cyjax.jacobian_embed#
- cyjax.jacobian_embed(grad_def, dependent)#
Jacobian of the defining embedding into ambient projective space.
Let \(X\) be a variety embedded in the ambient projective space \(\mathbb{P}^{d+1}\) as the zero-locus of a homogeneous polynomial \(Q(z)=0\), where \(z\) are ambient affine coordinates. Denote by \(x\) local coordinates on the variety. Explicitly, the embedding would be given by \(z(x)\). However, we can compute the Jacobian \(\partial z / \partial x\) needed for pullbacks without explicitly deriving this embedding.
Here, let \(x\) be the local coordinates obtained by omitting \(z[\mathrm{dep}]\). The output of this function is the matrix \((dz_i/dx_j)_{ij}\) computed by using
\[\frac{dz_{\mathrm{dep}}}{dx_j} = - \frac{dQ}{dz_j} \left( \frac{Q}{dz_{\mathrm{dep}}} \right)^{-1} \,.\]- Parameters:
grad_def (
Union[Array,ndarray,bool_,number]) – Derivative of defining equation with respect to affine coordinates; Array of lengthd.dependent (
int) – Index of dependent coordinate in affine coordinate vector.
- Return type:
Union[Array,ndarray,bool_,number]- Returns:
\((d-1 \times d)\) array. The Jacobian for the defining embedding.