(Blowup)
The blowup of \(\AA^n\) at the origin is \[ \Bl_0 \AA^n \da \ts{ (x, \ell) \in \AA^n \times \PP^{n-1} \st x \in \ell } , \] with \(\pi\) the first projection. For \(X\subseteq\AA^n\) a closed subvariety through \(p=0\), \(\Bl_p X\) is the closure of \(\inverseof{\pi}(X \sm \ts{p})\) inside \(\Bl_0 \AA^n\); for a variety \(X\) and \(p\in X\), it is defined through an affine neighbourhood of \(p\) embedded in some \(\AA^n\). The exceptional divisor is \(E \da \inverseof{\pi}(p)\cap\Bl_pX\), and the proper transform of a subvariety \(C \subseteq X\) is the closure of \(\inverseof{\pi}(C \sm \ts{p})\).
\(\pi\) is an isomorphism away from \(E\), hence birational, and \(\Bl_p X\) is again a variety, projective over \(X\). The exceptional divisor \(E\subseteq\PP^{n-1}\) is the projectivization of the tangent cone of \(X\) at \(p\). If \(p\) is a smooth point of \(X\) and \(\dim X=d\), then \(E\cong\PP^{d-1}\) is the space of tangent directions at \(p\); for \(X=\AA^n\), \(E\cong\PP^{n-1}\).
Since \(E\) parametrizes tangent directions at \(p\), the proper transforms of two smooth branches through \(p\) with distinct tangent lines are disjoint over \(p\). The nodal cubic \(y^2 = x^2(x+1)\) has two branches at the origin with tangent lines \(y=\pm x\), and its proper transform after one blowup is smooth. The cuspidal cubic \(y^2 = x^3\) has proper transform \(y_1^2 = x\) in the chart \(y = xy_1\), which is smooth and tangent to \(E\); two further blowups make the total transform a divisor with normal crossings.
The scheme-theoretic definition \(\Bl_Z X = \Proj \bigoplus_{d \geq 0} \mci_Z^d\) also applies to non-reduced centers and makes the exceptional divisor Cartier by construction.
Backlinks
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