Linearspan

[Pietro Beri]

Hi,

I need a way to find the linear span of some random points on a given variety.

I've seen some program using linearSpan {point X, point X}, but when I try it I obtain "error: expected a list of embedded projective varieties". How can I make my points embedded projective varieties?

Also, can you find the linear span of a whole variety? For example if X is a zero-dimensional variety, for which "point X" doesn't work (error: expected a positive dimensional scheme)?

Thanks!

Pietro

[jche...@gmail.com]

If you know the points explicitly (e.g. via coordinates), and want the defining ideal of their linear span, then this can be done just via linear algebra:

R = QQ[x_0..x_5]

pts = apply(4, i -> random(R^1,R^6)) -- 4 random points in P^5

L = ideal(vars R * gens ker transpose matrix{pts/transpose})

all(L_*, f -> all(pts, p -> sub(f, p) == 0))

Here L is the ideal of the linear span, which is a 3-plane in P^5. On the other hand, if you have a variety X (given by its ideal), then the linear span of X is defined by the degree-1 part of the ideal of X. Continuing the example above:

ptIdeals = apply(pts, p -> trim minors(2, vars R || p))

I = intersect ptIdeals;

linSpan = super basis(1, I)

L == ideal linSpan

Justin

[Pietro Beri]

I never realized that the linear span is just the degree-1 part of the ideal of X. Thank you!

But if I try to do, for example  in the example below

needsPackage "K3Surfaces"

idGr = Grassmannian(1,6,CoefficientRing => ZZ/65521, Variable => T)  -- Grassmannian of 2 vect. planes in CC^7, seen as a subvariety of PP^20 
Gr = variety idGr
R = ring ambient Gr

ell=(point idGr) * (point idGr) -- Using the package "K3Surfaces" I can choose a random point on the Grassmannian using "point" 

linSpan = super basis(1, ell) -- ell is the union of two points, linSpan should be a line

dim variety ideal linSpan -- but we check that the dimension of linSpan is 20 

 

linSpan isn't a line, but the whole space. Do you see what I'm getting wrong?

Thank you,

Pietro

[Pietro Beri]

Sorry, of course ell has to be the intersection of the ideals, not the product! Just a stupid mistake, now everything is clear.

Thank you again,

Pietro