This function returns the kernel of the matrix describing the morphism
$\Phi: S^{md-2}V \otimes O_{\PP^d} \to S^{(m-1)d}V \otimes O_{\PP^d)}(1)$
given by the projection
$S^dV \otimes S^{(m-1)d}V \to S^{md-2}V$
of the irreducible $SL(2)$-subrepresentation of highest weight $md-2$, where $\PP^d = \PP(S^dV)$ as $V=<v_0,v_1>$.
In the paper A construction of equivariant bundles on the space of symmetric forms, it is proved that the matrix $\Phi$ has constant co-rank 1, so that the kernel $W = ker \Phi$ turns out to be a vector bundle, and the entries of the matrix $\Phi$ are explicitely describred.
i1 : d = 3, m = 2 o1 = (3, 2) o1 : Sequence |
i2 : W = sl2EquivariantVectorBundle(d,m) o2 = cokernel {4} | 0 x_3 0 x_2 | {4} | x_1 0 x_0 0 | {4} | -x_2 x_0 0 0 | {4} | x_3 -3x_1 -3x_2 x_0 | {4} | 0 0 x_3 -x_1 | 5 o2 : coherent sheaf on Proj(QQ[x ..x ]), quotient of OO (-4) 0 3 Proj(QQ[x ..x ]) 0 3 |
By default, slEquivariantVectorBundle defines the vector bundle over a projective space whose coordinate ring has rational coefficients. The optional argument CoefficientRing allows one to change the coefficient ring.
i3 : d = 3, m = 2 o3 = (3, 2) o3 : Sequence |
i4 : W = sl2EquivariantVectorBundle(d,m,CoefficientRing=>ZZ/10007) o4 = cokernel {4} | 0 3336x_3 0 x_2 | {4} | x_1 0 x_0 0 | {4} | -x_2 x_0 0 0 | {4} | 3336x_3 -x_1 -x_2 x_0 | {4} | 0 0 3336x_3 -x_1 | ZZ 5 o4 : coherent sheaf on Proj(-----[x ..x ]), quotient of OO (-4) 10007 0 3 ZZ Proj(-----[x ..x ]) 10007 0 3 |
If the first argument is a polynomial ring R, then d = numgens R-1.
i5 : R = QQ[y_0..y_3]; |
i6 : m = 2 o6 = 2 |
i7 : W = sl2EquivariantVectorBundle(R,m) o7 = cokernel {4} | 0 y_3 0 y_2 | {4} | y_1 0 y_0 0 | {4} | -y_2 y_0 0 0 | {4} | y_3 -3y_1 -3y_2 y_0 | {4} | 0 0 y_3 -y_1 | 5 o7 : coherent sheaf on Proj R, quotient of OO (-4) Proj R |
The object sl2EquivariantVectorBundle is a method function with options.