5_2x

V.2: Ruled Surfaces

V.2.1. #to_work

If \(X\) is a birationally ruled surface, show that the curve \(C\), such that \(X\) is birationally equivalent to \(C \times \mathbf{P}^1\), is unique (up to isomorphism).

V.2.2. #to_work

Let \(X\) be the ruled surface \(\mathbf{P}(\mathcal{E})\) over a curve \(C\). Show that \(\mathcal{E}\) is decomposable if and only if there exist two sections \(C^{\prime}, C^{\prime \prime}\) of \(X\) such that \(C^{\prime} \cap C^{\prime \prime}=\varnothing\).

V.2.3. #to_work

• If \(\mathcal{E}\) is a locally free sheaf of rank \(r\) on a (nonsingular) curve \(C\), then there is a sequence \begin{align*} 0=\mathcal{E}_0 \subseteq \mathcal{E}_1 \subseteq \ldots \subseteq \mathcal{E}_r=\mathcal{E} \end{align*} of subsheaves such that \(\mathcal{E}_i / \mathcal{E}_{i-1}\) is an invertible sheaf for each \(i=1, \ldots, r\). We say that \(\mathcal{E}\) is a successive extension of invertible sheaves. ¹

• Show that this is false for varieties of dimension \(\geqslant 2\). In particular, the sheaf of differentials \(\Omega\) on \(\mathbf{P}^2\) is not an extension of invertible sheaves.

V.2.4. #to_work

Let \(C\) be a curve of genus \(g\), and let \(X\) be the ruled surface \(C \times \mathbf{P}^1\). We consider the question, for what integers \(s \in \mathbf{Z}\) does there exist a section \(D\) of \(X\) with \(D^2=s\) ? First show that \(s\) is always an even integer, say \(s=2 r\).

• Show that \(r=0\) and any \(r \geqslant g+1\) are always possible. Cf. (V, Ex. 6.8).

• If \(g=3\), show that \(r=1\) is not possible, and just one of the two values \(r=2,3\) is possible, depending on whether \(C\) is hyperelliptic or not.

V.2.5. Values of \(e\). #to_work

Let \(C\) be a curve of genus \(g \geqslant 1\).

• Show that for each \(0 \leqslant e \leqslant 2 g-2\) there is a ruled surface \(X\) over \(C\) with invariant \(e\), corresponding to an indecomposable \(\mathcal{E}\). Cf. (2.12).

• Let \(e<0\), let \(D\) be any divisor of degree \(d=-e\), and let \(\xi \in H^1(\mathcal{L}(-D))\) be a nonzero element defining an extension \begin{align*} 0 \rightarrow \mathcal{O}_C \rightarrow \mathcal{E} \rightarrow \mathcal{L}(D) \rightarrow 0 . \end{align*} Let \(H \subseteq|D+K|\) be the sublinear system of codimension 1 defined by ker \(\xi\), where \(\xi\) is considered as a linear functional on \(H^0(\mathcal{L}(D+K))\). For any effective divisor \(E\) of degree \(d-1\), let \(L_E \subseteq|D+K|\) be the sublinear system \(|D+K-E|+E\). Show that \(\mathcal{E}\) is normalized if and only if for each \(E\) as above, \(L_E \nsubseteq H\). Cf. proof of \((2.15)\).

• Now show that if \(-g \leqslant e<0\), there exists a ruled surface \(X\) over \(C\) with invariant \(e\). ²

• For \(g=2\), show that \(e \geqslant-2\) is also necessary for the existence of \(X\). ³

V.2.6. #to_work

Show that every locally free sheaf of finite rank on \(\mathbf{P}^1\) is isomorphic to a direct sum of invertible sheaves. ⁴

V.2.7. #to_work

On the elliptic ruled surface \(X\) of (2.11.6), show that the sections \(C_0\) with \(C_0^2=1\) form a one-dimensional algebraic family, parametrized by the points of the base curve \(C\), and that no two are linearly equivalent.

V.2.8. #to_work

A locally free sheaf \(\mathcal{E}\) on a curve \(C\) is said to be stable if for every quotient locally free sheaf \begin{align*} \mathcal{E} \rightarrow \mathcal{F} \rightarrow 0, \qquad \mathcal{F} \neq \mathcal{E}, \mathcal{F} \neq 0 ,\end{align*} we have \begin{align*} (\operatorname{deg} \mathcal{F}) / \operatorname{rank} \mathcal{F}>(\operatorname{deg} \mathcal{E}) / \operatorname{rank} \mathcal{E} .\end{align*} Replacing \(>\) by \(\geqslant\) defines semistable.

• A decomposable \(\mathcal{E}\) is never stable.

• If \(\mathcal{E}\) has rank 2 and is normalized, then \(\mathcal{E}\) is stable (respectively, semistable) if and only if \(\deg \mathcal{E}>0\) (respectively, \(\geqslant 0\) ).

• Show that the indecomposable locally free sheaves \(\mathcal{E}\) of rank 2 that are not semistable are classified, up to isomorphism, by giving

  • an integer \(0<e \leqslant\) \(2 g-2\),
  • an element \(\mathcal{L} \in \operatorname{Pic}C\) of degree \(-e\), and
  • a nonzero \(\xi \in H^1\left(\mathcal{L} {}^{ \vee }\right)\), determined up to a nonzero scalar multiple.

V.2.9. #to_work

Let \(Y\) be a nonsingular curve on a quadric cone \(X_0\) in \(\mathbf{P}^3\). Show that either

• \(Y\) is a complete intersection of \(X_0\) with a surface of degree \(a \geqslant 1\), in which case \(\deg Y=\) \(2 a, g(Y)=(a-1)^2\), or,
• \(\deg Y\) is odd, say \(2 a+1\), and \(g(Y)=a^2-a\). ⁵

V.2.10. #to_work

For any \(n>e \geqslant 0\), let \(X\) be the rational scroll of degree \(d=2 n-e\) in \(\mathbf{P}^{d+1}\) given by (2.19). If \(n \geqslant 2 e-2\), show that \(X\) contains a nonsingular curve \(Y\) of genus \(g=d+2\) which is a canonical curve in this embedding.

Conclude that for every \(g \geqslant 4\), there exists a nonhyperelliptic curve of genus \(g\) which has a \(g_3^1\). Cf. (V, \(\S 5\)).

V.2.11. #to_work

Let \(X\) be a ruled surface over the curve \(C\), defined by a normalized bundle \(\mathcal{E}\), and let \(\mathfrak e\) be the divisor on \(C\) for which \(\mathcal{L}(\mathfrak{e}) \cong \bigwedge^2 \mathcal{E}\) (See 2.8 .1). Let \(\mathfrak{b}\) be any divisor on \(C\).

• If \(|{\mathfrak{b}}|\) and \(|{\mathfrak{b}}+ {\mathfrak{e}}|\) have no base points, and if \({\mathfrak{b}}\) is nonspecial, then there is a section \(D \sim C_0+{\mathfrak{b}}f\), and \(|D|\) has no base points.

• If \({\mathfrak{b}}\) and \({\mathfrak{b}}+ {\mathfrak{e}}\) are very ample on \(C\), and for every point \(P \in C\), we have \(\mathfrak{b}-P\) and \(\mathfrak{b}+\mathfrak{e}-P\) nonspecial, then \(C_0+\mathfrak{b} f\) is very ample.

V.2.12. #to_work

Let \(X\) be a ruled surface with invariant \(e\) over an elliptic curve \(C\), and let \(\mathfrak{b}\) be a divisor on \(C\).

• If \(\deg \mathfrak{b} \geqslant e+2\), then there is a section \(D \sim C_0+\mathfrak{b} f\) such that \(|D|\) has no base points.

• The linear system \(\left|C_0+\mathfrak{b} f\right|\) is very ample if and only if \(\deg \mathfrak{b} \geqslant e+3\). Note. The case \(e=-1\) will require special attention.

V.2.13. #to_work

For every \(e \geqslant-1\) and \(n \geqslant e+3\), there is an elliptic scroll of degree \(d=2 n-e\) in \(\mathbf{P}^{d-1}\). In particular, there is an elliptic scroll of degree 5 in \(\mathbf{P}^4\).

V.2.14. #to_work

Let \(X\) be a ruled surface over a curve \(C\) of genus \(g\), with invariant \(e<0\), and assume that char \(k=p>0\) and \(g \geqslant 2\).

• If \(Y \equiv a C_0+b f\) is an irreducible curve \(\neq C_0, f\), then either

  • \(a=1, b \geqslant 0\), or
  • \(2 \leqslant a \leqslant p-1, b \geqslant \frac{1}{2} a e\), or
  • \(a \geqslant p, b \geqslant \frac{1}{2} a e+1-g\).

• If \(a>0\) and \(b>a\left(\frac{1}{2} e+(1 / p)(g-1)\right)\), then any divisor \(D \equiv a C_0+b f\) is ample. On the other hand, if \(D\) is ample, then \(a>0\) and \(b>\frac{1}{2} a e\).

V.2.15. Funny behavior in characteristic \(p\). #to_work

Let \(C\) be the plane curve \(x^3 y+y^3 z+z^3 x=0\) over a field \(k\) of characteristic 3 (V, Ex. 2.4).

• Show that the action of the \(k\)-linear Frobenius morphism \(f\) on \(H^1\left(C, \mathcal{O}_C\right)\) is identically 0 (Cf. (V, 4.21)).

• Fix a point \(P \in C\), and show that there is a nonzero \(\xi \in H^1(\mathcal{L}(-P))\) such that \(f^* \xi=0\) in \(H^1(\mathcal{L}(-3 P))\).

• Now let \(\mathcal{E}\) be defined by \(\xi\) as an extension \begin{align*} 0 \rightarrow \mathcal{O}_C \rightarrow \mathcal{E} \rightarrow \mathcal{L}(P) \rightarrow 0, \end{align*} and let \(X\) be the corresponding ruled surface over \(C\). Show that \(X\) contains a nonsingular curve \(Y \equiv 3 C_0-3 f\), such that \(\pi: Y \rightarrow C\) is purely inseparable.

  Show that the divisor \(D=2 C_0\) satisfies the hypotheses of (2.21.b), but is not ample.

V.2.16. #to_work

Let \(C\) be a nonsingular affine curve. Show that two locally free sheaves \(\mathcal{E}, \mathcal{E}^{\prime}\) of the same rank are isomorphic if and only if their classes in the Grothendieck group \(K(X)\) (II, Ex. 6.10) and (II, Ex. 6.11) are the same. This is false for a projective curve.

V.2.17 *. #to_work

• Let \(\varphi: \mathbf{P}_k^1 \rightarrow \mathbf{P}_k^3\) be the 3-uple embedding (I, Ex. 2.12). Let \(\mathcal{I}\) be the sheaf of ideals of the twisted cubic curve \(C\) which is the image of \(\varphi\). Then \(\mathcal{I} / \mathcal{I}^2\) is a locally free sheaf of rank 2 on \(C\), so \(\varphi^*\left(\mathcal{I} / \mathcal{I}^2\right)\) is a locally free sheaf of rank 2 on \(\mathbf{P}^1\). By (2.14), therefore, \begin{align*} \varphi^*\left(\mathcal{I} / \mathcal{I}^2\right) \cong \mathcal{O}(l) \oplus \mathcal{O}(m) .\end{align*} for some \(l, m \in \mathbf{Z}\). Determine \(l\) and \(m\).

• Repeat part (a) for the embedding \(\varphi: \mathbf{P}^1 \rightarrow \mathbf{P}^3\) given by \(x_0=t^4, x_1=t^3 u\), \(x_2=t u^3, x_3=u^4\), whose image is a nonsingular rational quartic curve. ⁶