Let \(X\) be a ruled surface over the curve \(C\), defined by a normalized bundle \(\mathcal{E}\), and let \(\mfe\) be the divisor on \(C\) for which \(\mathcal{L}(\mfe) \cong \bigwedge^2 \mathcal{E}\) (See 2.8.1). Let \(\mfb\) be any divisor on \(C\).
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.
If \(\mfb\) and \(\mfb + \mfe\) are very ample on \(C\), and for every point \(P \in C\), we have \(\mfb-P\) and \(\mfb+\mfe-P\) nonspecial, then \(C_0+\mfb f\) is very ample.
Solution
Write \[ \pi:X=\PP(\mathcal E)\longrightarrow C \] for the ruling and put \[ L=\OO_X(C_0+\mfb f). \] For a divisor \(A\) on \(C\), write \(Af=\pi^*A\).
1. If \(\mfb-A\) is nonspecial, then\[ h^0\bigl(X,L(-Af)\bigr) = h^0\bigl(C,\OO_C(\mfb-A)\bigr) + h^0\bigl(C,\OO_C(\mfb+\mfe-A)\bigr). \]
The normalized section \(C_0\) gives the exact sequence \[ 0 \longrightarrow \OO_X((\mfb-A)f) \longrightarrow \OO_X(C_0+(\mfb-A)f) \longrightarrow \OO_{C_0}(C_0+(\mfb-A)f) \longrightarrow0. \] Under the identification \(C_0\cong C\), the ruled-surface conventions of (2.8.1) give \[ \OO_{C_0}(C_0+(\mfb-A)f) \cong \OO_C(\mfb+\mfe-A). \] Also \[ \OO_X((\mfb-A)f)=\pi^*\OO_C(\mfb-A), \] so \[ H^i\bigl(X,\OO_X((\mfb-A)f)\bigr) \cong H^i\bigl(C,\OO_C(\mfb-A)\bigr), \] because \(\pi_*\OO_X=\OO_C\) and \(R^1\pi_*\OO_X=0\). The nonspeciality hypothesis says that the \(H^1\) group on the left term vanishes. Taking cohomology therefore gives the displayed sum of dimensions.
2. Under the hypotheses of part (a), for every point \(P\in C\),\[ h^0\bigl(X,L(-Pf)\bigr)=h^0(X,L)-2. \]
Since \(|\mfb|\) is base-point free, \[ h^0(\mfb-P)=h^0(\mfb)-1. \] The exact sequence \[ 0\to\OO_C(\mfb-P)\to\OO_C(\mfb)\to k(P)\to0 \] has surjective evaluation map on global sections. Since \(\mfb\) is nonspecial, its long exact sequence gives \[ H^1(C,\OO_C(\mfb-P))=0. \] Thus step 1 applies both to \(A=0\) and to \(A=P\).
Because \(|\mfb+\mfe|\) is also base-point free, \[ h^0(\mfb+\mfe-P)=h^0(\mfb+\mfe)-1. \] Subtracting the two formulas from step 1 therefore gives \[ h^0(X,L)-h^0\bigl(X,L(-Pf)\bigr)=2. \]
3. The complete linear system \(|L|\) restricts to the complete system\(|\OO_{\PP^1}(1)|\) on every fibre of \(\pi\). In particular, \(|L|\) has nobase points.
For the fibre \[ F_P=\pi^{-1}(P)\cong\PP^1, \] one has \[ L|_{F_P}\cong\OO_{\PP^1}(1). \] The exact sequence \[ 0\to L(-Pf)\to L\to L|_{F_P}\to0 \] shows that the kernel of the restriction map on global sections is \(H^0(X,L(-Pf))\). By step 2 the image has dimension \(2\), which equals \[ h^0(\PP^1,\OO(1)). \] Hence \[ H^0(X,L)\twoheadrightarrow H^0(F_P,\OO_{F_P}(1)) \] for every \(P\).
The system \(|\OO_{F_P}(1)|\) has no base point. Since every point of \(X\) lies on some fibre, \(|L|\) has no base points on \(X\).
4. Under the hypotheses of part (a), there is a member\[ D\in|C_0+\mfb f| \]which is a section of \(\pi\).
Put \[ V=H^0(X,L). \] By step 3, for each \(P\in C\) the kernel of \[ V\longrightarrow H^0(F_P,\OO_{F_P}(1)) \] has codimension \(2\). Consider the incidence set \[ I={(P,[s])\in C\times\PP(V):s|_{F_P}=0\}. \] Its fibre over \(P\) is the projectivization of that codimension-two kernel, so \[ \dim I\le\dim\PP(V)-1. \] The projection \(I\to\PP(V)\) has closed image because \(C\) is projective. Thus a section \(s\in V\) can be chosen whose divisor \(D=(s=0)\) contains no fibre.
Since \[ D\cdot f=(C_0+\mfb f)\cdot f=1, \] and \(D\) contains no fibre, every irreducible component of \(D\) dominates \(C\). An effective Cartier divisor on the nonsingular surface \(X\) is pure of dimension one, and the sum of the generic degrees of its horizontal components, counted with multiplicity, is \(D\cdot f=1\). Thus \(D\) has one horizontal component, with multiplicity one; in particular \(D\) is integral. The proper morphism \[ \pi|_D:D\longrightarrow C \] is therefore finite of degree one and birational. Since \(C\) is nonsingular, hence normal, a finite birational map to \(C\) is an isomorphism. Thus \(D\) is a section of the ruling. Together with step 3 this proves part (a).
5. Assume the hypotheses of part (b). Then \(\mfb\) is nonspecial, andpart (a) applies; in particular \(L\) is base-point free and restrictssurjectively to \(\OO_{\PP^1}(1)\) on every fibre.
Fix any point \(P\in C\). By hypothesis \[ H^1(C,\OO_C(\mfb-P))=0. \] The long exact sequence of \[ 0\to\OO_C(\mfb-P)\to\OO_C(\mfb)\to k(P)\to0 \] then gives \[ H^1(C,\OO_C(\mfb))=0. \] Thus \(\mfb\) is nonspecial. The divisors \(\mfb\) and \(\mfb+\mfe\) are very ample, hence base-point free, so all hypotheses of part (a) hold. Steps <1>2–<1>4 therefore apply.
6. Let \(A\) be any effective divisor of degree \(2\) on \(C\), including adouble point \(2P\). Then both\[ \mfb-A \qquad\text{and}\qquad \mfb+\mfe-A \]are nonspecial.
Choose a point \(P\) contained in \(A\), and write \[ A=P+Q, \] allowing \(Q=P\).
Since \(\mfb\) is very ample, every length-two divisor imposes two independent conditions, so \[ h^0(\mfb-A)=h^0(\mfb)-2. \] Also \(\mfb\) is base-point free, hence \[ h^0(\mfb-P)=h^0(\mfb)-1. \] By hypothesis \(\mfb-P\) is nonspecial. In the exact sequence \[ 0\to\OO_C(\mfb-A) \to\OO_C(\mfb-P) \to\OO_Q(\mfb-P) \to0, \] the global-section dimensions show that the last evaluation map is surjective. The resulting long exact sequence therefore gives \[ H^1(C,\OO_C(\mfb-A))=0. \]
The identical argument, using the very ampleness of \(\mfb+\mfe\) and the assumed nonspeciality of \(\mfb+\mfe-P\), gives \[ H^1(C,\OO_C(\mfb+\mfe-A))=0. \]
7. For every effective degree-two divisor \(A\) on \(C\), restriction issurjective:\[ H^0(X,L) \twoheadrightarrow H^0\bigl(\pi^{-1}(A),L|_{\pi^{-1}(A)}\bigr). \]
Apply the normalized-section exact sequence from step 1 to \[ L(-Af)=\OO_X(C_0+(\mfb-A)f). \] Step 6 makes the \(H^1\) groups of both the left term \[ \OO_X((\mfb-A)f) \] and the quotient on \(C_0\) \[ \OO_C(\mfb+\mfe-A) \] vanish. Hence \[ H^1(X,L(-Af))=0. \]
Now use \[ 0\to L(-Af) \to L \to L|_{\pi^{-1}(A)} \to0. \] The vanishing just proved makes the restriction map on \(H^0\) surjective.
8. The line bundle \(L\) separates every zero-dimensional subscheme of\(X\) of length \(2\).
Let \(Z\subset X\) have length \(2\).
If \(Z\) is contained in one fibre \(F_P\), step 5 gives a surjection \[ H^0(X,L)\twoheadrightarrow H^0(F_P,\OO_{F_P}(1)). \] Since \(\OO_{\PP^1}(1)\) separates every length-two subscheme of \(\PP^1\), the composite map onto \(H^0(Z,L|_Z)\) is surjective.
Suppose \(Z\) is not contained in a fibre. Its scheme-theoretic image on the nonsingular curve \(C\) is then a degree-two effective divisor \(A\); this is either the sum of the two base points of two distinct fibres or the double point \(2P\) when \(Z\) is a tangent direction transverse to the fibre. Moreover the definition of scheme-theoretic image makes \[ \OO_A\longrightarrow (\pi|_Z)_*\OO_Z \] injective. Both sides have length \(2\), so this map is an isomorphism and \(Z\cong A\).
Step 7 gives \[ H^0(X,L)\twoheadrightarrow H^0\bigl(\pi^{-1}(A),L|_{\pi^{-1}(A)}\bigr). \] Over the Artinian scheme \(A\), the surface \(\pi^{-1}(A)\) is the projective bundle \(\PP(\mathcal E|_A)\) and \(L\) is its relative \(\OO(1)\) tensored by a line bundle from \(A\). The subscheme \(Z\cong A\) is a section, hence corresponds to a quotient of the rank-two bundle on \(A\) by an invertible sheaf. Since \(A\) is affine, taking global sections of that quotient is surjective. Thus the restriction map from \(H^0(\pi^{-1}(A),L)\) to \(H^0(Z,L|_Z)\) is surjective, and so is the composite from \(H^0(X,L)\).
9. Under the hypotheses of part (b),\[ \boxed{C_0+\mfb f\text{ is very ample}.} \]
A line bundle on a projective scheme is very ample exactly when its complete linear system separates every zero-dimensional subscheme of length \(2\); equivalently, it separates distinct points and tangent vectors, as in T-DIVMAPPN. Step 8 proves this criterion for \(L\). Hence \(L\) is very ample, proving part (b).
10. Q.E.D.
Steps <1>1–<1>4 prove part (a), and steps <1>5–<1>9 prove part (b).