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# ToricCycle + ToricCycle -- perform arithmetic on toric cycles

## Synopsis

• Operator: +
• Usage:
C1 + C2
C1 - C2
5*C1
(1/2)*C1
-C1
• Inputs:
• C1, ,
• C2, ,
• Outputs:

## 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 multplication 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