4_2x

IV.2: Hurwitz

2.1

Use (2.5.3) to show that \(\mathbf{P}^n\) is simply connected.

2.2 Classification of Curves of Genus 2 .

Fix an algebraically closed field \(k\) of characteristic \(\neq 2\).

• If \(X\) is a curve of genus 2 over \(k\), the canonical linear system \(|K|\) determines a finite morphism \(f: X \rightarrow \mathbf{P}^1\) of degree 2 (Ex. 1.7). Show that it is ramified at exactly 6 points, with ramification index 2 at each one. Note that \(f\) is uniquely determined, up to an automorphism of \(\mathbf{P}^1\), so \(X\) determines an (unordered) set of 6 points of \(\mathbf{P}^1\), up to an automorphism of \(\mathbf{P}^1\).

• Conversely, given six distinct elements \(\alpha_1, \ldots, \alpha_6 \in k\), let \(K\) be the extension of \(k(x)\) determined by the equation \(z^2=\left(x-\alpha_1\right) \cdots\left(x-\alpha_6\right)\). Let \(f: X \rightarrow \mathbf{P}^1\) be the corresponding morphism of curves. Show that \(g(X)=2\), the map \(f\) is the same as the one determined by the canonical linear system, and \(f\) is ramified over the six points \(x=\alpha_i\) of \(\mathbf{P}^1\), and nowhere else. (Cf. (II, Ex. 6.4).)

• Using (I, Ex. 6.6), show that if \(P_1, P_2, P_3\) are three distinct points of \(\mathbf{P}^1\), then there exists a unique \(\varphi \in\) Aut \(\mathbf{P}^1\) such that \(\varphi\left(P_1\right)=0, \varphi\left(P_2\right)=1, \varphi\left(P_3\right)=\infty\). Thus in (a), if we order the six points of \(\mathbf{P}^1\), and then normalize by sending the first three to \(0,1, x\), respectively, we may assume that \(X\) is ramified over \(0,1, \infty, \beta_1, \beta_2, \beta_3\), where \(\beta_1, \beta_2, \beta_3\) are three distinct elements of \(k, \neq 0,1\).

• Let \(\Sigma_6\) be the symmetric group on 6 letters. Define an action of \(\Sigma_6\) on sets of three distinct elements \(\beta_1, \beta_2, \beta_3\) of \(k, \neq 0,1\), as follows: reorder the set \(0,1, \infty, \beta_1, \beta_2, \beta_3\) according to a given element \(\sigma \in \Sigma_6\), then renormalise as in (c) so that the first three become \(0,1, \infty\) again. Then the last three are the new \(\beta_1^{\prime}, \beta_2^{\prime}, \beta_3^{\prime}\).

• Summing up, conclude that there is a one-to-one correspondence between the set of isomorphism classes of curves of genus 2 over \(k\), and triples of distinct elements \(\beta_1, \beta_2, \beta_3\) of \(k, \neq 0,1\), modulo the action of \(\Sigma_6\) described in (d). In particular, there are many non-isomorphic curves of genus 2 . We say that curves of genus 2 depend on three parameters, since they correspond to the points of an open subset of \(\mathbf{A}_k^3\) modulo a finite group.

2.3 Plane Curves.

Let \(X\) be a curve of degree \(d\) in \(\mathbf{P}^2\). For each point \(P \in X\), let \(T_P(X)\) be the tangent line to \(X\) at \(P\) (I, Ex. 7.3). Considering \(T_P(X)\) as a point of the dual projective plane \(\left(\mathbf{P}^2\right)^*\), the map \(P \rightarrow T_P(X)\) gives a morphism of \(X\) to its dual curve \(X^*\) in \(\left(\mathbf{P}^2\right)^*\) (I, Ex. 7.3).

Note that even though \(X\) is nonsingular, \(X^*\) in general will have singularities. We assume char \(k=0\) below.

• Fix a line \(L \subseteq \mathbf{P}^2\) which is not tangent to \(X\). Define a morphism \(\varphi: X \rightarrow L\) by \(\varphi(P)=T_P(X) \cap L\), for each point \(P \in X\). Show that \(\varphi\) is ramified at \(P\) if and only if either

  • \(P \in L\), or
  • \(P\) is an inflection point of \(X\), which means that the intersection multiplicity (I, Ex. 5.4) of \(T_P(X)\) with \(X\) at \(P\) is \(\geqslant 3\). Conclude that \(X\) has only finitely many inflection points.

• A line of \(\mathbf{P}^2\) is a multiple tangent of \(X\) if it is tangent to \(X\) at more than one point. It is a bitangent if it is tangent to \(X\) at exactly two points. If \(L\) is a multiple tangent of \(X\), tangent to \(X\) at the points \(P_1, \ldots, P_r\), and if none of the \(P_i\) is an inflection point, show that the corresponding point of the dual curve \(X^*\) is an ordinary \(r\)-fold point, which means a point of multiplicity \(r\) with distinct tangent directions (I, Ex. 5.3). Conclude that \(X\) has only finitely many multiple tangents.

• Let \(O \in \mathbf{P}^2\) be a point which is not on \(X\), nor on any inflectional or multiple tangent of \(X\). Let \(L\) be a line not containing \(O\). Let \(\psi: X \rightarrow L\) be the morphism defined by projection from \(O\). Show that \(\psi\) is ramified at a point \(\mathrm{P} \in X\) if and only if the line \(O P\) is tangent to \(X\) at \(P\), and in that case the ramification index is 2. Use Hurwitz’s theorem and (I, Ex. 7.2) to conclude that there are exactly \(d(d-1)\) tangents of \(X\) passing through \(O\). Hence the degree of the dual curve (sometimes called the class of \(X)\) is \(d(d-1)\).

• Show that for all but a finite number of points of \(X\), a point \(O\) of \(X\) lies on exactly \((d+1)(d-2)\) tangents of \(X\), not counting the tangent at \(O\).

• Show that the degree of the morphism \(\varphi\) of a. is \(d(d-1)\). Conclude that if \(d \geqslant 2\), then \(X\) has \(3 d(d-2)\) inflection points, properly counted. (If \(T_P(X)\) has intersection multiplicity \(r\) with \(X\) at \(P\), then \(P\) should be counted \(r-2\) times as an inflection point. If \(r=3\) we call it an ordinary inflection point.) Show that an ordinary inflection point of \(X\) corresponds to an ordinary cusp of the dual curve \(X^*\).

• Now let \(X\) be a plane curve of degree \(d \geqslant 2\), and assume that the dual curve \(X^*\) has only nodes and ordinary cusps as singularities (which should be true for sufficiently general \(X)\). Then show that \(X\) has exactly \(\frac{1}{2} d(d-2)(d-3)(d+3)\) bitangents. ¹

• For example, a plane cubic curve has exactly 9 inflection points, all ordinary. The line joining any two of them intersects the curve in a third one.

• A plane quartic curve has exactly 28 bitangents. (This holds even if the curve has a tangent with four-fold contact, in which case the dual curve \(X^*\) has a tacnode.)

2.4 A Funny Curve in Characteristic \(p\).

Let \(X\) be the plane quartic curve \(x^3 y+y^3 z+ z^3 x = 0\) over a field of characteristic 3 . Show that \(X\) is nonsingular, every point of \(X\) is an inflection point, the dual curve \(X^*\) is isomorphic to \(X\), but the natural map \(X \rightarrow X^*\) is purely inseparable.

2.5 Automorphisms of a Curve of Genus \(\geqslant 2\).

Prove the theorem of Hurwitz that a curve \(X\) of genus \(g \geqslant 2\) over a field of characteristic 0 has at most \(84(g-1)\) automorphisms.

We will see later (Ex. 5.2) or (V, Ex. 1.11) that the group \(G=\) Aut \(X\) is finite. So let \(G\) have order \(n\). Then \(G\) acts on the function field \(K(X)\). Let \(L\) be the fixed field. Then the field extension \(L \subseteq K(X)\) corresponds to a finite morphism of curves \(f: X \rightarrow Y\) of degree \(n\).

• If \(P \in X\) is a ramification point, and \(e_P=r\), show that \(f^{-1} f(P)\) consists of exactly \(n / r\) points, each having ramification index \(r\). Let \(P_1, \ldots, P_s\) be a maximal set of ramification points of \(X\) lying over distinct points of \(Y\), and let \(e_{P_1}=r_i\). Then show that Hurwitz’s theorem implies that \begin{align*} {2 g-2 \over n } =2 g(Y)-2+\sum_{i=1}^s\left(1- {1\over r_i}\right) \end{align*}

• Since \(g \geqslant 2\), the left hand side of the equation is \(>0\). Show that if \(g(Y) \geqslant 0\), \(s \geqslant 0, r_i \geqslant 2, i=1, \ldots, s\) are integers such that \begin{align*} 2g(Y)-2+\sum_{i=1}^s\left(1- {1\over r_i}\right)>0, \end{align*} then the minimum value of this expression is \(1 / 42\). Conclude that \(n \leqslant 84(g-1)\). ²

2.6 \(f_*\) for Divisors.

Let \(f: X \rightarrow Y\) be a finite morphism of curves of degree \(n\). We define a homomorphism \(f_*: \operatorname{Div} X \rightarrow\) Div \(Y\) by \(f_*\left(\sum n_i P_i\right)=\sum n_i f\left(P_i\right)\) for any divisor \(D=\sum n_i P_i\) on \(X\).

• For any locally free sheaf \(\mathscr{E}\) on \(Y\), of rank \(r\), we define \(\operatorname{det}\mathscr{E}=\wedge^r \mathscr{E} \in \operatorname{Pic}Y\) (II, Ex. 6.11). In particular, for any invertible sheaf \(\mathscr{M}\) on \(X, f_* \mathscr{M}\) is locally free of rank \(n\) on \(Y\), so we can consider \(\operatorname{det}f_* \mathscr{M} \in \operatorname{Pic}Y\). Show that for any divisor \(D\) on \(X\), ³ \begin{align*} \operatorname{det}\left(f_* \mathscr{L}(D)\right) \cong\left(\operatorname{det} f_*\left({\mathcal{O}}_X\right) \otimes \mathscr{L}\left(f_* D\right) .\right. \end{align*} Note in particular that \(\operatorname{det}\left(f_* \mathscr{L}(D)\right) \neq \mathscr{L}\left(f_* D\right)\) in general!

• Conclude that \(f_* D\) depends only on the linear equivalence class of \(D\), so there is an induced homomorphism \(f_*: \operatorname{Pic}X \rightarrow \operatorname{Pic}Y\). Show that \(f_* f^*: \operatorname{Pic}Y \rightarrow \operatorname{Pic}Y\) is just multiplication by \(n\).

• Use duality for a finite flat morphism (III, Ex. 6.10) and (III, Ex. 7.2) to show that \begin{align*}\operatorname{det}f_* \Omega_X \cong \qty{ \operatorname{det}f_* {\mathcal{O}}_X}^{-1}\otimes \Omega_Y^{\otimes n}.\end{align*}

• Now assume that \(f\) is separable, so we have the ramification divisor \(R\). We define the branch divisor \(B\) to be the divisor \(f_* R\) on \(Y\). Show that \begin{align*}\left(\operatorname{det} f_* \mathcal{O}_X\right)^2 \cong \mathscr{L}(-B).\end{align*}

2.7 Etale Covers of Degree 2.

Let \(Y\) be a curve over a field \(k\) of characteristic \(\neq 2\). We show there is a one-to-one correspondence between finite étale morphisms \(f: X \rightarrow Y\) of degree 2, and 2-torsion elements of \(\operatorname{Pic}Y\), i.e., invertible sheaves \(\mathscr{L}\) on \(Y\) with \(\mathscr{L}^2 \cong \mathcal{O}_Y\).

• Given an étale morphism \(f: X \rightarrow Y\) of degree 2, there is a natural map \({\mathcal{O}}_Y \rightarrow\) \(f_* {\mathcal{O}}_X\). Let \(\mathscr{L}\) be the cokernel. Then \(\mathscr{L}\) is an invertible sheaf on \(Y, \mathscr{L} \cong \operatorname{det} f_* {\mathcal{O}}_X\), and so \(\mathscr{L}^2 \cong \mathcal{O}_Y\) by (Ex. 2.6). Thus an étale cover of degree 2 determines a 2-torsion element in \(\operatorname{Pic}Y\).

• Conversely, given a 2-torsion element \(\mathscr{L}\) in \(\operatorname{Pic}Y\), define an \({\mathcal{O}}_Y{\hbox{-}}\)algebra structure on \(\mathcal{O}_Y \oplus \mathscr{L}\) by \(\langle a, b\rangle \cdot\left\langle a^{\prime}, b^{\prime}\right\rangle=\left\langle a a^{\prime}+\varphi\left(b \otimes b^{\prime}\right), a b^{\prime}+a^{\prime} b\right\rangle\), where \(\varphi\) is an isomorphism of \(\mathscr{L} \otimes \mathscr{L} \rightarrow \mathcal{O}_Y\). Then take \(X=\operatorname{Spec}\left({\mathcal{O}}_Y \oplus \mathscr{L}\right)\) (II, Ex. 5.17). Show that \(X\) is an étale cover of \(Y\).

• Show that these two processes are inverse to each other. ^{4 5}