# random(List,MultiprojectiveVariety) -- get a random hypersurface of given multi-degree containing a multi-projective variety

## Synopsis

• Function: random
• Usage:
random(d,X)
• Inputs:
• d, a list, a list of $n$ nonnegative integers
• X, , a subvariety of $\mathbb{P}^{k_1}\times\cdots\times\mathbb{P}^{k_n}$
• Optional inputs:
• Outputs:
• , a random hypersurface in $\mathbb{P}^{k_1}\times\cdots\times\mathbb{P}^{k_n}$ of multi-degree $d$ containing $X$

## Description

More generally, if d is a list of multi-degrees, then the output is the intersection of the hypersurfaces random(d_i,X).

 i1 : X = PP_(ZZ/65521)^(1,3); -- twisted cubic curve o1 : ProjectiveVariety, curve in PP^3 i2 : random({2},X); o2 : ProjectiveVariety, surface in PP^3 i3 : ideal oo 2 2 o3 = ideal(t - t t - 68t t - 20741t + 68t t + 20741t t ) 1 0 2 1 2 2 0 3 1 3 ZZ o3 : Ideal of -----[t ..t ] 65521 0 3 i4 : random({{2},{2}},X); o4 : ProjectiveVariety, curve in PP^3 i5 : ideal oo 2 2 2 o5 = ideal (t t + 21021t - t t - 21021t t , t - t t + 29086t - 1 2 2 0 3 1 3 1 0 2 2 ------------------------------------------------------------------------ 29086t t ) 1 3 ZZ o5 : Ideal of -----[t ..t ] 65521 0 3 i6 : X = X^2; o6 : ProjectiveVariety, X x X i7 : random({1,2},X); o7 : ProjectiveVariety, hypersurface in PP^3 x PP^3 i8 : ideal oo 2 2 2 2 o8 = ideal(x0 x1 + 29294x0 x1 + 16820x0 x1 + 1316x0 x1 - x0 x1 x1 - 0 1 1 1 2 1 3 1 0 0 2 ------------------------------------------------------------------------ 29294x0 x1 x1 - 16820x0 x1 x1 - 1316x0 x1 x1 - 3472x0 x1 x1 - 1 0 2 2 0 2 3 0 2 0 1 2 ------------------------------------------------------------------------ 2 29829x0 x1 x1 + 2976x0 x1 x1 - 23867x0 x1 x1 - 32075x0 x1 + 1 1 2 2 1 2 3 1 2 0 2 ------------------------------------------------------------------------ 2 2 2 17896x0 x1 - 7591x0 x1 - 19385x0 x1 + 3472x0 x1 x1 + 29829x0 x1 x1 1 2 2 2 3 2 0 0 3 1 0 3 ------------------------------------------------------------------------ - 2976x0 x1 x1 + 23867x0 x1 x1 + 32075x0 x1 x1 - 17896x0 x1 x1 + 2 0 3 3 0 3 0 1 3 1 1 3 ------------------------------------------------------------------------ 7591x0 x1 x1 + 19385x0 x1 x1 ) 2 1 3 3 1 3 ZZ o8 : Ideal of -----[x0 ..x0 , x1 ..x1 ] 65521 0 3 0 3 i9 : random({{1,2},{1,2},{2,0}},X); o9 : ProjectiveVariety, threefold in PP^3 x PP^3 i10 : degrees oo o10 = {({2, 0}, 1), ({1, 2}, 2)} o10 : List