Lemma 11.13.18FE2AABC7334

Proof. Over a trivializing chart of the base, sections of $\mathcal O_ X(aS)$ are homogeneous polynomials of degree $a$ in the fiber coordinates. Their transition law gives

and twisting by the base gives the formula. The first cohomology of $\mathcal O_{\mathbb P^1}(a)$ on each fiber vanishes, which gives the assertion on the higher direct image. If $a<0$, restriction to a general ruling fiber excludes a nonzero section. For $a\geq 0$, the direct-image formula gives a nonzero section exactly when $b\geq 0$.

Suppose that $a\geq 1$ and $b>ae$. All line bundles in the direct image then have degree at least one. Fiber polynomials of degree $a$ separate points and tangent directions within a fiber. To separate points over different base points, choose a fiber monomial that is nonzero at the first point and a coefficient section vanishing at the second base point but not at the first. At any point, choose such a nonzero monomial and a coefficient section with a simple zero at its base point; the resulting global section vanishes there, has nonzero derivative in a prescribed base direction and has zero vertical derivative. The complete linear system therefore separates points and first jets, so it defines an embedding. The formulas for $H^2$ and $h^0(H)$ follow from the intersection numbers and from the direct image $\mathcal O(3e+1)\oplus \mathcal O(2e+1)$. Finally, the relative Euler sequence gives $K_{X/\mathbb P^1}=-2S-ef$, and adding the pullback of the canonical class of $\mathbb P^1$ proves the last identity. $\square$