Sections of self-intersection one on the elliptic ruled surface

For the elliptic ruled surface of (2.11.6), the invariant is \[ e=-1. \] By the definition of the invariant of a normalized ruled surface, \[ e=-\deg\det\mathcal E. \] Hence \[ \deg\mathcal E=\deg\det\mathcal E=1. \]

Set \[ \mathcal F_P=\mathcal E^\vee\tensor\OO_C(P). \] Since \(\mathcal E\) has rank two and degree one, \[ \deg\mathcal F_P=-1+2=1. \] On the elliptic curve \(C\), Riemann–Roch gives \[ \chi(\mathcal F_P)=\deg\mathcal F_P=1. \] Also \(K_C\cong\OO_C\), so Serre duality gives \[ H^1(C,\mathcal F_P)^\vee \cong H^0\qty(C,\mathcal F_P^\vee) = H^0\qty(C,\mathcal E\tensor\OO_C(-P)). \] The line bundle \(\OO_C(-P)\) has negative degree, and \(\mathcal E\) is normalized. Therefore the last space is zero. Hence \[ h^0(C,\mathcal F_P)=\chi(\mathcal F_P)=1. \] Finally, \[ H^0(C,\mathcal F_P) = \Hom\qty(\mathcal E,\OO_C(P)). \]

Let the image be \[ \OO_C(P-Z)\subseteq\OO_C(P) \] for an effective divisor \(Z\). If \(Z\ne0\), the kernel is an invertible subsheaf \(\mathcal N\subseteq\mathcal E\) with \[ \deg\mathcal N = \deg\mathcal E-\deg\OO_C(P-Z) = 1-(1-\deg Z) = \deg Z>0. \] Then the inclusion \(\mathcal N\hookrightarrow\mathcal E\) gives a nonzero section of \[ \mathcal E\tensor\mathcal N^{-1}, \] where \(\deg\mathcal N^{-1}<0\), contradicting normalization. Thus \(Z=0\), and the map is surjective.

By steps <1>2–<1>3, there is a surjection \[ \mathcal E\twoheadrightarrow\OO_C(P), \] unique up to a nonzero scalar. Under the quotient convention for \(\PP(\mathcal E)\), a line-bundle quotient of \(\mathcal E\) gives a section of \(\pi\). Multiplying the quotient map by a scalar does not change the section, so the resulting section \(C_P\) is unique.

The normal bundle of the section determined by the quotient \(\mathcal E\twoheadrightarrow\mathcal L\) is \[ \mathcal N^\vee\tensor\mathcal L. \] Indeed this follows fibrewise from the tangent space to the projective line of one-dimensional quotients, and the local splittings of the exact sequence glue the resulting identification. Therefore \[ D^2 = \deg\qty(\mathcal N^\vee\tensor\mathcal L) = \deg\mathcal L-\deg\mathcal N. \] Since \[ \deg\mathcal N+\deg\mathcal L=\deg\mathcal E=1 \] by step 1, one obtains \[ D^2=2\deg\mathcal L-1. \]

If \(D^2=1\), step 5 gives \[ 1=2\deg\mathcal L-1, \] so \(\deg\mathcal L=1\). On an elliptic curve, Riemann–Roch and Serre duality give \[ h^0(C,\mathcal L)=1 \] for every degree-one line bundle \(\mathcal L\). Its unique nonzero section has an effective divisor consisting of one point \(P\), so \[ \mathcal L\cong\OO_C(P). \] Thus \(D=C_P\) by uniqueness in step 4.

Conversely, for the quotient to \(\OO_C(P)\) one has \(\deg\mathcal L=1\), so step 5 gives \[ C_P^2=1. \]

Let \(p,q:C\times C\to C\) be the first and second projections, and let \(\Delta\subseteq C\times C\) be the diagonal. The line bundle \[ \OO_{C\times C}(\Delta) \] restricts on \(C\times\{P\}\) to \(\OO_C(P)\). Put \[ \mathcal G = p^*\mathcal E^\vee\tensor\OO_{C\times C}(\Delta). \] By step 2 and its proof, every fibre of \(q\) satisfies \[ h^0(\mathcal G|_{C\times\{P\}})=1, \qquad h^1(\mathcal G|_{C\times\{P\}})=0. \] Cohomology and base change therefore makes \[ \mathcal R=q_*\mathcal G \] an invertible sheaf on the parameter curve \(C\). The universal evaluation map is equivalently a morphism \[ p^*\mathcal E\tensor q^*\mathcal R \longrightarrow \OO_{C\times C}(\Delta) \] whose restriction over \(P\) is the unique nonzero map \(\mathcal E\to\OO_C(P)\) up to scalar. By step 3 every fibre map is surjective, hence this is a relative line-bundle quotient. Projectivizing it gives a family of sections of \(X\to C\) over the second copy of \(C\), with fibre \(C_P\) over \(P\). Thus the self-intersection-one sections form a one-dimensional algebraic family parametrized by \(C\).

Let \[ 0\longrightarrow\mathcal N_P \longrightarrow\mathcal E \longrightarrow\OO_C(P) \longrightarrow0 \] be the quotient defining \(C_P\). The composite \[ \pi^*\mathcal N_P \longrightarrow \pi^*\mathcal E \longrightarrow \OO_X(1) \] is a section of \[ \OO_X(1)\tensor\pi^*\mathcal N_P^{-1}, \] and its zero divisor is exactly \(C_P\). Hence \[ \OO_X(C_P) \cong \OO_X(1)\tensor\pi^*\mathcal N_P^{-1}. \] Taking determinants in the defining exact sequence gives \[ \mathcal N_P \cong \det\mathcal E\tensor\OO_C(-P). \] Thus \[ \OO_X(C_P) \cong \OO_X(1)\tensor\pi^*\qty((\det\mathcal E)^{-1}\tensor\OO_C(P)). \] Dividing the formulas for \(P\) and \(Q\) gives the displayed identity.

Suppose \[ C_P\sim C_Q. \] Step 8 gives \[ \pi^*\OO_C(P-Q)\cong\OO_X. \] Pulling back along any section of \(\pi\) shows \[ \OO_C(P-Q)\cong\OO_C. \] Thus \[ \OO_C(P)\cong\OO_C(Q). \] Each degree-one line bundle on the elliptic curve has a one-dimensional space of sections, so its unique effective divisor is determined by the line bundle. Hence \(P=Q\). Therefore distinct parameters give non-linearly-equivalent sections.

Steps <1>4–<1>7 identify the self-intersection-one sections with an algebraic family parametrized by \(C\), and steps <1>8–<1>9 prove that no two distinct members are linearly equivalent.