Use Tsen’s theorem to show that given a flat family \(X\to Y\) with \(Y\) a curve where the fibers smooth curves of genus 0, this determines a Zariski \(\PP^1\)-bundle iff there exists a relative degree 1 line bundle on \(X\) over \(Y\).
Solution
Let \(X\) and \(Y\) be integral schemes, let \(Y\) be a smooth curve over an algebraically closed field \(k\), and let \[ f:X\longrightarrow Y \] be projective and flat with smooth genus-zero fibres. Here a Zariski \(\PP^1\)-bundle means a projective bundle \(\PP_Y(\mce)\) for a rank-two locally free sheaf \(\mce\). A relative degree-one line bundle is an invertible sheaf \(\mcl\) on \(X\) such that \[ \deg(\mcl|_{X_y})=1 \] for every \(y\in Y\). Since a projective morphism is of finite presentation, flatness together with smooth geometric fibres implies that \(f\) is smooth.
1. If\[ X\cong\PP_Y(\mce) \]for a rank-two vector bundle \(\mce\) on \(Y\), then \(X\) has a relativedegree-one line bundle.
The projective bundle carries its tautological quotient line bundle \[ \OO_{\PP_Y(\mce)}(1). \] On every fibre \[ \PP(\mce_y)\cong\PP^1_{\kappa(y)}, \] its restriction is \(\OO_{\PP^1}(1)\), which has degree \(1\).
2. Conversely, suppose \(\mcl\) is a relative degree-one line bundle.For every fibre \(C=X_y\),\[ h^0(C,\mcl_y)=2, \qquad h^1(C,\mcl_y)=0. \]
The curve \(C\) is smooth proper of genus \(0\), and \[ \deg\mcl_y=1. \] Riemann–Roch gives \[ h^0(C,\mcl_y)-h^1(C,\mcl_y) = \deg\mcl_y+1-g =2. \] By Serre duality, \[ h^1(C,\mcl_y) = h^0(C,\omega_C\tensor\mcl_y^{-1}). \] The line bundle on the right has degree \[ -2-1=-3, \] so it has no nonzero section. Hence \(h^1=0\) and \(h^0=2\).
3. The sheaf\[ \mce=f_*\mcl \]is locally free of rank \(2\), and its formation commutes with base change:\[ \mce\tensor\kappa(y) \cong H^0(X_y,\mcl_y). \]
The line bundle \(\mcl\) is flat over \(Y\) because it is locally free over \(\OO_X\) and \(X\) is flat over \(Y\). By step 2, the fibre dimensions \[ h^0(X_y,\mcl_y)=2 \] are constant. Cohomology and base change T-COHBC therefore gives that \(f_*\mcl\) is locally free of rank \(2\) and commutes with restriction to every fibre.
4. For every fibre \(C=X_y\), the line bundle \(\mcl_y\) is isomorphic to\(\OO_{\PP^1}(1)\) after identifying \(C\cong\PP^1_{\kappa(y)}\), and itscomplete linear system defines an isomorphism\[ C\xrightarrow{\sim}\PP\bigl(H^0(C,\mcl_y)\bigr). \]
By step 2, \(\mcl_y\) has a nonzero section. Its zero divisor is an effective divisor of degree \(1\), hence a single \(\kappa(y)\)-rational point \(p\). A smooth proper genus-zero curve with a rational point is \(\PP^1_{\kappa(y)}\) D-VARSEVBRAUER, and under such an identification \[ \mcl_y\cong\OO_{\PP^1}(p)\cong\OO_{\PP^1}(1). \] The complete linear system of \(\OO_{\PP^1}(1)\) is the standard isomorphism to projective one-space.
5. The evaluation homomorphism\[ f^*\mce\longrightarrow\mcl \]is surjective and defines a \(Y\)-morphism\[ g:X\longrightarrow\PP_Y(\mce) \]whose restriction to every fibre is an isomorphism.
By step 3, restriction of the evaluation map to \(X_y\) is \[ H^0(X_y,\mcl_y)\tensor\OO_{X_y} \longrightarrow \mcl_y. \] By step 4 this is the usual evaluation map for \(\OO_{\PP^1}(1)\), so it is surjective. The cokernel of the global evaluation map is coherent and has zero restriction to every fibre; Nakayama’s lemma therefore implies that the cokernel is zero.
A quotient \[ f^*\mce\twoheadrightarrow\mcl \] defines the displayed morphism to the projective bundle. Base change in step 3 identifies its fibre map with the complete-linear-system map in step 4, hence \[ g_y:X_y\xrightarrow{\sim}\PP(\mce_y) \] for every \(y\).
6. The morphism \(g\) is an isomorphism. Thus a relative degree-one linebundle makes \(X\) a Zariski \(\PP^1\)-bundle.
The morphism \(g\) is proper because \(X\) is proper over \(Y\) and \(\PP_Y(\mce)\) is separated over \(Y\). Since every fibre map \(g_y\) is an isomorphism, \(g\) is quasi-finite. Proper and quasi-finite implies finite.
The map on the generic fibre is an isomorphism by step 5, so \(g\) is birational. Since \(Y\) is smooth and \(\mce\) is locally free, \[ \PP_Y(\mce) \] is smooth over the regular curve \(Y\), hence regular and in particular normal. A finite birational morphism onto a normal integral scheme is an isomorphism. Therefore \[ g:X\xrightarrow{\sim}\PP_Y(\mce). \]
7. Hence\[ \boxed{ X\cong\PP_Y(\mce)\text{ for a rank-two }\mce \iff X\text{ carries a relative degree-one line bundle}. } \]
Step 1 proves the forward implication and steps <1>2–<1>6 prove the reverse implication.
8. Tsen’s theorem supplies such a degree-one line bundle for thegeneric genus-zero fibre.
Let \[ K=k(Y) \] and let \(X_\eta\) be the generic fibre. It is a smooth proper genus-zero curve over \(K\), hence a one-dimensional Severi–Brauer variety. Tsen’s theorem gives \[ \operatorname{Br}(K)=0, \] so D-VARSEVBRAUER yields \[ X_\eta\cong\PP^1_K. \] In particular \(X_\eta\) has a \(K\)-rational point, which is a rational section \[ s_\eta:\eta\dashrightarrow X \] of \(f\).
For a closed point \(y\in Y\), the local ring \(\OO_{Y,y}\) is a DVR with fraction field \(K\). Properness of \(f\) and the valuative criterion extend \(s_\eta\) uniquely over \(\Spec\OO_{Y,y}\). Since \(X\to Y\) is of finite presentation, this morphism from the local scheme descends to a section over some open neighborhood of \(y\). The resulting local sections all agree on the generic point; separatedness of \(f\) makes them agree on their overlaps. They therefore glue to a global section \[ s:Y\longrightarrow X. \] Since \(f\) is a smooth relative curve, its section \(s\) is a regular closed immersion of codimension one. Thus \(s(Y)\) is an effective Cartier divisor meeting every fibre in one point. Therefore \[ \OO_X(s(Y)) \] has relative degree \(1\).
9. In particular, over a smooth curve \(Y\) over an algebraically closedfield, every such smooth genus-zero family is a Zariski \(\PP^1\)-bundle.
Step 8 constructs a relative degree-one line bundle, and step 7 applies the criterion.
10. Q.E.D.
Step 7 proves the requested equivalence, and steps <1>8–<1>9 use Tsen’s theorem to produce the relative degree-one line bundle.
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Dependencies
T-COHBCSemicontinuity, constancy of $\chi$, and cohomology and base changeuses