# ToricMap * ToricMap -- make the composition of two toric maps

## Synopsis

• Operator: *
• Usage:
g * f
• Inputs:
• f, , a toric map between toric varieties
• g, , a toric map between toric varieties
• Outputs:
• , the composition g * f from source f to target g

## Description

Given two toric maps with the target of first equal to the source of second, this method returns the composite map from source of the first to target of the second.

We illustrate this construction with the projection from a smooth Fano Toric threefold to the first Hirzebruch surface and a projection from this Hirzebruch surface to the projective line.

 i1 : X = smoothFanoToricVariety(3,14); i2 : Y = hirzebruchSurface 1; i3 : f = map(Y, X, matrix{{1,0,0},{0,1,0}}) o3 = | 1 0 0 | | 0 1 0 | o3 : ToricMap Y <--- X i4 : assert (isWellDefined f and source f === X and target f === Y) i5 : Z = toricProjectiveSpace 1; i6 : g = map(Z, Y, matrix{{1, 0}}) o6 = | 1 0 | o6 : ToricMap Z <--- Y i7 : assert (isWellDefined g and source g === Y and target g === Z) i8 : h = g * f o8 = | 1 0 0 | o8 : ToricMap Z <--- X i9 : assert (isWellDefined h and source h === X and target h === Z) i10 : X = hirzebruchSurface 1;

Composing diagonal maps and canonical projections yields identity maps.

 i11 : X2 = X ** X; i12 : delta = diagonalToricMap X o12 = | 1 0 | | 0 1 | | 1 0 | | 0 1 | o12 : ToricMap X2 <--- X i13 : assert (X2^ * delta == id_X and X2^ * delta == id_X)