Pushforward and vanishing for point blowups on a surface

The calculation is local on the base, by the sheafification description of higher direct images and the locality of the blowup. In the regular-surface alternative, the local dimension at a closed point is positive: if it were zero, the regular local domain would be a field and the closed point would also be the generic point of its ambient irreducible component, forcing that component to be a point, contrary to the pure dimension two convention for a surface. If its dimension is one, its maximal ideal has a regular generator by [F8]. Lift this generator to an affine Noetherian neighborhood; shrinking kills the finite quotient of the point ideal by that generator and the finite kernel of multiplication by it, just as below. The point center is then effective Cartier on that neighborhood, so its blowup is the identity there by [F9], and it is also the identity off the center by [F5]. Thus the asserted pushforward and vanishing follow in this case. The more general alternative in the statement already assumes local dimension two at the center. It remains to treat local dimension two. Near $p$, choose an affine Noetherian neighborhood $\mathrm{Spec}R$. Lift regular parameters of $R_{{\mathfrak{m}}_p}$ to functions $x,y$ after inverting denominators not vanishing at $p$. The ideal of $p$ is finite; its quotient by $(x,y)$ has zero stalk at $p$, so shrinking kills this finite module. Likewise the kernels of multiplication by $x$ on $R$ and by $y$ on $R\mathrm{/}(x)$ are finite modules with zero stalk at $p$, and another shrinking kills them. Thus $(x,y)$ is a regular sequence generating the point ideal on this affine neighborhood, with nonzero quotient $\kappa (p)$.

The affine regular-sequence calculation applies on this neighborhood and gives the asserted direct images. On the complement of $p$ the blowup is the identity, with the same direct images. These local results give ${\pi }_{\ast }\mathcal{O}=\mathcal{O}$ and $R^q{\pi }_{\ast }\mathcal{O}=0$ globally. The argument uses only local Noetherianity and the two-dimensional regular local ring at the center.

For a finite composition of the point blowups just considered, write it as $f\circ g$, where $g$ is the last step. Assume by induction the conclusions for $f$. For every affine open $V$ in the original base, apply the vanishing-direct-image comparison to $g$ restricted over $f^{-1}(V)$. Step 2.1 makes its higher direct images zero and its degree-zero image the structure sheaf. Hence $H^q(g^{-1}{f}^{-1}V,\mathcal{O})=H^q(f^{-1}V,\mathcal{O})$. A second comparison for $f$ over $V$, followed by affine acyclicity, identifies the latter with $\mathrm{\Gamma }(V,\mathcal{O})$ for $q=0$ and zero for $q>0$. Sheafifying these identifications proves the same direct-image conclusions for the composition, completing the induction.