Let \(C\) be a curve of genus \(g\), and let \(X\) be the ruled surface \(C \times \PP^1\). We consider the question, for what integers \(s \in \ZZ\) does there exist a section \(D\) of \(X\) with \(D^2=s\)? First show that \(s\) is always an even integer, say \(s=2 r\).
Show that \(r=0\) and any \(r \geqslant g+1\) are always possible. Cf. (V, Ex. 6.8).
If \(g=3\), show that \(r=1\) is not possible, and just one of the two values \(r=2,3\) is possible, depending on whether \(C\) is hyperelliptic or not.
Solution
Let \[ p_1:C\times\PP^1\longrightarrow C \] be the first projection.
1. Every section \(D\) of \(p_1\) is the graph \(\Gamma_u\) of a unique morphism\[ u:C\longrightarrow\PP^1. \]
If \[ \sigma:C\longrightarrow C\times\PP^1 \] is a section, then \[ p_1\circ\sigma=\id_C. \] Writing \[ u=p_2\circ\sigma, \] one has \[ \sigma(P)=(P,u(P)), \] so the image of \(\sigma\) is the graph of \(u\). Conversely every graph gives a section. Uniqueness is immediate from the second projection.
2. For \(D=\Gamma_u\),\[ \mcn_{D/(C\times\PP^1)}\cong u^*T_{\PP^1}. \]
Restrict the tangent bundle of the product to the graph. Under the identification \(D\cong C\) this gives \[ T_{C\times\PP^1}|_D \cong T_C\oplus u^*T_{\PP^1}. \] The differential of the graph embedding is \[ T_C\longrightarrow T_C\oplus u^*T_{\PP^1}, \qquad v\longmapsto(v,du(v)). \] The map \[ T_C\oplus u^*T_{\PP^1}\longrightarrow u^*T_{\PP^1}, \qquad (v,w)\longmapsto w-du(v) \] is surjective and has exactly this graph as its kernel. Hence the quotient in the normal sequence is \(u^*T_{\PP^1}\).
3. Every section has even, nonnegative self-intersection, namely\[ \boxed{D^2=2\deg u}. \]
For a smooth curve on a smooth surface, self-intersection is the degree of the normal bundle. By step 2, \[ D^2 = \deg u^*T_{\PP^1}. \] Since \[ T_{\PP^1}\cong\OO_{\PP^1}(2), \] one obtains \[ D^2=2\deg u. \] For a constant map set \(\deg u=0\); for a nonconstant map the degree is a positive integer. Thus every possible self-intersection is of the form \(s=2r\) with \(r\ge0\).
4. The value \(r=0\) always occurs.
Take \(u:C\to\PP^1\) to be constant. Its graph is the horizontal section \[ C\times\{Q\}, \] and step 3 gives self-intersection zero.
5. For every integer\[ r\ge g+1 \]there is a morphism \(u:C\to\PP^1\) of degree \(r\).
Exercise IV.6.8, proved on P-AGH468BASEPOINTFREENONSPECIAL, gives a nonspecial line bundle \(L\) of degree \(r\) whose complete linear system is base-point free. Riemann–Roch gives \[ h^0(C,L)=r+1-g\ge2. \]
Choose a nonzero section \(s_0\in H^0(C,L)\). Its zero divisor has finite support. For each point \(P\) in that support, global generation says that the subspace of sections vanishing at \(P\) is a proper hyperplane in \(H^0(C,L)\). Because the ground field is infinite, a finite union of proper hyperplanes cannot cover \(H^0(C,L)\). Hence one can choose \(s_1\) nonzero at every zero of \(s_0\). The two-dimensional space \[ V=\langle s_0,s_1\rangle \] is therefore base-point free.
By the degree statement for pencil maps T-DIVMAPPN, \(V\) defines a finite morphism \[ u:C\longrightarrow\PP^1 \] of degree \[ \deg L=r. \]
6. The values\[ \boxed{r=0\quad\text{and every }r\ge g+1} \]always occur.
3. This proves part (a).
7. Assume now \(g=3\). The value \(r=1\) is impossible.
If \(r=1\) occurred, step 3 would give a degree-one morphism \[ u:C\longrightarrow\PP^1. \] A finite degree-one morphism of nonsingular projective curves is birational, so uniqueness of the smooth projective model T-CRVMINMOD would imply \[ C\cong\PP^1, \] contradicting \(g(C)=3\).
8. If \(C\) is hyperelliptic of genus three, then \(r=2\) occurs.
By definition, a hyperelliptic curve has a degree-two morphism \[ h:C\longrightarrow\PP^1 \] D-CRVHYP. Its graph therefore has self-intersection four by step 3, so \(r=2\) occurs.
9. If \(C\) is hyperelliptic of genus three, then \(r=3\) does not occur.
Let \(A\) denote the unique hyperelliptic \(g^1_2\). Suppose for contradiction that there were a degree-three morphism \(u:C\to\PP^1\). Then \[ L=u^*\OO_{\PP^1}(1) \] has degree three and is generated by the pulled-back two-dimensional space of sections. In particular \[ h^0(C,L)\ge2. \]
Riemann–Roch on the genus-three curve gives \[ h^0(C,L)-h^0(C,K_C\tensor L^{-1})=1. \] Hence \(K_C\tensor L^{-1}\) has a nonzero section. It has degree one, so \[ K_C\tensor L^{-1}\cong\OO_C(Q) \] for some point \(Q\in C\). Therefore \[ L\cong K_C(-Q). \]
For a genus-three hyperelliptic curve, D-CRVHYP gives \[ K_C\sim2A. \] Let \(Q'\) be the residual point in the hyperelliptic fibre through \(Q\), so \[ Q+Q'\sim A \] (possibly \(Q'=Q\) at a ramification point). Then \[ L\sim2A-Q\sim A+Q'. \] Now \(h^0(C,A)=2\). Since \(K_C\tensor L^{-1}\cong\OO_C(Q)\) has exactly one section on the nonrational curve \(C\), Riemann–Roch gives \[ h^0(C,L)=2. \] The natural inclusion \[ H^0(C,A)\hookrightarrow H^0(C,A+Q')=H^0(C,L) \] is therefore an isomorphism. Every section of \(L\) consequently vanishes at the added point \(Q'\), so \(Q'\) is a base point of \(|L|\). This contradicts the fact that the two pulled-back sections from \(\PP^1\) generate \(L\). Thus no degree-three morphism exists.
10. If \(C\) is nonhyperelliptic of genus three, then \(r=2\) does not occur.
A degree-two morphism to \(\PP^1\) is exactly a \(g^1_2\), hence would make \(C\) hyperelliptic by D-CRVHYP. Thus no such morphism exists.
11. If \(C\) is nonhyperelliptic of genus three, then \(r=3\) occurs.
By the genus-three canonical model FE-CRVLOWG, the canonical linear system embeds \(C\) as a smooth plane quartic \[ C\hookrightarrow\PP^2. \] Fix a point \(P\in C\) and consider the pencil of lines through \(P\). On \(C\) each line cuts a canonical divisor of degree four containing \(P\), so after removing the fixed point \(P\) the pencil has degree three.
It has no base point. For \(Q\ne P\), choose a line through \(P\) that does not pass through \(Q\). At \(P\), choose a line through \(P\) that is not tangent to the smooth quartic; its intersection multiplicity with \(C\) at \(P\) is one, so after subtracting the fixed copy of \(P\) the residual divisor does not contain \(P\). Thus this is a base-point-free \(g^1_3\), hence by T-DIVMAPPN it defines a degree-three morphism \[ C\longrightarrow\PP^1. \] Step 3 then gives a section of self-intersection six, so \(r=3\) occurs.
12. For genus three, exactly one of \(r=2,3\) occurs:\[ \boxed{ \begin{array}{c|cc} &r=2&r=3\\ \hline C\text{ hyperelliptic}&\text{yes}&\text{no}\\ C\text{ nonhyperelliptic}&\text{no}&\text{yes} \end{array}} \]
This is exactly the combination of steps <1>8–<1>11. Together with step 7, it proves part (b).
13. Q.E.D.
Steps <1>1–<1>3 identify possible self-intersections with twice the degrees of maps to \(\PP^1\); steps <1>4–<1>6 prove part (a), and steps <1>7–<1>12 prove part (b).