# pullback(ToricMap,ToricDivisor) -- make the pullback of a Cartier divisor under a toric map

## Synopsis

• Function: pullback
• Usage:
pullback(f, D)
• Inputs:
• f, ,
• D, , on the target of f that is Cartier
• Outputs:
• , the pullback of the divisor D under the map f

## Description

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