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 \(X^{\prime}\).
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 \(\widetilde{D}^2=D^2-1\). 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 \(\PP(\mathcal{E})\) with \(\mathcal{E}\) decomposable behaves under \(\operatorname{elm}_P\).
Solution
Choose rank-two locally free sheaves \[ \mathcal E,\mathcal E' \] on \(C\) such that \[ X=\PP(\mathcal E), \qquad X'=\PP(\mathcal E'). \] We use Hartshorne’s quotient convention for projective bundles.
1. Let \(p\in C\), let\[ q:\mathcal E\otimes k(p)\twoheadrightarrow k(p) \]be a one-dimensional quotient, and let\[ \mathcal F = \ker\!\left( \mathcal E\longrightarrow \mathcal E\otimes k(p) \xrightarrow{q} k(p) \right). \]Then \(\mathcal F\) is locally free of rank two, and\[ \boxed{ \operatorname{elm}_{P}\PP(\mathcal E) \cong \PP(\mathcal F), } \]where \(P\in\PP(\mathcal E_p)\) is the point represented by \(q\).
Away from \(p\), the inclusion \[ \mathcal F\hookrightarrow\mathcal E \] is an isomorphism. Since \(\mathcal F\) is a torsion-free coherent sheaf on the nonsingular curve \(C\), it is locally free.
It remains to identify the birational map near \(p\). Let \[ R=\OO_{C,p} \] be the discrete valuation ring, choose a uniformizer \(t\), and choose a basis \(e_1,e_2\) of \(\mathcal E_p\) such that \(q\) is reduction modulo \(t\) followed by projection to the \(e_2\)-coordinate. Then \[ \mathcal E_p=Re_1\oplus Re_2, \qquad \mathcal F_p=Re_1\oplus Rt e_2. \]
With quotient coordinates \([x:y]\) on \(\PP(\mathcal E)\) and \([u:v]\) on \(\PP(\mathcal F)\), restriction of a quotient of \(\mathcal E\) to \(\mathcal F\) gives the rational map \[ [x:y]\dashrightarrow[x:ty]. \] The point at which it is undefined is \[ t=0,\qquad[x:y]=[0:1], \] namely \(P\).
On the chart \(y\ne0\), put \[ z=x/y. \] Then the map is \[ (t,z)\dashrightarrow[z:t]\in\PP^1. \] The closure of its graph is the blowup of the \((t,z)\)-plane at the origin. On that blowup, the strict transform of the old fibre \(t=0\) maps to the single point \([1:0]\), while the exceptional curve maps isomorphically to the new fibre. Thus the resolved birational map is exactly: blow up \(P\) and contract the strict transform of the fibre through \(P\). This is the elementary transformation of (5.7.1), and its target is \(\PP(\mathcal F)\).
2. The inverse of an elementary transformation is again an elementarytransformation.
In the local coordinates of step 1, the generic inverse of \[ [x:y]\dashrightarrow[x:ty] \] is \[ [u:v]\dashrightarrow[tu:v]. \] On the new fibre \(t=0\), this inverse is undefined only at \[ [u:v]=[1:0]. \] Resolving it blows up that point and contracts the strict transform of the new fibre, exactly the same elementary-transformation construction. Hence the operation is reversible by another elementary transformation.
3. If \(D\subseteq X\) is a section containing the centre \(P\) and\(\widetilde D\) is its transform under \(\operatorname{elm}_P\), then\[ \boxed{\widetilde D^{\,2}=D^2-1.} \]
Let \[ b:Z\longrightarrow X \] be the blowup at \(P\), with exceptional curve \(E\), and let \(F\) be the ruling fibre through \(P\). Since \(D\) is a section, \(D\) and \(F\) are smooth and meet transversely at \(P\).
The strict transform \(\bar D\) therefore satisfies the usual blowup formula \[ \bar D^{\,2}=D^2-1. \] The strict transforms \(\bar D\) and \(\bar F\) are disjoint, because the blowup separates their two tangent directions at \(P\).
The elementary transformation next contracts the \((-1)\)-curve \(\bar F\). Because \(\bar D\) is disjoint from \(\bar F\), the contraction is an isomorphism in a neighbourhood of \(\bar D\), and the pullback of its image \(\widetilde D\) is exactly \(\bar D\). Hence \[ \widetilde D^{\,2} = \bar D^{\,2} = D^2-1. \] This is the first calculation requested in the hint.
4. Let\[ K=k(C) \]be the function field. After identifying the generic fibres\[ \mathcal E_\eta \cong \mathcal E'_\eta \cong K^2, \]there is an effective divisor \(A\) on \(C\) and an injective morphism\[ \boxed{ \varphi:\mathcal E \hookrightarrow \mathcal E'(A) } \]whose cokernel is a finite-length torsion sheaf.
Choose any \(K\)-linear isomorphism \[ \varphi_\eta:\mathcal E_\eta\xrightarrow{\sim}\mathcal E'_\eta. \] Viewed in local trivializations, its matrix entries are rational functions on \(C\). They have only finitely many poles. Choose an effective divisor \(A\) whose multiplicities are large enough to clear all those poles. Then \(\varphi_\eta\) extends to a morphism \[ \varphi:\mathcal E\longrightarrow\mathcal E'(A). \]
The map is an isomorphism at the generic point. Its kernel is therefore a torsion subsheaf of the locally free sheaf \(\mathcal E\), hence is zero. Thus \(\varphi\) is injective.
Source and target have the same rank, so the cokernel \[ \mathcal T=\operatorname{coker}\varphi \] has rank zero. On the noetherian projective curve \(C\), a coherent rank-zero sheaf is supported on finitely many closed points and has finite length.
5. There are rank-two locally free sheaves\[ \mathcal F_0,\mathcal F_1,\ldots,\mathcal F_N \]with\[ \mathcal F_0=\mathcal E'(A), \qquad \mathcal F_N=\mathcal E, \]such that for every \(0\le i<N\) there is an exact sequence\[ \boxed{ 0 \longrightarrow \mathcal F_{i+1} \longrightarrow \mathcal F_i \longrightarrow k(p_i) \longrightarrow 0 } \]for some point \(p_i\in C\).
The finite-length torsion sheaf \[ \mathcal T=\mathcal E'(A)/\mathcal E \] from step 4 has a composition series \[ \mathcal T=\mathcal T_0 \supset \mathcal T_1 \supset \cdots \supset \mathcal T_N=0 \] whose successive quotients are simple torsion sheaves. Since the ground field is algebraically closed, every such simple quotient is \[ \mathcal T_i/\mathcal T_{i+1}\cong k(p_i) \] for some closed point \(p_i\).
Let \(\mathcal F_i\) be the inverse image of \(\mathcal T_i\) under \[ \mathcal E'(A)\twoheadrightarrow\mathcal T. \] Then \[ \mathcal F_0=\mathcal E'(A), \qquad \mathcal F_N=\mathcal E, \] and the successive quotients are the displayed skyscraper sheaves.
Each \(\mathcal F_i\) is a torsion-free coherent sheaf of rank two on the nonsingular curve \(C\), hence is locally free of rank two.
6. For every \(0\le i<N\),\[ \PP(\mathcal F_{i+1}) \]is obtained from\[ \PP(\mathcal F_i) \]by one elementary transformation.
The quotient in step 5, \[ \mathcal F_i\twoheadrightarrow k(p_i), \] induces a nonzero one-dimensional quotient of the fibre \[ \mathcal F_i\otimes k(p_i). \] Let \[ P_i\in\PP((\mathcal F_i)_{p_i}) \] be the corresponding point. By construction, \[ \mathcal F_{i+1} = \ker\bigl(\mathcal F_i\to k(p_i)\bigr). \] Step 1 therefore gives \[ \operatorname{elm}_{P_i}\PP(\mathcal F_i) \cong \PP(\mathcal F_{i+1}). \]
7. There is a finite sequence of elementary transformations taking\[ \PP(\mathcal E) \]to\[ \PP(\mathcal E'(A)). \]
Step 6 gives a finite sequence \[ \PP(\mathcal E'(A)) = \PP(\mathcal F_0) \dashrightarrow \PP(\mathcal F_1) \dashrightarrow \cdots \dashrightarrow \PP(\mathcal F_N) = \PP(\mathcal E), \] in which every arrow is an elementary transformation.
By step 2, every arrow can be reversed by another elementary transformation. Reversing the finite sequence therefore gives elementary transformations from \(\PP(\mathcal E)\) to \(\PP(\mathcal E'(A))\).
8. Twisting by a line bundle does not change a projective bundle:\[ \boxed{ \PP(\mathcal E'(A)) \cong \PP(\mathcal E') } \]over \(C\).
For every \(C\)-scheme \(T\), tensoring with the pullback of \(\OO_C(A)\) gives a bijection between invertible quotients of \(\mathcal E'_T\) and invertible quotients of \[ \mathcal E'_T\otimes\OO_T(A). \] This identifies the two projective-bundle functors and hence gives an isomorphism \[ \PP(\mathcal E') \xrightarrow{\sim} \PP(\mathcal E'(A)). \] Equivalently, this is the standard identity \[ \PP(\mathcal E'\otimes\mathcal L)\cong\PP(\mathcal E') \] for an invertible sheaf \(\mathcal L\).
9. There is a finite sequence of elementary transformations transforming\(X\) into \(X'\):\[ \boxed{ X=\PP(\mathcal E) \dashrightarrow \cdots \dashrightarrow \PP(\mathcal E'(A)) \cong \PP(\mathcal E')=X'. } \]
Step 7 supplies the finite elementary-transformation sequence from \(\PP(\mathcal E)\) to \(\PP(\mathcal E'(A))\), and step 8 identifies the latter ruled surface over \(C\) with \(X'\). Thus the two given geometrically ruled surfaces are connected by finitely many elementary transformations.
10. Q.E.D.
Steps <1>1–<1>3 identify the geometric elementary transformation with a length-one modification of the defining rank-two bundle and verify the self-intersection calculation from the hint. Steps <1>4–<1>6 factor the difference between the two bundles into finitely many such modifications, and steps <1>7–<1>9 reverse the resulting chain and remove the harmless line-bundle twist.