# specialCubicTransformation -- special cubic transformations whose base locus has dimension at most three

## Synopsis

• Usage:
specialCubicTransformation i
specialCubicTransformation(i,K)
• Inputs:
• i, an integer, an integer between 1 and 9
• K, a ring, the ground field (optional, the default value is QQ)
• Outputs:

## Description

The field K is required to be large enough.

 i1 : time specialCubicTransformation 9 -- used 0.173758 seconds o1 = -- rational map -- source: Proj(QQ[x , x , x , x , x , x , x ]) 0 1 2 3 4 5 6 target: subvariety of Proj(QQ[t , t , t , t , t , t , t , t , t , t ]) defined by 0 1 2 3 4 5 6 7 8 9 { 2 2 2 3t t + 10t t + 10t t - 3t t - 10t t + 6t t + 13t t + 13t - 10t t - 10t t + t t + 10t t - 2t t + 10t + 4t t - 7t t + 4t t - 2t t - 8t t + 29t t - 4t + 10t t - 4t t - 13t t + 13t t + 18t t - 4t t + 4t t - 7t t + 4t t - 2t t - 21t t + 19t t - 4t t - 4t t , 2 4 3 4 0 5 1 5 2 5 3 5 4 5 5 0 6 1 6 2 6 4 6 5 6 6 0 7 1 7 2 7 4 7 5 7 6 7 7 1 8 3 8 4 8 5 8 6 8 7 8 0 9 1 9 2 9 4 9 5 9 6 9 7 9 8 9 2 2 2 3t t - 8t t - 5t t + 8t t - 8t t - 8t + 8t t + 8t t + t t - 8t t + t t - 8t - 2t t - t t - 2t t + 3t t - 2t t - 5t t - 13t t - t - 3t t - 8t t + 2t t + 8t t - 8t t - 3t t - t t - 2t t - t t - 2t t + 3t t - 2t t + 3t t - 5t t - t t - t t , 0 2 3 4 0 5 2 5 4 5 5 0 6 1 6 2 6 4 6 5 6 6 0 7 1 7 2 7 3 7 4 7 5 7 6 7 7 0 8 1 8 3 8 4 8 5 8 6 8 7 8 0 9 1 9 2 9 3 9 4 9 5 9 6 9 7 9 8 9 2 2 2 t t - t t - 2t t - 2t t + 2t t - 2t t - 2t + 2t t + 2t t + t t - 3t t - 3t - t t - t t + t t - 4t t + t - 2t t + t t + 2t t - 2t t - 3t t + t t - t t - t t + t t + 2t t - t t + t t + t t 0 1 0 4 3 4 0 5 2 5 4 5 5 0 6 1 6 3 6 4 6 6 0 7 2 7 4 7 6 7 7 1 8 3 8 4 8 5 8 6 8 7 8 0 9 2 9 4 9 5 9 6 9 7 9 8 9 } defining forms: { 2 2 2 2 3 2 2 2 2 2 2 2 2 2 2 2 2 2 2 3 2 2 2 2 2 2 2 2 2 2 3 - 6x x - 2x x x + 8x x + 5x x - 6x x + x + 2x x x - 16x x x + 7x x + 8x x - 5x x x - 2x x x + 3x x + 2x x x - x x + 6x x - 2x x x - 4x x - x x x + 8x x x - 3x x + 2x x x + 8x x x - 9x x x - 4x x + 5x x x - x x x + x x + x x - 6x x + 2x x + 6x x + x x - 2x + 4x x x - 4x x - 11x x x - 2x x x + 6x x - 4x x x + 8x x x + 2x x x - 4x x + 2x x x - 5x x x - 2x x x + 4x x x + 4x x x - 2x x x - 4x x x + x x x + 5x x + 2x x + 2x x - 6x x - 2x x + x x - 3x x + 2x , 0 2 0 1 2 1 2 0 2 1 2 2 0 2 3 1 2 3 2 3 2 3 0 2 4 1 2 4 2 4 2 3 4 2 4 0 5 0 1 5 1 5 0 2 5 1 2 5 2 5 0 3 5 1 3 5 2 3 5 3 5 0 4 5 2 4 5 4 5 0 5 1 5 2 5 3 5 4 5 5 0 1 6 1 6 0 2 6 1 2 6 2 6 0 3 6 1 3 6 2 3 6 3 6 1 4 6 2 4 6 3 4 6 0 5 6 1 5 6 2 5 6 3 5 6 4 5 6 5 6 0 6 1 6 2 6 3 6 4 6 5 6 6 3 2 2 2 2 2 2 2 2 2 2 2 2 3 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 - 6x - 8x x - 6x x + 4x x + 7x x x + 6x x + 2x x + x x + 12x x + 6x x x - 6x x - 6x x x - 9x x x - 2x x - 2x x + 10x x + 2x x - 4x - x x - 3x x x + x x x + 3x x x + 7x x x - x x x - 4x x x + x x - x x + x x - 10x x - 9x x x + 8x x x + 9x x x + 2x x + 8x x x - 8x x x - 6x x x + 6x x - 3x x x + 3x x x + x x x - 4x x + 4x x - 2x x + 4x x - x x x - 6x x - 2x x x + 4x x x + 14x x x - 8x x + 5x x x - x x x - 2x x x + 2x x x + x x + 2x x x - 8x x x + 10x x x + x x x - 2x x + 4x x + 4x x - 4x x + 2x x + 4x x , 0 0 1 0 1 0 2 0 1 2 1 2 0 2 1 2 0 3 0 1 3 1 3 0 2 3 1 2 3 2 3 0 3 1 3 2 3 3 0 4 0 1 4 0 2 4 1 2 4 0 3 4 1 3 4 2 3 4 0 4 2 4 3 4 0 5 0 1 5 0 2 5 1 2 5 2 5 0 3 5 1 3 5 2 3 5 3 5 0 4 5 2 4 5 3 4 5 0 5 2 5 3 5 0 6 0 1 6 1 6 1 2 6 0 3 6 1 3 6 3 6 0 4 6 1 4 6 2 4 6 3 4 6 4 6 0 5 6 1 5 6 3 5 6 4 5 6 5 6 0 6 1 6 3 6 4 6 5 6 3 2 2 2 2 2 2 2 2 2 2 2 2 2 3 2 2 2 2 2 2 2 2 2 3 2 2 2 2 2 2 2 6x + 4x x - 10x x - 14x x - 12x x x - 6x x - 10x x + 6x x x + 18x x x + 10x x x + 4x x - 4x x + 11x x + x x x - 2x x - 5x x x - 5x x x - 6x x x - 2x x x + 8x x x + 4x x + 6x x - x x + x x + x + 4x x - 13x x x - 14x x - 22x x x - 11x x x + 2x x x + 18x x x + 12x x x - 4x x + 10x x x - 4x x x - 7x x x + 4x x - 6x x - 15x x - 8x x + 8x x - x x - 4x + 18x x + 28x x x + 10x x - 4x x x - 30x x x - 22x x x + 4x x x + 12x x + 11x x x + 7x x x + 2x x x - 8x x x + x x + 30x x x + 27x x x - 4x x x - 24x x x + 13x x x + 12x x - 2x x - 2x x , 0 0 1 0 1 0 2 0 1 2 1 2 0 3 0 1 3 0 2 3 1 2 3 0 3 2 3 0 4 0 1 4 1 4 0 2 4 1 2 4 0 3 4 1 3 4 2 3 4 3 4 0 4 1 4 2 4 4 0 5 0 1 5 1 5 0 2 5 1 2 5 0 3 5 1 3 5 2 3 5 3 5 0 4 5 1 4 5 2 4 5 4 5 0 5 1 5 2 5 3 5 4 5 5 0 6 0 1 6 1 6 0 2 6 0 3 6 1 3 6 2 3 6 3 6 0 4 6 1 4 6 2 4 6 3 4 6 4 6 0 5 6 1 5 6 2 5 6 3 5 6 4 5 6 5 6 1 6 4 6 3 2 2 2 2 2 3 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 3 2 2 2 2 2 2 2 2 2 - 12x - 8x x + 7x x + 6x x x + 8x x + 3x x - 6x x + 2x + 8x x - 8x x x + 5x x x - 16x x x + 8x x + 8x x + 8x x - 10x x - 4x x x + 6x x x + 2x x x + 2x x + 4x x x - 4x x x + 3x x x + 4x x - 2x x + x x - 8x x - 6x x x - 4x x + 12x x x + 12x x x - 2x x + 7x x x + 4x x x - 7x x x - x x x + 5x x x + x x x + x x - 3x x - 6x x + 6x x + 5x x + x x - 2x - 3x x - 8x x x - 4x x + 7x x x + 2x x x + 3x x + 4x x x + 4x x x + x x x - x x x - 2x x x + x x x + 4x x x + 5x x x + x x x + 2x x x + 4x x - 3x x - 2x x - 2x x - x x - 2x x , 0 0 1 0 2 0 1 2 1 2 0 2 1 2 2 0 3 0 1 3 0 2 3 1 2 3 2 3 0 3 2 3 0 4 0 1 4 0 2 4 1 2 4 2 4 0 3 4 1 3 4 2 3 4 3 4 0 4 2 4 0 5 0 1 5 1 5 0 2 5 1 2 5 2 5 0 3 5 1 3 5 2 3 5 0 4 5 2 4 5 3 4 5 4 5 0 5 1 5 2 5 3 5 4 5 5 0 6 0 1 6 1 6 0 2 6 1 2 6 2 6 0 3 6 1 3 6 2 3 6 0 4 6 1 4 6 2 4 6 0 5 6 2 5 6 3 5 6 4 5 6 5 6 0 6 1 6 2 6 4 6 5 6 3 2 2 3 2 2 2 3 2 2 2 2 2 2 3 2 2 2 2 2 2 2 3 2 2 2 2 2 2 2 2 2 2 3 2 2 2 2 2 2 2 2 2 2 2 2 3 12x - 12x x - 12x x + 12x + 6x x x - 6x x - 3x x + 4x x - x + 14x x + 24x x x - 38x x - 7x x x + 14x x x - 5x x - 10x x + 38x x - 8x x - 12x + 24x x - 16x x x - 8x x + 3x x x - 2x x + 17x x x + 17x x x - 5x x x - 8x x + 15x x - 5x x + 5x x + 3x - 4x x - 16x x x + 20x x + 2x x x - 5x x x + 2x x + 15x x x - 41x x x + 6x x x + 20x x - 4x x x - 12x x x + x x x + 12x x x - x x - 5x x + 11x x - x x - 11x x - 4x x + 2x + 20x x + 8x x x - 28x x - 6x x x + 12x x x - 4x x - 6x x x + 58x x x - 14x x x - 28x x + 26x x x + 6x x x - 4x x x - 6x x x + 8x x + 8x x x - 32x x x + 5x x x + 32x x x + 7x x x - 9x x + 4x x + 20x x - 6x x - 20x x + 2x x + 12x x - 4x , 0 0 1 0 1 1 0 1 2 1 2 0 2 1 2 2 0 3 0 1 3 1 3 0 2 3 1 2 3 2 3 0 3 1 3 2 3 3 0 4 0 1 4 1 4 1 2 4 2 4 0 3 4 1 3 4 2 3 4 3 4 0 4 1 4 3 4 4 0 5 0 1 5 1 5 0 2 5 1 2 5 2 5 0 3 5 1 3 5 2 3 5 3 5 0 4 5 1 4 5 2 4 5 3 4 5 4 5 0 5 1 5 2 5 3 5 4 5 5 0 6 0 1 6 1 6 0 2 6 1 2 6 2 6 0 3 6 1 3 6 2 3 6 3 6 0 4 6 1 4 6 2 4 6 3 4 6 4 6 0 5 6 1 5 6 2 5 6 3 5 6 4 5 6 5 6 0 6 1 6 2 6 3 6 4 6 5 6 6 3 2 2 3 2 2 2 2 2 2 2 2 2 3 2 2 2 2 2 2 2 2 3 2 2 2 2 2 2 2 2 2 3 2 2 2 2 2 2 2 2 2 2 2 2 - 6x + 20x x - 2x x - 12x - x x - 2x x x + 8x x - x x - 4x x - 14x x + 2x x x + 32x x + 2x x x - 16x x x + 4x x - 28x x + 8x x + 8x - 11x x + 17x x x + 4x x - 6x x x + x x - 12x x x - 8x x x + 6x x x + 4x x - 6x x + 3x x + x x - 2x x - x + 5x x + 7x x x - 22x x - 2x x x + 10x x x - 3x x - 6x x x + 38x x x - 10x x x - 16x x + 3x x x + 9x x x - 4x x x - 8x x x + 3x x - 12x x + 3x x + 10x x + 3x x - 2x - 18x x + 18x x x + 20x x - 12x x x + 2x x - 12x x x - 36x x x + 12x x x + 16x x - 18x x x + 2x x x + 4x x x - 4x x + 28x x x - 8x x x - 24x x x - 4x x x + 8x x - 12x x - 8x x + 4x x + 8x x - 4x x - 8x x , 0 0 1 0 1 1 0 2 0 1 2 1 2 0 2 1 2 0 3 0 1 3 1 3 0 2 3 1 2 3 2 3 1 3 2 3 3 0 4 0 1 4 1 4 1 2 4 2 4 0 3 4 1 3 4 2 3 4 3 4 0 4 1 4 2 4 3 4 4 0 5 0 1 5 1 5 0 2 5 1 2 5 2 5 0 3 5 1 3 5 2 3 5 3 5 0 4 5 1 4 5 2 4 5 3 4 5 0 5 1 5 2 5 3 5 4 5 5 0 6 0 1 6 1 6 1 2 6 2 6 0 3 6 1 3 6 2 3 6 3 6 0 4 6 1 4 6 2 4 6 4 6 1 5 6 2 5 6 3 5 6 4 5 6 5 6 0 6 1 6 2 6 3 6 4 6 5 6 3 2 2 2 2 2 2 3 2 2 2 2 2 2 2 2 2 3 2 2 2 2 2 2 2 2 2 2 3 2 2 2 2 2 2 2 2 2 2 2 2 - 24x + 8x x + 16x x - 4x x - 4x x x + 8x x + 3x x - 4x x + x - 8x x - 32x x x + 8x x x - 16x x x + 5x x + 16x x + 8x x - 32x x + 4x x x + 8x x - 2x x x - 4x x x + 2x x - 4x x x - 16x x x + 6x x x + 8x x - 14x x - 2x - 10x x + 26x x x + 4x x - 4x x x + 10x x x - 2x x - 26x x x - 8x x x - 9x x x + 4x x - 11x x x + 12x x x - 3x x x - 12x x x - 3x x + 7x x + 6x x + 3x x - 6x x + 3x x + 2x - 30x x - 14x x x + 4x x + 3x x x - 10x x x + 4x x + 14x x x - 8x x x + 12x x x + 4x x - 25x x x - 8x x x + 3x x x + 8x x x - 5x x - 23x x x + 2x x x - 6x x x - 2x x x - 11x x x - 2x x - 6x x - 4x x + 4x x + 4x x - 2x x - 4x x , 0 0 1 0 1 0 2 0 1 2 1 2 0 2 1 2 2 0 3 0 1 3 0 2 3 1 2 3 2 3 0 3 2 3 0 4 0 1 4 1 4 0 2 4 1 2 4 2 4 0 3 4 1 3 4 2 3 4 3 4 0 4 4 0 5 0 1 5 1 5 0 2 5 1 2 5 2 5 0 3 5 1 3 5 2 3 5 3 5 0 4 5 1 4 5 2 4 5 3 4 5 4 5 0 5 1 5 2 5 3 5 4 5 5 0 6 0 1 6 1 6 0 2 6 1 2 6 2 6 0 3 6 1 3 6 2 3 6 3 6 0 4 6 1 4 6 2 4 6 3 4 6 4 6 0 5 6 1 5 6 2 5 6 3 5 6 4 5 6 5 6 0 6 1 6 2 6 3 6 4 6 5 6 2 2 3 2 2 2 2 3 2 2 2 2 2 2 3 2 2 2 2 2 2 2 2 3 2 2 2 2 2 2 2 2 2 2 3 2 2 2 2 2 2 2 2 2 2 2 3 8x x - 4x x - 24x - 7x x + 5x x x + 24x x + 4x x - 9x x + 3x - 10x x + 4x x x + 70x x - x x x - 47x x x + 11x x - 2x x - 66x x + 22x x + 20x - 2x x + 18x x x + 2x x - 8x x x + 2x x x + 2x x - 19x x x - 7x x x + x x x + 4x x - 3x x + 7x x - 3x x - 7x x - x + 9x x - 12x x x - 42x x + 7x x x + 28x x x - 4x x + 13x x x + 82x x x - 24x x x - 38x x + 16x x x + 4x x x + 6x x - 3x x - 27x x + 9x x + 27x x + x x - 6x + 16x x + 9x x x + 24x x - 13x x x - 22x x x + 5x x - 16x x x - 54x x x + 21x x x + 28x x + 7x x x + 4x x x - 6x x x - 6x x x - x x + 17x x x + 29x x x - 13x x x - 33x x x + 9x x x + 12x x + 14x x - 6x x + 6x x + 3x x - 2x x + 4x , 0 1 0 1 1 0 2 0 1 2 1 2 0 2 1 2 2 0 3 0 1 3 1 3 0 2 3 1 2 3 2 3 0 3 1 3 2 3 3 0 4 0 1 4 1 4 0 2 4 1 2 4 2 4 0 3 4 1 3 4 2 3 4 3 4 0 4 1 4 2 4 3 4 4 0 5 0 1 5 1 5 0 2 5 1 2 5 2 5 0 3 5 1 3 5 2 3 5 3 5 0 4 5 2 4 5 4 5 0 5 1 5 2 5 3 5 4 5 5 0 6 0 1 6 1 6 0 2 6 1 2 6 2 6 0 3 6 1 3 6 2 3 6 3 6 0 4 6 1 4 6 2 4 6 3 4 6 4 6 0 5 6 1 5 6 2 5 6 3 5 6 4 5 6 5 6 0 6 1 6 3 6 4 6 5 6 6 2 3 2 2 2 2 2 2 2 2 2 2 3 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 - 12x x + 12x + 12x x - 4x x x - 14x x + 8x x + 6x x + 12x x - 4x x x - 32x x + 10x x x + 26x x x - 6x x + 4x x + 28x x - 12x x - 8x - 10x x x - 2x x + 10x x x + 3x x x + x x + 10x x x + 2x x x - 2x x + 2x x + 2x x + 12x x + 4x x x - 17x x x + 7x x + 2x x x - 24x x x + 18x x x + 12x x - x x x + 3x x x + 2x x x + 3x x - 5x x - 4x x - 6x x - 12x x x - 6x x + 9x x x + 10x x x + 10x x x + 10x x x - 6x x x - 4x x - 5x x x - 5x x x + 4x x x + 4x x x - x x - x x x + x x x + 8x x x + 2x x x - x x x + 2x x - 6x x - 6x x + 4x x - 2x x - 4x x , 0 1 1 0 2 0 1 2 1 2 0 2 1 2 0 3 0 1 3 1 3 0 2 3 1 2 3 2 3 0 3 1 3 2 3 3 0 1 4 1 4 0 2 4 1 2 4 2 4 0 3 4 1 3 4 1 4 2 4 3 4 1 5 0 2 5 1 2 5 2 5 0 3 5 1 3 5 2 3 5 3 5 1 4 5 2 4 5 3 4 5 1 5 2 5 3 5 0 6 0 1 6 1 6 0 2 6 1 2 6 0 3 6 1 3 6 2 3 6 3 6 0 4 6 1 4 6 2 4 6 3 4 6 4 6 0 5 6 1 5 6 2 5 6 3 5 6 4 5 6 5 6 0 6 1 6 3 6 4 6 5 6 2 2 3 2 2 2 2 3 2 2 2 2 2 2 3 2 2 2 2 2 2 2 2 3 2 2 2 2 2 2 2 2 2 2 3 2 2 2 2 2 2 2 2 2 2 2 2 8x x - 12x x + 24x - 11x x + 17x x x - 24x x - 10x x + 11x x - 3x - 6x x + 28x x x - 70x x - 21x x x + 47x x x - 13x x - 14x x + 66x x - 22x x - 20x + 2x x - 2x x x - 10x x - 11x x x + 8x x x - 5x x + 3x x x + 23x x x - 11x x x - 12x x + 3x x - 3x x - 2x x + 3x x + x - 11x x + 14x x x + 34x x - 6x x x - 16x x x + 3x x - 15x x x - 66x x x + 12x x x + 30x x - 19x x x + 2x x x - 5x x x - 2x x x - 7x x + 6x x + 21x x - 3x x - 21x x + x x + 5x - 8x x + 7x x x - 32x x - 13x x x + 28x x x - 9x x + 70x x x - 27x x x - 36x x + x x x + 4x x x - 7x x x - 2x x x + 3x x - 25x x x - 23x x x + 4x x x + 27x x x - 14x x x - 9x x - 2x x + 10x x - 6x x - 10x x + 3x x - 2x x 0 1 0 1 1 0 2 0 1 2 1 2 0 2 1 2 2 0 3 0 1 3 1 3 0 2 3 1 2 3 2 3 0 3 1 3 2 3 3 0 4 0 1 4 1 4 0 2 4 1 2 4 2 4 0 3 4 1 3 4 2 3 4 3 4 0 4 1 4 2 4 3 4 4 0 5 0 1 5 1 5 0 2 5 1 2 5 2 5 0 3 5 1 3 5 2 3 5 3 5 0 4 5 1 4 5 2 4 5 3 4 5 4 5 0 5 1 5 2 5 3 5 4 5 5 0 6 0 1 6 1 6 0 2 6 1 2 6 2 6 1 3 6 2 3 6 3 6 0 4 6 1 4 6 2 4 6 3 4 6 4 6 0 5 6 1 5 6 2 5 6 3 5 6 4 5 6 5 6 0 6 1 6 2 6 3 6 4 6 5 6 } o1 : RationalMap (cubic birational map from PP^6 to 6-dimensional subvariety of PP^9) i2 : time describe oo -- used 0.0158778 seconds o2 = rational map defined by forms of degree 3 source variety: PP^6 target variety: complete intersection of type (2,2,2) in PP^9 dominance: true birationality: true projective degrees: {1, 3, 9, 17, 21, 16, 8} number of minimal representatives: 1 dimension base locus: 3 degree base locus: 10 coefficient ring: QQ