# plucker -- get the Plücker coordinates of a linear subspace

## Synopsis

• Usage:
plucker L
plucker p
• Inputs:
• L, an ideal, the ideal of a $k$-dimensional linear subspace $V(L)\subset\mathbb{P}^n$
• p, an ideal, the ideal of the corresponding point of V(L) in the Grassmannian $\mathbb{G}(k,\mathbb{P}^n)$ (embedded via the Plücker embedding)
• Optional inputs:
• AffineChartGrass => ..., default value true, use an affine chart on the Grassmannian
• Variable => ..., default value null, specify a name for a variable
• Outputs:
• an ideal, p if the input is L, and L if the input is p

## Description

 i1 : P4 = Grass(0,4,ZZ/33331,Variable=>x); G'1'4 = Grass(1,4,ZZ/33331,Variable=>x); i3 : L = trim ideal apply(3,i->random(1,P4)) -- a line in P^4 o3 = ideal (x + 8480x - 11656x , x - 6727x + 14853x , x + 15777x - 2 3 4 1 3 4 0 3 ------------------------------------------------------------------------ 664x ) 4 o3 : Ideal of P4 i4 : time p = plucker L -- used 0.0139507 seconds o4 = ideal (x + 8480x , x - 6727x , x + 15777x , x + 2,4 3,4 1,4 3,4 0,4 3,4 2,3 ------------------------------------------------------------------------ 11656x , x - 14853x , x + 664x , x + 13522x , x + 3,4 1,3 3,4 0,3 3,4 1,2 3,4 0,2 ------------------------------------------------------------------------ 11804x , x + 14854x ) 3,4 0,1 3,4 o4 : Ideal of G'1'4 i5 : time L' = plucker p -- used 0.124926 seconds o5 = ideal (x + 8480x - 11656x , x - 6727x + 14853x , x + 15777x - 2 3 4 1 3 4 0 3 ------------------------------------------------------------------------ 664x ) 4 o5 : Ideal of P4 i6 : assert(L' == L)

More generally, if the input is the ideal of a subvariety $Y\subset\mathbb{G}(k,\mathbb{P}^n)$, then the method returns the ideal of the variety $W\subset\mathbb{P}^n$ swept out by the linear spaces corresponding to points of $Y$. As an example, we now compute a surface scroll $W\subset\mathbb{P}^4$ over an elliptic curve $Y\subset\mathbb{G}(1,\mathbb{P}^4)$.

 i7 : Y = ideal apply(5,i->random(1,G'1'4)); -- an elliptic curve o7 : Ideal of G'1'4 i8 : time W = plucker Y; -- surface swept out by the lines of Y -- used 0.141346 seconds o8 : Ideal of P4 i9 : (codim W,degree W) o9 = (2, 5) o9 : Sequence

In this example, we can recover the subvariety $Y\subset\mathbb{G}(k,\mathbb{P}^n)$ by computing the Fano variety of $k$-planes contained in $W$.

 i10 : time Y' = plucker(W,1); -- variety of lines contained in W -- used 0.695275 seconds o10 : Ideal of G'1'4 i11 : assert(Y' == Y)

Warning: Notice that, by default, the computation is done on a randomly chosen affine chart on the Grassmannian. To change this behavior, you can use the AffineChartGrass option.

## Ways to use plucker :

• "plucker(Ideal)"
• "plucker(Ideal,ZZ)"

## For the programmer

The object plucker is .