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

Synopsis

Function: pullback
Usage:

pullback(f, D)
Inputs:
- f, a toric map,
- D, a toric divisor, on the target of f that is Cartier
Outputs:
- a toric divisor, 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

Ways to use this method: