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NormalToricVarieties :: pullback(ToricMap,ToricDivisor)

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

Synopsis

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

See also