Base points and very ampleness on a ruled surface over an elliptic curve

Since \[ K_C\cong\OO_C, \] Serre duality gives \[ H^1(C,M)^\vee\cong H^0(C,M^{-1}). \] If \(\deg M>0\), then \(M^{-1}\) has negative degree and no nonzero section, so \(M\) is nonspecial.

The base-point-free and very-ample degree bounds are the genus-one cases of the standard curve criteria T-D8TUX: degree at least \(2g=2\) is base-point free, and degree at least \(2g+1=3\) is very ample.

One has \[ \deg\mfb=m\ge2 \] and \[ \deg(\mfb+\mfe)=m-e\ge2. \] By step 1, both complete linear systems are base-point free, and \(\mfb\) is nonspecial. Hartshorne V.2.11, proved on P-AGH5211VERYAMPLESECT, therefore gives a section \[ D\sim C_0+\mfb f \] and says that \(|L|\) has no base points.

By definition of the invariant of a normalized ruled surface, \[ e=-\deg\det\mathcal E. \] Thus \(e=-1\) gives \[ \deg\mathcal E=\deg\det\mathcal E=1. \] Hartshorne V.2.8, proved on P-AGH528STABLEBUNDLE, says that a normalized rank-two bundle is stable exactly when its degree is positive. Hence \(\mathcal E\) is stable.

Normalization gives an exact sequence \[ 0\longrightarrow\OO_C \longrightarrow\mathcal E \longrightarrow\det\mathcal E \longrightarrow0. \] The corresponding quotient section is \(C_0\). The divisor formula for a section of a projective bundle, as used in P-AGH527ELLRULEDSECT, gives \[ \OO_X(C_0)\cong\OO_X(1) \] because the kernel is \(\OO_C\). Therefore \[ L \cong \OO_X(1)\tensor\pi^*\OO_C(\mfb). \] For Hartshorne’s quotient convention, \[ \pi_*\OO_X(1)\cong\mathcal E, \] so the projection formula gives the displayed formula for \(\pi_*L\).

Serre duality and \(K_C\cong\OO_C\) give \[ H^1(C,F)^\vee\cong H^0(C,F^\vee). \] The dual bundle \(F^\vee\) is stable of negative degree. A nonzero section would give a subsheaf \(\OO_C\subseteq F^\vee\); after saturation this would give a line subbundle of degree at least \(0\), contradicting stability because \[ 0>\mu(F^\vee). \] Thus \(H^0(C,F^\vee)=0\) and hence \(H^1(C,F)=0\).

Put \[ F=\mathcal E\tensor\OO_C(\mfb). \] Twisting preserves stability, and \[ \deg F = \deg\mathcal E+2m = 1+2m. \] For every point \(P\in C\), the stable bundle \[ F(-P) \] has degree \[ 1+2m-2=2m-1>0. \] Step 4 therefore gives \[ H^1(C,F(-P))=0. \] The exact sequence \[ 0\to F(-P)\to F\to F|_P\to0 \] shows that \[ H^0(C,F)\twoheadrightarrow F|_P \] for every \(P\).

By step 3 and cohomology and base change, \[ F|_P \cong H^0\bigl(F_P,L|_{F_P}\bigr), \qquad L|_{F_P}\cong\OO_{\PP^1}(1). \] Thus the complete system \(|L|\) restricts to the complete \(|\OO_{\PP^1}(1)|\) on every ruling fibre. In particular \(|L|\) has no base points.

The incidence argument of P-AGH5211VERYAMPLESECT, step 4, now applies verbatim: the sections vanishing identically on a fixed fibre form a codimension-two subspace of \(H^0(X,L)\), so a general divisor in \(|L|\) contains no fibre. Since \[ L\cdot f=1, \] such a divisor is a section of \(\pi\). This proves part (a) in the exceptional \(e=-1\) case.

The two line bundles on the base have degrees \[ \deg\mfb=m\ge3, \qquad \deg(\mfb+\mfe)=m-e\ge3. \] Hence both are very ample by step 1.

For every point \(P\in C\), \[ \deg(\mfb-P)=m-1>0 \] and \[ \deg(\mfb+\mfe-P)=m-e-1>0. \] Step 1 makes both line bundles nonspecial. Hartshorne V.2.11(b), proved on P-AGH5211VERYAMPLESECT, therefore says that \[ C_0+\mfb f \] is very ample.

First let \(Z\) be a length-two subscheme contained in one fibre \(F_P\). The bundle \(F(-P)\) has positive degree \(2m-1\), so step 4 gives \[ H^1(C,F(-P))=0. \] Thus global sections of \(L\) restrict surjectively to \[ H^0(F_P,\OO_{F_P}(1)), \] and \(\OO_{\PP^1}(1)\) separates \(Z\).

Over the Artinian scheme \(A\), the subscheme \(Z\cong A\) is a section of \[ \PP(\mathcal E|_A)\longrightarrow A. \] It therefore corresponds to an invertible quotient of \[ F|_A=\mathcal E|_A\tensor\OO_A(\mfb). \] Since \(A\) is affine, taking global sections of that quotient is surjective. Consequently \[ H^0(X,L)\longrightarrow H^0(Z,L|_Z) \] is surjective. Thus every length-two subscheme is separated.

By step 8, the complete linear system of \(L\) separates every zero-dimensional subscheme of length \(2\). The closed-immersion criterion T-DIVMAPPN therefore makes \(L\) very ample.

The restriction of a very ample line bundle to a closed subscheme is very ample. In particular, \[ L|_{C_0} \cong \OO_C(\mfb+\mfe) \] must be very ample on the elliptic curve \(C_0\cong C\).

Its degree is \[ m-e. \] A very ample line bundle on an elliptic curve has degree at least \(3\): Riemann–Roch gives \(h^0(M)=\deg M\) for positive-degree \(M\), while an embedding of a positive-genus curve requires at least three independent sections. Hence \[ m-e\ge3, \] or equivalently \[ m\ge e+3. \]

Step 6 proves part (a), and steps <1>7–<1>11 prove part (b).