Blowup of a complex manifold at a point

The blowup replaces a point of a complex $n$-manifold by the projective space of complex tangent directions ${\mathbb{CP}}^{n-1}$. A biholomorphism carrying one center to another lifts to a biholomorphism of their blowups.

It is the point-center case of blowup of an algebraic variety, also interpreted analytically.

The holomorphic map $[Z_0:Z_1:Z_2]\mapsto [Z_1:Z_2]$ on ${\mathbb{CP}}^2\setminus \{[1:0:0]\}$ records the line through the missing point and the input point. Its graph closure is the incidence surface $Z_{1}b=Z_{2}a$ in ${\mathbb{CP}}^2\times {\mathbb{CP}}^1$, the blowup of a complex manifold at a point. Projection to $[a:b]$ extends the pencil holomorphically. On the exceptional divisor it is the identity map on the projective tangent directions.

For the blowup of a complex manifold at a point in complex dimension $n$, the differential gives a section of $K_{\tilde{M}}\otimes ({\varphi }^{\ast }{K}_M{)}^{-1}$ whose divisor on a complex manifold is $(n-1)E$. Indeed, a local chart has $x_i=u$ and $x_j=u{t}_j$ for $j\ne i$, so its Jacobian determinant is $\pm u^{n-1}$. The holomorphic line bundle associated to a divisor then gives the formula for the canonical bundle.

Let $Y$ be a smooth complex analytic hypersurface through the blowup centre with $[-Y]\cong K_X$. Its multiplicity at that point is one, so ${\sigma }^{\ast }Y=\tilde{Y}+E$. Combining this with the canonical bundle formula for a point blowup gives $[-\tilde{Y}]\cong K_{\tilde{X}}\otimes [E{]}^{\otimes (2-n)}$. The nontrivial powers of the exceptional line bundle of a point blowup prove the equivalence for $n\ge 2$. On a compact complex curve the premise cannot hold for an irreducible hypersurface: it is one point, so its negative divisor has degree $-1$, whereas the canonical bundle has even degree $2g-2$.

The exceptional divisor of the blowup of a complex manifold at a point $p$ is the fiber over $p$. In complex dimension $n$ it is naturally the projective space $\mathbb{P}(T_{p}X)\cong {\mathbb{CP}}^{n-1}$ of tangent directions at the center.

For the blowup of a complex manifold at a point in dimension $n\ge 2$, the exceptional divisor is ${\mathbb{CP}}^{n-1}$ and $[E]{|}_E={\mathcal{O}}_E(-1)$. The local blowup is the total space of the tautological line bundle, so its zero section has that normal bundle. Restricting a power to a projective line inside $E$ gives degree $-r$; a trivial bundle has degree zero. This proves that no nonzero power of $[E]$ is trivial, without assuming global projectivity of the original manifold.

The strict transform of a subvariety $Y$ under a blowup $\sigma :\tilde{X}\to X$ is the closure of ${\sigma }^{-1}(Y\setminus Z)$, where $Z$ is the center. Its total transform also contains the exceptional divisor with the multiplicity with which $Y$ passes through $Z$.

A pencil generated by cubic forms $F_0,F_1$ is the one-parameter family $s{F}_0+t{F}_1=0$ indexed by $[s:t]\in {\mathbb{CP}}^1$. A general pair has nine transverse base points.

Blowing up the nine base points of a general pencil of plane cubics produces the rational elliptic surface

Projection to $[s:t]$ gives an elliptic fibration with connected cubic fibers. Each of the nine exceptional divisors maps isomorphically to ${\mathbb{CP}}^1$ and is therefore a holomorphic section.

Writing $H$ for the pullback of a line and $E_1,\dots ,E_9$ for the exceptional classes, a fiber has class $F=3H-{\sum }_i{E}_i$, while the blowup formula gives

If a smooth connected complex curve $C\subset E(1)$ satisfies $C^2=k$ and has degree $d=C\cdot F$ over the base, the adjunction formula gives