# lineSchemeFourDim -- Compute the line scheme of a four-dimensional AS regular algebra

## Synopsis

• Usage:
L = lineSchemeFourDim B
• Inputs:
• Outputs:

## Description

This method computes the scheme that parametrizes the set of line modules over an AS-regular algebra B due to Shelton and Vancliff. More precisely, it computes the image of this scheme under the Plücker embedding.

As a first example, we see that the line scheme of the commutative polynomial ring is just the image of the Grassmannian Gr(4,2) in $\mathbb{P}^5$:

 i1 : S = skewPolynomialRing(QQ,1_QQ,{x_1,x_2,x_3,x_4}) Using numthreads = 0 o1 = S o1 : FreeAlgebraQuotient i2 : L = lineSchemeFourDim(S,M); Using numthreads = 0 Using numthreads = 0 o2 : Ideal of QQ[M ..M , M ..M , M ] 1,2 1,4 2,3 2,4 3,4 i3 : netList minimalPrimes L +-------------------------------------+ o3 = |ideal(M M - M M + M M )| | 1,4 2,3 1,3 2,4 1,2 3,4 | +-------------------------------------+

Next, we compute the line scheme of a (-1)-skew polynomial ring. We see that it is a union of four planes and three quadric surfaces.

 i4 : S = skewPolynomialRing(QQ,(-1)_QQ,{x_1,x_2,x_3,x_4}) Using numthreads = 0 o4 = S o4 : FreeAlgebraQuotient i5 : L = lineSchemeFourDim(S,M); Using numthreads = 0 Using numthreads = 0 o5 : Ideal of QQ[M ..M , M ..M , M ] 1,2 1,4 2,3 2,4 3,4 i6 : netList minimalPrimes L +---------------------------------------+ o6 = |ideal (M , M , M ) | | 3,4 2,4 1,4 | +---------------------------------------+ |ideal (M , M , M ) | | 3,4 2,3 1,3 | +---------------------------------------+ |ideal (M , M , M M - M M )| | 3,4 1,2 1,4 2,3 1,3 2,4 | +---------------------------------------+ |ideal (M , M , M ) | | 2,4 2,3 1,2 | +---------------------------------------+ |ideal (M , M , M M + M M )| | 2,4 1,3 1,4 2,3 1,2 3,4 | +---------------------------------------+ |ideal (M , M , M M - M M )| | 2,3 1,4 1,3 2,4 1,2 3,4 | +---------------------------------------+ |ideal (M , M , M ) | | 1,4 1,3 1,2 | +---------------------------------------+

Finally, we consider the following AS-regular algebra of dimension four. Its line scheme is dimension one and degree 20, and is a union of 10 conics.

 i7 : R = QQ <|x_4,x_1,x_2,x_3|> o7 = R o7 : FreeAlgebra i8 : I = ideal {x_3^2 - x_1*x_2, x_4^2 - x_2*x_1, x_1*x_3 - x_2*x_4, x_3*x_1 - x_2*x_3, x_1*x_4 - x_4*x_2, x_4*x_1 - x_3*x_2} 2 2 o8 = ideal (- x x + x , x - x x , x x - x x , - x x + x x , - x x + 1 2 3 4 2 1 1 3 2 4 2 3 3 1 4 2 ------------------------------------------------------------------------ x x , x x - x x ) 1 4 4 1 3 2 o8 : Ideal of R i9 : Igb = NCGB(I, 10); Using numthreads = 0 1 9 o9 : Matrix R <--- R i10 : S = R/I o10 = S o10 : FreeAlgebraQuotient i11 : L = lineSchemeFourDim(S,M); Using numthreads = 0 Using numthreads = 0 o11 : Ideal of QQ[M ..M , M ..M , M ] 1,2 1,4 2,3 2,4 3,4 i12 : netList minimalPrimes L +-----------------------------------------------------------------+ | 2 | o12 = |ideal (M , M - M , M , M - M M ) | | 3,4 1,4 2,3 1,2 2,3 1,3 2,4 | +-----------------------------------------------------------------+ | 2 | |ideal (M , M + M , M , M + M M ) | | 3,4 1,4 2,3 1,2 2,3 1,3 2,4 | +-----------------------------------------------------------------+ | 2 | |ideal (M , M , M - M , M M + M ) | | 2,4 1,3 1,2 3,4 1,4 2,3 3,4 | +-----------------------------------------------------------------+ | 2 | |ideal (M , M , M + M , M M - M ) | | 2,4 1,3 1,2 3,4 1,4 2,3 3,4 | +-----------------------------------------------------------------+ | 2 | |ideal (M , M , M - M , M - M M ) | | 2,3 1,4 1,3 2,4 2,4 1,2 3,4 | +-----------------------------------------------------------------+ | 2 | |ideal (M , M , M + M , M + M M ) | | 2,3 1,4 1,3 2,4 2,4 1,2 3,4 | +-----------------------------------------------------------------+ | 2 2 2 | |ideal (M - M , M - M , M + M , M - M - M )| | 1,4 2,3 1,3 2,4 1,2 3,4 2,3 2,4 3,4 | +-----------------------------------------------------------------+ | 2 2 2 | |ideal (M - M , M + M , M - M , M + M + M )| | 1,4 2,3 1,3 2,4 1,2 3,4 2,3 2,4 3,4 | +-----------------------------------------------------------------+ | 2 2 2 | |ideal (M + M , M - M , M - M , M + M - M )| | 1,4 2,3 1,3 2,4 1,2 3,4 2,3 2,4 3,4 | +-----------------------------------------------------------------+ | 2 2 2 | |ideal (M + M , M + M , M + M , M - M + M )| | 1,4 2,3 1,3 2,4 1,2 3,4 2,3 2,4 3,4 | +-----------------------------------------------------------------+

## Ways to use lineSchemeFourDim :

• "lineSchemeFourDim(FreeAlgebraQuotient,Symbol)"

## For the programmer

The object lineSchemeFourDim is .