classGroup(ToricMap) -- make the induced map between class groups

Synopsis

Function: classGroup
Usage:

classGroup f
Inputs:
- f, a toric map, with a smooth target
Outputs:
- a matrix, representing the map of abelian groups between the corresponding class groups

Description

Given a toric map $f : X \to Y$ where $Y$ a smooth toric variety, this method returns the induced map of abelian groups from the class group of $Y$ to the class group of $X$. For arbitrary normal toric varieties, the classGroup is not a functor. However, classGroup is a contravariant functor on the category of smooth normal toric varieties.

We illustrate this method on the projection from the first Hirzebruch surface to the projective line.

i1 : X = hirzebruchSurface 1;

i2 : Y = toricProjectiveSpace 1;

i3 : f = map(Y, X, matrix {{1, 0}})

o3 = | 1 0 |

o3 : ToricMap Y <--- X

i4 : f' = classGroup f

o4 = | 1 |
     | 0 |

              2       1
o4 : Matrix ZZ  <-- ZZ

i5 : assert (isWellDefined f and source f' == classGroup Y and
         target f' == classGroup X)

The induced map between the class groups is compatible with the induced map between the groups of torus-invariant Weil divisors.

i6 : f'' = weilDivisorGroup f

o6 = | 0 1 |
     | 0 0 |
     | 1 0 |
     | 0 0 |

              4       2
o6 : Matrix ZZ  <-- ZZ

i7 : assert(f' * fromWDivToCl Y  == fromWDivToCl X  * f'')

The source of the toric map need not be smooth.

i8 : Z = toricBlowup({0, 1}, X, {1,2});

i9 : assert (isWellDefined Z and not isSmooth Z)

i10 : g = map(Y, Z, matrix{{1, 0}})

o10 = | 1 0 |

o10 : ToricMap Y <--- Z

i11 : g' = classGroup g

o11 = | 1 |
      | 0 |
      | 0 |

               3       1
o11 : Matrix ZZ  <-- ZZ

i12 : g'' = weilDivisorGroup g

o12 = | 0 1 |
      | 0 0 |
      | 1 0 |
      | 0 0 |
      | 0 1 |

               5       2
o12 : Matrix ZZ  <-- ZZ

i13 : assert(g' * fromWDivToCl Y == fromWDivToCl Z  * g'')

i14 : assert (isWellDefined g and source g' == classGroup Y and
          target g' == classGroup Z)

Ways to use this method: