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SRdeformations :: mirrorSphere

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  |
     | 0  0  1  0  |
     | 1  0  1  0  |
     | -1 -1 -1 0  |
     | 0  0  0  1  |
     | 1  0  0  1  |
     | -1 -1 0  -1 |
     | -1 0  -1 -1 |
     | -1 -1 -1 -1 |

              11        4
o8 : Matrix QQ   <--- QQ
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 OldPolyhedra to compute convex hulls and its faces instead of ConvexInterface you are limited to rather simple examples.

See also

For the programmer

The object mirrorSphere is a symbol.