# projectiveVariety(MultidimensionalMatrix) -- the multi-projective variety defined by a multi-dimensional matrix

## Synopsis

• Function: projectiveVariety
• Usage:
projectiveVariety A
• Inputs:
• A, , an $n$-dimensional matrix of shape $(k_1+1)\times\cdots\times (k_n+1)$
• Optional inputs:
• MinimalGenerators => ..., default value true, whether to trim the ideal (intended for internal use only)
• Saturate => ..., default value true, whether to compute the multi-saturation of the ideal (intended for internal use only)
• Outputs:
• , the corresponding hypersurface of multi-degree $(1,\ldots,1)$ on the product of projective spaces $\mathbb{P}^{k_1}\times\cdots\times\mathbb{P}^{k_n}$

## Description

In particular, we have det A != 0 if and only if dim singularLocus(projectiveVariety A) == -1.

 i1 : K = ZZ/33331; i2 : A = randomMultidimensionalMatrix({2,2,3},CoefficientRing=>K) o2 = {{{-456, -6898, 3783}, {-6635, 8570, 16659}}, {{8444, -9579, 5071}, ------------------------------------------------------------------------ {-7531, -10808, 5864}}} o2 : 3-dimensional matrix of shape 2 x 2 x 3 over K i3 : det A o3 = -12772 o3 : K i4 : X = projectiveVariety A; o4 : ProjectiveVariety, hypersurface in PP^1 x PP^1 x PP^2 i5 : dim singularLocus X o5 = -1 i6 : B = multidimensionalMatrix {{{9492_K, 13628, -9292}, {9311, -5201, -16439}}, {{11828, -16301, 8162}, {15287, 8345, -2094}}} o6 = {{{9492, 13628, -9292}, {9311, -5201, -16439}}, {{11828, -16301, 8162}, ------------------------------------------------------------------------ {15287, 8345, -2094}}} o6 : 3-dimensional matrix of shape 2 x 2 x 3 over K i7 : det B o7 = 0 o7 : K i8 : Y = projectiveVariety B; o8 : ProjectiveVariety, hypersurface in PP^1 x PP^1 x PP^2 i9 : dim singularLocus Y o9 = 0