck's Toric Varieties.
As a first example, we consider the projection from a product of two projective lines onto the first factor. The pullback of a point is just a fibre in the product.
i1 : P = toricProjectiveSpace 1; |
i2 : X = P ** P; |
i3 : f = X^[0] o3 = | 1 0 | o3 : ToricMap P <--- X |
i4 : pullback(f, P_0) o4 = X 0 o4 : ToricDivisor on X |
i5 : pullback(f, 2*P_0 - 6*P_1) o5 = 2*X - 6*X 0 1 o5 : ToricDivisor on X |
i6 : assert (isWellDefined f and f == map(P, X, matrix {{1,0}})) |
The next example illustrates that the pullback of a line through the origin in affine plane under the blowup map is a line together with the exceptional divisor.
i7 : A = affineSpace 2, max A o7 = (A, {{0, 1}}) o7 : Sequence |
i8 : B = toricBlowup({0,1}, A); |
i9 : g = B^[] o9 = | 1 0 | | 0 1 | o9 : ToricMap A <--- B |
i10 : pullback(g, A_0) o10 = B + B 0 2 o10 : ToricDivisor on B |
i11 : pullback(g, -3*A_0 + 7*A_1) o11 = - 3*B + 7*B + 4*B 0 1 2 o11 : ToricDivisor on B |