5_5x – AG Notes

V.5: Birational Transformations

V.5.1. #to_work

Let \(f\) be a rational function on the surface \(X\). Show that it is possible to "resolve the singularities of \(f^{\prime \prime}\) in the following sense: there is a birational morphism \(g\) : \(X^{\prime} \rightarrow X\) so that \(f\) induces a morphism of \(X^{\prime}\) to \(\mathbf{P}^1\). ¹

V.5.2. #to_work

Let \(Y \cong \mathbf{P}^1\) be a curve in a surface \(X\), with \(Y^2<0\). Show that \(Y\) is contractible (5.7.2) to a point on a projective variety \(X_0\) (in general singular).

V.5.3. #to_work

If \(\pi: \tilde{X} \rightarrow X\) is a monoidal transformation with center \(P\), show that \(H^1\left(\tilde{X}, \Omega_{\tilde{X}}\right) \cong\) \(H^1\left(X, \Omega_X\right) \oplus k\). This gives another proof of (5.8). ²

V.5.4. #to_work

Let \(f: X \rightarrow X^{\prime}\) be a birational morphism of nonsingular surfaces.

If \(Y \subseteq X\) is an irreducible curve such that \(f(Y)\) is a point, then \(Y \cong \mathbf{P}^1\) and \(Y^2<0\)
Let \(P^{\prime} \in X^{\prime}\) be a fundamental point of \(f^{-1}\), and let \(Y_1, \ldots, Y_r\) be the irreducible components of \(f^{-1}\left(P^{\prime}\right)\). Show that the matrix \(\left|Y_i . Y_j\right|\) is negative definite.

V.5.5. #to_work

Let \(C\) be a curve, and let \(\pi: X \rightarrow C\) and \(\pi^{\prime}: X^{\prime} \rightarrow C\) be two geometrically ruled surfaces over \(C\). Show that there is a finite sequence of elementary transformations (5.7.1) which transform \(X\) into \(X^{\prime}\). ³

V.5.6. #to_work

Let \(X\) be a surface with function field \(K\). Show that every valuation \(\operatorname{ring} R\) of \(K / k\) is one of the three kinds described in (II, Ex. 4.12). ⁴

V.5.7. #to_work

Let \(Y\) be an irreducible curve on a surface \(X\), and suppose there is a morphism \(f: X \rightarrow X_0\) to a projective variety \(X_0\) of dimension 2 , such that \(f(Y)\) is a point \(P\) and \(f^{-1}(P)=Y\). Then show that \(Y^2<0\). ⁵

V.5.8. A surface singularity. #to_work

Let \(k\) be an algebraically closed field, and let \(X\) be the surface in \(\mathbf{A}_k^3\) defined by the equation \(x^2+y^3+z^5=0\). It has an isolated singularity at the origin \(P=(0,0,0)\).

Show that the affine ring \(A=k[x, y, z] /\left(x^2+y^3+z^5\right)\) of \(X\) is a unique factorization domain, as follows. Let \(t=z^{-1} ; u=t^3 x\), and \(v=t^2 y\). Show that \(z\) is irreducible in \(A ; t \in k[u, v]\), and \(A\left[z^{-1}\right]=k\left[u, v, t^{-1}\right]\). Conclude that \(A\) is a UFD.
Show that the singularity at \(P\) can be resolved by eight successive blowings-up. If \(\tilde{X}\) is the resulting nonsingular surface, then the inverse image of \(P\) is a union of eight projective lines, which intersect each other according to the Dynkin \(\operatorname{diagram} \mathbf{E}_8\) :

Footnotes

Hints: Write the divisor of \(f\) as \((f)=\sum n_i C_i\). Then apply embedded resolution (3.9) to the curve \(Y=\bigcup C_i\). Then blow up further as necessary whenever a curve of zeros meets a curve of poles until the zeros and poles of \(f\) are disjoint.

Hints: Use the projection formula (III, Ex. 8.3) and (III, Ex. 8.1) to show that \(H^i\left(X, \Omega_X\right) \cong H^i\left(\tilde{X}, \pi^* \Omega_X\right)\) for each i. Next use the exact sequence \begin{align*} 0 \rightarrow \pi^* \Omega_X \rightarrow \Omega_{\tilde{X}} \rightarrow \Omega_{\tilde{X} / X} \rightarrow 0 \end{align*} and a local calculation with coordinates to show that there is a natural isomorphism \(\Omega_{\tilde{X} / X} \cong \Omega_E\), where \(E\) is the exceptional curve. Now use the cohomology sequence of the above sequence (you will need every term) and Serre duality to get the result.

Hints: First show if \(D \subseteq X\) is a section of \(\pi\) containing a point \(P\), and if \(\tilde{D}\) is the strict transform of \(D\) by \(\mathrm{elm}_P\), then \(\widetilde{D}^2=D^2-1\) (Fig. 23).
Next show that \(X\) can be transformed into a geometrically ruled surface \(X^{\prime \prime}\) with invariant \(e \gg 0\). Then use (2.12), and study how the ruled surface \(\mathbf{P}(\mathcal{E})\) with \(\mathcal{E}\) decomposable behaves under \(\operatorname{elm}_P\).

Hint: In case (3), let \(f \in R\). Use (Ex. 5.1) to show that for all \(i \gg 0, f \in \mathcal{O}_{X_i}\), so in fact \(f \in R_0\).

Hint: Let \(|H|\) be a very ample (Cartier) divisor class on \(X_0\), let \(H_0 \in|H|\) be a divisor containing \(P\), and let \(H_1 \in|H|\) be a divisor not containing \(P\). Then consider \(f^* H_0, f^* H_1\) and \(\tilde{H}_0=f^*\left(H_0-P\right)^{-}\).