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})
o1 = S
o1 : FreeAlgebraQuotient
|
i2 : L = lineSchemeFourDim(S,M);
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})
o4 = S
o4 : FreeAlgebraQuotient
|
i5 : L = lineSchemeFourDim(S,M);
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);
1 9
o9 : Matrix R <--- R
|
i10 : S = R/I o10 = S o10 : FreeAlgebraQuotient |
i11 : L = lineSchemeFourDim(S,M);
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 |
+-----------------------------------------------------------------+
|
The object lineSchemeFourDim is a method function.