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mirrorSphere -- Example how to compute the mirror sphere.

Description

Example how to compute the mirror sphere as an Complex.

This is work in progress. Many interesting pieces are not yet implemented.

i1 : R=QQ[x_0..x_4]

o1 = R

o1 : PolynomialRing
i2 : I=ideal(x_0*x_1,x_2*x_3*x_4)

o2 = ideal (x x , x x x )
             0 1   2 3 4

o2 : Ideal of R
i3 : C=idealToComplex I

o3 = 2: x x x  x x x  x x x  x x x  x x x  x x x  
         0 2 3  1 2 3  0 2 4  1 2 4  0 3 4  1 3 4

o3 : complex of dim 2 embedded in dim 4 (printing facets)
     equidimensional, simplicial, F-vector {1, 5, 9, 6, 0, 0}, Euler = 1
i4 : PT1C=PT1 C

o4 = 4: y y y y y y y y y y  
         0 1 2 3 4 5 6 7 8 9

o4 : complex of dim 4 embedded in dim 4 (printing facets)
     equidimensional, non-simplicial, F-vector {1, 10, 24, 25, 11, 1}, Euler = 0
i5 : tropDefC=tropDef(C,PT1C)

o5 = 1: y y  y y  y y  y y  y y  
         0 4  8 9  3 7  2 6  1 5

o5 : co-complex of dim 1 embedded in dim 4 (printing facets)
     equidimensional, non-simplicial, F-vector {0, 0, 5, 9, 6, 1}, Euler = -1
i6 : tropDefC.grading

o6 = | -1 0  0  0  |
     | 1  0  0  0  |
     | -1 2  0  0  |
     | -1 0  2  0  |
     | 0  -1 -1 -1 |
     | 3  -1 -1 -1 |
     | 0  2  -1 -1 |
     | 0  -1 2  -1 |
     | -1 0  0  2  |
     | 0  -1 -1 2  |

              10        4
o6 : Matrix ZZ   <--- ZZ
i7 : B=dualize tropDefC

o7 = 2: v v v  v v v v  v v v v  v v v v  v v v  
         2 4 7  2 4 8 9  2 5 7 9  4 5 7 8  5 8 9

o7 : complex of dim 2 embedded in dim 4 (printing facets)
     equidimensional, non-simplicial, F-vector {1, 6, 9, 5, 0, 0}, Euler = 1
i8 : B.grading

o8 = | -1 0  0  0  |
     | 0  -1 0  0  |
     | -1 -1 0  0  |
     | 1  1  1  0  |
     | 0  0  -1 0  |
     | -1 0  -1 0  |
     | 1  1  0  1  |
     | 1  0  1  1  |
     | 1  1  1  1  |
     | 0  0  0  -1 |
     | -1 0  0  -1 |

              11        4
o8 : Matrix ZZ   <--- ZZ
i9 : fvector C

o9 = {1, 5, 9, 6, 0, 0}

o9 : List
i10 : fvector B

o10 = {1, 6, 9, 5, 0, 0}

o10 : List

Caveat

The implementation of testing whether a face is tropical so far uses a trick to emulate higher order. For very complicated (non-complete intersections and non-Pfaffians) examples this may lead to an incorrect result. Use with care. This will be fixed at some point.

If using Polyhedra to compute convex hulls and its faces instead of ConvexInterface you are limited to rather simple examples.

See also

  • PT1 -- Compute the deformation polytope associated to a Stanley-Reisner complex.
  • tropDef -- The co-complex of tropical faces of the deformation polytope.
  • HH Complex -- Compute the homology of a complex.

For the programmer

The object mirrorSphere is a symbol.