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NormalToricVarieties :: isWellDefined(ToricMap)

isWellDefined(ToricMap) -- whether a toric map is well defined

Synopsis

Description

Let $X$ and $Y$ be normal toric varieties whose underlying lattices are $N_X$ and $N_Y$ respectively. Every toric map $f : X \to Y$ corresponds to a unique map $g : N_X \to N_Y$ of lattices such that, for any cone $\sigma$ in the fan of $X$, there is a cone in the fan of $Y$ that contains the image $g(\sigma)$. For more information on this correspondence, see Theorem 3.3.4 in Cox-Little-Schenck's Toric Varieties. This method determines whether the underlying map of lattices defines a toric map.

We illustrate this test with the projection from the second Hirzebruch surface to the projective line.

i1 : X = hirzebruchSurface 2;
 
i2 : Y = toricProjectiveSpace 1;
 
i3 : f = map (Y, X, matrix {{1, 0}})

o3 = | 1 0 |

o3 : ToricMap Y <--- X
 
i4 : source f

o4 = X

o4 : NormalToricVariety
 
i5 : target f

o5 = Y

o5 : NormalToricVariety
 
i6 : matrix f

o6 = | 1 0 |

              1        2
o6 : Matrix ZZ  <--- ZZ
 
i7 : assert (isWellDefined f and source f === X and
         target f === Y and matrix f === matrix {{1, 0}})

The second example illustrates two attempts to define a toric map from the projective plane to a weighted projective space. The first, corresponding to the identity on the lattices, is not well-defined. The second, corresponding to a stretch in the lattices, is well-defined. By making the current debugging level greater than one, one gets some addition information about the nature of the failure.

i8 : debugLevel = 1;
 
i9 : Z = toricProjectiveSpace 2;
 
i10 : W = weightedProjectiveSpace {1, 1, 2};
 
i11 : g = map (W, Z, 1)

o11 = | 1 0 |
      | 0 1 |

o11 : ToricMap W <--- Z
 
i12 : assert not isWellDefined g
-- expected image of each maximal cone to be contained in some maximal cone
 
i13 : h = map (W, Z, matrix {{1, 0}, {0, 2}})

o13 = | 1 0 |
      | 0 2 |

o13 : ToricMap W <--- Z
 
i14 : assert isWellDefined h

This method also checks the following aspects of the data structure:

  • the underlying HashTable has the expected keys, namely source, target, matrix, and cache,
  • the value of the source key is a NormalToricVariety,
  • the value of the target key is a NormalToricVariety,
  • the value of the matrix key is a Matrix,
  • the underling ring of the matrix is ZZ,
  • the rank of the source of the matrix equal dimension of the source variety,
  • the rank of the target of the matrix equal dimension of the target variety,
  • the value of the cache key is a CacheTable.

See also