IV: Curves

IV.1: Riemann-Roch

1.1.

Let \(X\) be a curve, and let \(P \in X\) be a point. Then there exists a nonconstant rational function \(f \in K(X)\), which is regular everywhere except at \(P\).

1.2.

Again let \(X\) be a curve, and let \(P_1, \ldots P_r \in X\) be points. Then there is a rational function \(f \in K(X)\) having poles (of some order) at each of the \(P_1\), and regular elsewhere.

1.3.

Let \(X\) be an integral, separated, regular, one-dimensional scheme of finite type over \(k\), which is not proper over \(k\). Then \(X\) is affine. ¹

1.4.

Show that a separated, one-dimensional scheme of finite type over \(k\), none of whose irreducible components is proper over \(k\), is affine. ²

1.5.

For an effective divisor \(D\) on a curve \(X\) of genus \(g\), show that \(\operatorname{dim}|D| \leqslant \operatorname{deg} D\). Furthermore, equality holds if and only if \(D=0\) or \(g=0\).

1.6.

Let \(X\) be a curve of genus \(g\). Show that there is a finite morphism \(f: X \rightarrow \mathbf{P}^1\) of degree ³ \(\leqslant g+1\).

1.7.

A curve \(X\) is called hyperelliptic if \(g \geqslant 2\) and there exists a finite morphism \(f: X \rightarrow \mathbf{P}^1\) of degree 2.

• If \(X\) is a curve of genus \(g=2\), show that the canonical divisor defines a complete linear system \(|K|\) of degree 2 and dimension 1, without base points. Use (II, 7.8.1) to conclude that \(X\) is hyperelliptic.

• Show that the curves constructed in (1.1.1) all admit a morphism of degree 2 to \(\mathbf{P}^1\). Thus there exist hyperelliptic curves of any genus \(g \geqslant 2\). ⁴

1.8. \(p_a\) of a Singular Curve.

Let \(X\) be an integral projective scheme of dimension 1 over \(k\), and let \(\tilde{X}\) be its normalization (II, Ex. 3.8). Then there is an exact sequence of sheaves on \(X\), \begin{align*} 0 \rightarrow {\mathcal{O}}_X \rightarrow f_* {\mathcal{O}}_X \rightarrow \sum_{P \in X} \tilde{\mathcal{O}}_P/{\mathcal{O}}_P \rightarrow 0 \end{align*} where \(\tilde{\mathcal{O}}_P\) is the integral closure of \({\mathcal{O}}_P\). For each \(P \in X\), let \(\delta_P=\operatorname{length}(\tilde{\mathcal{O}}_P/{\mathcal{O}}_P)\).

• Show that \(p_a(X)=p_a(\tilde{X})+\sum_{p \in X} \delta_p\). ⁵

• If \(p_a(X)=0\), show that \(X\) is already nonsingular and in fact isomorphic to \(\mathbf{P}^1\). ⁶

• * If \(P\) is a node or an ordinary cusp (I, Ex. 5.6, Ex. 5.14), show that \(\delta_P=1\). ⁷

1.9. * Riemann-Roch for Singular Curves.

Let \(X\) be an integral projective scheme of dimension 1 over \(k\). Let \(X_{\mathrm{reg}}\) be the set of regular points of \(X\).

• Let \(D=\sum n_i P_i\) be a divisor with support in \(X_{\mathrm{reg}}\), i.e., all \(P_i \in X_{\mathrm{reg}}\). Then define deg \(D=\sum n_i\). Let \(\mathscr{L}(D)\) be the associated invertible sheaf on \(X\), and show that \begin{align*} \chi(\mathscr{L}(D))=\operatorname{deg} D+1-p_a . \end{align*}

• Show that any Cartier divisor on \(X\) is the difference of two very ample Cartier divisors. ⁸

• Conclude that every invertible sheaf \(\mathscr{L}\) on \(X\) is isomorphic to \(\mathscr{L}(D)\) for some divisor \(D\) with support in \(X_{\mathrm{reg}}\).

• Assume furthermore that \(X\) is a locally complete intersection in some projective space. Then by (III, 7.11) the dualizing sheaf \(\omega_X\) is an invertible sheaf on \(X\), so we can define the canonical divisor \(K\) to be a divisor with support in \(X_{\mathrm{reg}}\) corresponding to \(\omega_X\). Then the formula of a. becomes \begin{align*} l(D)-l(K-D)=\operatorname{deg} D+1-p_a \end{align*}

1.10.

Let \(X\) be an integral projective scheme of dimension 1 over \(k\), which is locally complete intersection, and has \(p_a=1\). Fix a point \(P_0 \in X_{\text {reg. }}\). Imitate (1.3.7) to show that the map \(P \rightarrow \mathscr{L}\left(P-P_0\right)\) gives a one-to-one correspondence between the points of \(X_{\mathrm{reg}}\) and the elements of the group \(\operatorname{Pic}X\). ⁹