= Differential of the projective quotient map
{title2=$d\pi_X(v)=\sum_{j=1}^n(v_j/X_0-X_jv_0/X_0^2)\partial_{x_j}$}
In the chart $x_j=X_j/X_0$ of <Complex projective space>, differentiate the quotient coordinates to obtain this formula for $\pi:\mathbb C^{n+1}\setminus\{0\}\to\mathbf P^n$. The kernel is the radial line $\mathbb CX$. A degree-one homogeneous vector field on the punctured cone descends to a <holomorphic vector field>, since its scaling cancels the inverse scaling of $d\pi$. This gives the fiberwise construction of the <Euler sequence on complex projective space>.
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