## 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 a. If $\mathcal{E}$ is a locally free sheaf of rank $r$ on a (nonsingular) curve $C$, then there is a sequence \[ 0=\mathcal{E}_0 \subseteq \mathcal{E}_1 \subseteq \ldots \subseteq \mathcal{E}_r=\mathcal{E} \] 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.[^hint.4.2.3.a] b. 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. [^hint.4.2.3.a]: Hint: Use (II, Ex. 8.2). ### 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 $s=2 r$. a. Show that $r=0$ and any $r \geqslant g+1$ are always possible. Cf. (V, Ex. 6.8). b. 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 $g \geqslant 1$. a. 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). b. 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 \[ 0 \rightarrow \mathcal{O}_C \rightarrow \mathcal{E} \rightarrow \mathcal{L}(D) \rightarrow 0 . \] 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 $(2.15)$. c. Now show that if $-g \leqslant e<0$, there exists a ruled surface $X$ over $C$ with invariant $e$.[^hint.5.2222.c] d. For $g=2$, show that $e \geqslant-2$ is also necessary for the existence of $X$.[^nagata_note_v.2.5] [^nagata_note_v.2.5]: Note. It has been shown that $e \geqslant-g$ for any ruled surface (Nagata $[8]$). [^hint.5.2222.c]: Hint: For any given $D$ in (b), show that a suitable $\xi$ exists, using an argument similar to the proof of (II, 8.18). ### 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.[^hint.5.2.6] [^hint.5.2.6]: Hint: Choose a subinvertible sheaf of maximal degree, and use induction on the rank. ### 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 \[ \mathcal{E} \rightarrow \mathcal{F} \rightarrow 0, \qquad \mathcal{F} \neq \mathcal{E}, \mathcal{F} \neq 0 ,\] we have \[ (\operatorname{deg} \mathcal{F}) / \operatorname{rank} \mathcal{F}>(\operatorname{deg} \mathcal{E}) / \operatorname{rank} \mathcal{E} .\] Replacing $>$ by $\geqslant$ defines **semistable**. a. A decomposable $\mathcal{E}$ is never stable. b. 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, $\geqslant 0$ ). c. Show that the indecomposable locally free sheaves $\mathcal{E}$ of rank 2 that are not semistable are classified, up to isomorphism, by giving (1) an integer $0e \geqslant 0$, 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 $C$. a. If $|\mfb|$ and $|\mfb + \mfe|$ have no base points, and if $\mfb$ is nonspecial, then there is a section $D \sim C_0+\mfb f$, and $|D|$ has no base points. b. If $\mfb$ and $\mfb + \mfe$ 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 $C$. a. 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. b. 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 $g \geqslant 2$. a. 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 - $a \geqslant p, b \geqslant \frac{1}{2} a e+1-g$. b. 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). a. 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)). b. 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 $H^1(\mathcal{L}(-3 P))$. c. Now let $\mathcal{E}$ be defined by $\xi$ as an extension \[ 0 \rightarrow \mathcal{O}_C \rightarrow \mathcal{E} \rightarrow \mathcal{L}(P) \rightarrow 0, \] 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 a. 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, \[ \varphi^*\left(\mathcal{I} / \mathcal{I}^2\right) \cong \mathcal{O}(l) \oplus \mathcal{O}(m) .\] for some $l, m \in \mathbf{Z}$. Determine $l$ and $m$. b. 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.[^answer.5.2.17.b] [^answer.5.2.17.b]: Answer: If char $k \neq 2$, then $l=m=-7$; if char $k=2$, then $l, m=-6,-8$.