ToricCycle + ToricCycle -- perform arithmetic on toric cycles

Operator: +
Usage:

C1 + C2

C1 - C2

5*C1

(1/2)*C1

-C1
Inputs:
- C1, a toric cycle,
- C2, a toric cycle,
Outputs:
- a toric cycle

Description

The set of torus-invariant cycles forms an abelian group under addition. The basic operations arising from this structure, including addition, subtraction, negation, and scalar multiplication by integers, are available.

We illustrate a few of the possibilities on one variety.

i1 : rayList={{1,0},{0,1},{-1,-1},{0,-1}}

o1 = {{1, 0}, {0, 1}, {-1, -1}, {0, -1}}

o1 : List

i2 : coneList={{0,1},{1,2},{2,3},{3,0}}

o2 = {{0, 1}, {1, 2}, {2, 3}, {3, 0}}

o2 : List

i3 : X = normalToricVariety(rayList,coneList)

o3 = X

o3 : NormalToricVariety

i4 : cyc = toricCycle({{2,3} => 1,{3,0} => 4},X)

o4 = X       + 4*X
      {2, 3}      {3, 0}

o4 : ToricCycle on X

i5 : altcyc = (-2)*cyc

o5 = - 2*X       - 8*X
          {2, 3}      {3, 0}

o5 : ToricCycle on X

i6 : cyc + altcyc

o6 = - X       - 4*X
        {2, 3}      {3, 0}

o6 : ToricCycle on X

i7 : cyc - altcyc

o7 = 3*X       + 12*X
        {2, 3}       {3, 0}

o7 : ToricCycle on X

i8 : -cyc

o8 = - X       - 4*X
        {2, 3}      {3, 0}

o8 : ToricCycle on X

i9 : X_{0} + X_{1}

o9 = X    + X
      {0}    {1}

o9 : ToricCycle on X

i10 : 2*X_{0,1} - X_{0,2}

o10 = 2*X       - X
         {0, 1}    {0, 2}

o10 : ToricCycle on X

Ways to use this method:

ToricCycle + ToricCycle -- perform arithmetic on toric cycles
ToricCycle + ToricDivisor
ToricDivisor + ToricCycle

The source of this document is in TropicalToric/ToricCycleDoc.m2:344:0.