cohomCalg(NormalToricVariety) -- locally stashed cohomology vectors from CohomCalg

Synopsis

Function: cohomCalg
Usage:

H = cohomCalg X
Inputs:
- X, a normal toric variety,
Optional inputs:
- Silent => a Boolean value, default value null, Not used in this particular method
Outputs:
- H, a mutable hash table,
Consequences:
- The mutable hash table is stashed into X.cache#CohomCalg. If it doesn't exist yet, it is created.

Description

The keys of this hash table are the divisor classes (degrees) whose cohomology vector has already been computed. The value of the hash table for this key is a list of two things: the cohomology vector, and a list representing the denominators which appear for this degree.

i1 : needsPackage "ReflexivePolytopesDB"

o1 = ReflexivePolytopesDB

o1 : Package

i2 : topes = kreuzerSkarke(5, Limit => 20);
using offline data file: ks5-n50.txt

i3 : A = matrix topes_15

o3 = | 1 1 0 1 -1 -2 1  |
     | 0 2 0 0 -4 0  6  |
     | 0 0 1 0 2  -1 -4 |
     | 0 0 0 2 -2 0  0  |

              4       7
o3 : Matrix ZZ  <-- ZZ

i4 : P = convexHull A

o4 = P

o4 : Polyhedron

i5 : X = normalToricVariety P

o5 = X

o5 : NormalToricVariety

i6 : H = cohomCalg X

o6 = MutableHashTable{}

o6 : MutableHashTable

Notice that the hash table H is empty, as we haven't tried computing any cohomology vectors yet.

i7 : cohomCalg(X, {-4, 10, -9})

o7 = {0, 0, 0, 12960, 0}

o7 : List

i8 : for i from 0 to dim X list rank HH^i(X, OO_X(-4, 10, -9))

o8 = {0, 0, 0, 12960, 0}

o8 : List

i9 : peek cohomCalg X

o9 = MutableHashTable{{-4, 10, -9} => {{0, 0, 0, 12960, 0}, {{3,
     ------------------------------------------------------------------------
     1x0*x1*x2*x6}, {3, 1x0*x1*x2*x3*x6}, {3, 1x0*x1*x2*x4*x6}}}}

Ways to use this method: