Vanishing points of an ideal I am trying to find vanishing points of a homogeneous ideal.
I tried to use $\mathtt{COCOA}$ and $\mathtt{Singular}$ to find but it seems there is not any function.
Is there anyone who knows to do it? what about $\mathtt{Sagemath}$?, Has anyone tried it?
$$I=(wxy + x^2y + xy^2 + xyz, w^2y + wxy + wy^2 + wyz, w^2x + wx^2 + wxy + wxz, wxy) $$
 A: I calculated the primary decomposition with Macaulay2. It gives 6 linear subspaces of dimension 2 (projectively of dimension 1, that is straight lines):
i3 : R=QQ[x,y,z,w]

o3 = R

o3 : PolynomialRing

i4 : idI1 = ideal(w*x*y+x^2*y+x*y^2+x*y*z,w^2*y+w*x*y+w*y^2+w*y*z,w^2*x+w*x^2+w*x*y+w*x*z,w*x*y)

             2       2                           2               2   2                       2
o4 = ideal (x y + x*y  + x*y*z + x*y*w, x*y*w + y w + y*z*w + y*w , x w + x*y*w + x*z*w + x*w , x*y*w)

o4 : Ideal of R

i5 : primaryDecomposition idI1

o5 = {ideal (y + z + w, x), ideal (y, x + z + w), ideal (w, y), ideal (y, x), ideal (w, x + y + z),
     ---------------------------------------------------------------------------------------------------
     ideal (w, x)}

o5 : List

i6 : pdlis = oo

o6 = {ideal (y + z + w, x), ideal (y, x + z + w), ideal (w, y), ideal (y, x), ideal (w, x + y + z),
     ---------------------------------------------------------------------------------------------------
     ideal (w, x)}

o6 : List

i7 : apply(pdlis,zz->dim(R/zz))

o7 = {2, 2, 2, 2, 2, 2}

o7 : List

