# random(MultiprojectiveVariety) -- apply a random automorphism of the ambient multi-projective space

## Synopsis

• Function: random
• Usage:
random X
• Inputs:
• X, , a subvariety of $\mathbb{P}^{k_1}\times\cdots\times\mathbb{P}^{k_n}$
• Optional inputs:
• Outputs:
• , the image of $X$ under the action of a random element of $\mathrm{Aut}(\mathbb{P}^{k_1})\times\cdots\times\mathrm{Aut}(\mathbb{P}^{k_n})$

## Description

 i1 : K = ZZ/65521; i2 : X = PP_K^({1,1},{2,3}); o2 : ProjectiveVariety, surface in PP^2 x PP^3 i3 : ideal X 2 2 2 o3 = ideal (x1 - x1 x1 , x1 x1 - x1 x1 , x1 - x1 x1 , x0 - x0 x0 ) 2 1 3 1 2 0 3 1 0 2 1 0 2 o3 : Ideal of K[x0 ..x0 , x1 ..x1 ] 0 2 0 3 i4 : Y = random X; o4 : ProjectiveVariety, surface in PP^2 x PP^3 i5 : ideal Y 2 2 o5 = ideal (x1 + 20603x1 x1 - 6053x1 x1 + 6350x1 - 2214x1 x1 - 1 0 2 1 2 2 0 3 ------------------------------------------------------------------------ 2 25885x1 x1 + 10120x1 x1 - 10487x1 , x1 x1 + 32257x1 x1 + 14135x1 x1 1 3 2 3 3 0 1 0 2 1 2 ------------------------------------------------------------------------ 2 2 2 + 9059x1 - 14139x1 x1 - 18793x1 x1 + 10289x1 x1 + 21002x1 , x1 - 2 0 3 1 3 2 3 3 0 ------------------------------------------------------------------------ 2 14850x1 x1 - 1416x1 x1 - 4590x1 - 29865x1 x1 + 2365x1 x1 + 0 2 1 2 2 0 3 1 3 ------------------------------------------------------------------------ 2 2 2 2855x1 x1 - 16986x1 , x0 - 9730x0 x0 - 2767x0 - 29185x0 x0 - 2 3 3 0 0 1 1 0 2 ------------------------------------------------------------------------ 2 15865x0 x0 + 11692x0 ) 1 2 2 o5 : Ideal of K[x0 ..x0 , x1 ..x1 ] 0 2 0 3