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computeEigencharacters -- compute the eigencharacters

Description

This method is the implementation of Algorithm A.5 in "Reduced \v{C}ech complexes and computing higher direct images under toric maps". More precisely, the higher direct image sheaf inherits a trivial torus action by the kernel torus $T_K :\!= \ker \phi\vert_{T_X}$, and so splits into a direct sum of sheaves indexed by eigencharacters. This method computes these characters, as well as a pair of divisors $(D,E)$ for use in Algorithm A.8 which computes the higher direct image sheaf.

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

o3 : ToricMap Y <--- X
i4 : D = {0,-3,0,0};
i5 : computeEigencharacters(phi,1,D)

o5 = HashTable{| 0 | => (| 0  |, {{1}})}
               | 1 |     | -2 |
                         | 0  |
                         | -1 |
               | 0 | => (| 0  |, {{2}})
               | 2 |     | -1 |
                         | 0  |
                         | -2 |

o5 : HashTable

One might hope that the set of eigencharacters is a convex set, but this is not always the case. For instance, when $X$ is the blowup of $\mathbb{P}^1 \times \mathbb{P}^1$ at the four torus fixed points and phi is the projection to $\mathbb{P}^1$, we can find an example of a line bundle whose eigencharacters are not convex.

i6 : Z = normalToricVariety({{1,0},{1,1},{0,1},{-1,1},{-1,0},{-1,-1},
             {0,-1},{1,-1}},{{0,1},{1,2},{2,3},{3,4},{4,5},{5,6},{6,7},{0,7}});
i7 : phi = map(Y,Z,matrix{{1,0}});

o7 : ToricMap Y <--- Z
i8 : D = toricDivisor({0,-2,0,0,0,-2,1,0},Z);

o8 : ToricDivisor on Z
i9 : keys computeEigencharacters(phi,1,D)

o9 = {| 0  |, | 0 |}
      | -1 |  | 1 |

o9 : List

See also

Ways to use computeEigencharacters:

  • computeEigencharacters(ToricMap,ZZ,List)
  • computeEigencharacters(ToricMap,ZZ,ToricDivisor)

For the programmer

The object computeEigencharacters is a method function.


The source of this document is in /build/reproducible-path/macaulay2-1.25.06+ds/M2/Macaulay2/packages/ToricHigherDirectImages.m2:875:0.