Chapter 8

Let \(X=C=\mathbb{P}^{1}, k(X)=k(t)\), where \(t=X_{1} / X_{2}, X_{1}, X_{2}\) homogeneous coordinates on \(\mathbb{P}^{1}\). (a) Calculate \(\operatorname{div}(t)\). (b) Calculate \(\operatorname{div}(f / g), f, g\) relatively prime in \(k[t]\). (c) Prove Proposition 1 directly in this case.

Let \(X=C=V\left(Y^{2} Z-X(X-Z)(X-\lambda Z)\right) \subset \mathbb{P}^{2}, \lambda \in k, \lambda \neq 0,1\). Let \(x=X / Z\), \(y=Y / Z \in K ; K=k(x, y) .\) Calculate \(\operatorname{div}(x)\) and \(\operatorname{div}(y)\).

Let \(C=X\) be a nonsingular cubic. (a) Let \(P, Q \in C\). Show that \(P \equiv Q\) if and only if \(P=Q\). (Hint: Lines are adjoints of degree 1.) (b) Let \(P, Q, R, S \in C\). Show that \(P+Q \equiv R+S\) if and only if the line through \(P\) and \(Q\) intersects the line through \(R\) and \(S\) in a point on \(C\) (if \(P=Q\) use the tangent line). (c) Let \(P_{0}\) be a fixed point on \(C\), thus defining an addition \(\oplus\) on \(C\) (Chapter 5, Section 6). Show that \(P \oplus Q=R\) if and only if \(P+Q=R+P_{0}\). Use this to give another proof of Proposition 4 of \(\$ 5.6\).

Let \(C\) be a nonsingular quartic, \(P_{1}, P_{2}, P_{3} \in C\). Let \(D=P_{1}+P_{2}+P_{3} .\) Let \(L\) and \(L^{\prime}\) be lines such that \(L . \cdot C=P_{1}+P_{2}+P_{4}+P_{5}, L^{\prime} \cdot C=P_{1}+P_{3}+P_{6}+P_{7} .\) Suppose these seven points are distinct. Show that \(D\) is not linearly equivalent to any other effective divisor. (Hint: Apply the residue theorem to the conic \(L L^{\prime} .\) ) Investigate in a similar way other divisors of small degree on quartics with various types of multiple points.

Let \(D(X)\) be the group of divisors on \(X, D_{0}(X)\) the subgroup consisting of divisors of degree zero, and \(P(X)\) the subgroup of \(D_{0}(X)\) consisting of divisors of rational functions. Let \(C_{0}(X)=D_{0}(X) / P(X)\) be the quotient group. It is the divisor class group on \(X\). (a) If \(X=\mathbb{P}^{1}\), then \(C_{0}(X)=0\). (b) Let \(X=C\) be a nonsingular cubic. Pick \(P_{0} \in C\), defining \(\oplus\) on \(C\). Show that the map from \(C\) to \(C_{0}(X)\) that sends \(P\) to the residue class of the divisor \(P-P_{0}\) is an isomorphism from \((C, \oplus)\) onto \(C_{0}(X)\).

If \(D \leq D^{\prime}\), then \(l\left(D^{\prime}\right) \leq l(D)+\operatorname{deg}\left(D^{\prime}-D\right)\), i.e., \(\operatorname{deg}(D)-l(D) \leq \operatorname{deg}\left(D^{\prime}\right)-l\left(D^{\prime}\right)\).

Let \(X=\mathbb{P}^{1}, t\) as in Problem 8.1. Calculate \(L\left(r(t)_{0}\right)\) explicitly, and show that \(l\left(r(t)_{0}\right)=r+1\)

Let \(D\) be a divisor. Show that \(l(D)>0\) if and only if \(D\) is linearly equivalent to an effective divisor.

Show that \(\operatorname{deg}(D)=0\) and \(l(D)>0\) are true if and only if \(D \equiv 0\).

Suppose \(l(D)>0\), and let \(f \neq 0, f \in L(D)\). Show that \(f \notin L(D-P)\) for all but a finite number of \(P\). So \(l(D-P)=l(D)-1\) for all but a finite number of \(P\).

Calculate the genus of each of the following curves: (a) \(X^{2} Y^{2}-Z^{2}\left(X^{2}+Y^{2}\right)\) (b) \(\left(X^{3}+Y^{3}\right) Z^{2}+X^{3} Y^{2}-X^{2} Y^{3}\) (c) The two curves of Problem \(7.12\). (d) \(\left(X^{2}-Z^{2}\right)^{2}-2 Y^{3} Z-3 Y^{2} Z^{2}\)

Let \(X, Y\) be nonsingular projective curves, \(f:: X \rightarrow Y\) a dominating morphism. Prove that \(f(X)=Y\). (Hint: If \(P \in Y \backslash f(X)\), then \(\tilde{f}(\Gamma(Y \backslash\\{P\\})) \subset \Gamma(X)=k\); apply Problem 8.16.)

Show that a morphism from a projective curve \(X\) to a curve \(Y\) is either constant or surjective; if it is surjective, \(Y\) must be projective.

If \(f: C \rightarrow V\) is a morphism from a projective curve to a variety \(V\), then \(f(C)\) is a closed subvariety of \(V\). (Hint: Consider \(C^{\prime}=\) closure of \(f(C)\) in \(V\).)

Suppose \(D\) and \(D^{\prime}\) are divisors, and \(D+D^{\prime}=W\) is a canonical divisor. Then \(l(D)-l\left(D^{\prime}\right)=\frac{1}{2}\left(\operatorname{deg}(D)-\operatorname{deg}\left(D^{\prime}\right)\right)\).

Let \(D\) be a divisor with \(\operatorname{deg}(D)=2 g-2\) and \(l(D)=g\). Show that \(D\) is a canonical divisor. So these properties characterize canonical divisors.

Let \(P_{1}, \ldots, P_{m} \in \mathbb{P}^{2}, r_{1}, \ldots, r_{m}\) nonnegative integers. Let \(V\left(d ; r_{1} P_{1}, \ldots, r_{m} P_{m}\right)\) be the projective space of curves \(F\) of degree \(d\) with \(m_{P_{i}}(F) \geq r_{i} .\) Suppose there is a curve \(C\) of degree \(n\) with ordinary multiple points \(P_{1}, \ldots, P_{m}\), and \(m_{P_{i}}(C)=r_{i}+1\) and suppose \(d \geq n-3\). Show that $$ \operatorname{dim} V\left(d ; r_{1} P_{1}, \ldots, r_{m} P_{m}\right)=\frac{d(d+3)}{2}-\sum \frac{\left(r_{i}+1\right) r_{i}}{2} $$ Compare with Theorem 1 of $$\$ 5.2$$.

Let \(D\) be a divisor, and let \(V\) be a subspace of \(L(D)\) (as a vector space). The set of effective divisors \(\\{\operatorname{div}(f)+D \mid f \in V, f \neq 0\\}\) is called a linear series. If \(f_{1}, \ldots, f_{r+1}\) is a basis for \(V\), then the correspondence \(\operatorname{div}\left(\sum \lambda_{i} f_{i}\right)+D \mapsto\left(\lambda_{1}, \ldots, \lambda_{r+1}\right)\) sets up a one-to-one correspondence between the linear series and \(\mathbb{P}^{r}\). If \(\operatorname{deg}(D)=\) \(n\), the series is often called a \(g_{n}^{r}\). The series is called complete if \(V=L(D)\), i.e., every effective divisor linearly equivalent to \(D\) appears. (a) Show that, with \(C, E\) as in Section 1, the series \(\\{\operatorname{div}(G)-E \mid G\) is an adjoint of degree \(n\) not containing \(C\\}\) is complete. (b) Assume that there is no \(P\) in \(X\) such that \(\operatorname{div}(f)+D \geq P\) for all nonzero \(f\) in \(V\). (This can always be achieved by replacing \(D\) by a divisor \(\left.D^{\prime} \leq D .\right)\) For each \(P \in X\), let \(H_{P}=\\{f \in V \mid \operatorname{div}(f)+D \geq P\) or \(f=\) \(0\\}\), a hyperplane in \(V\). Show that the mapping \(P \mapsto H_{P}\) is a morphism \(\varphi_{V}\) from \(X\) to the projective space \(\mathbb{P}^{*}(V)\) of hyperplanes in \(V .\) (c) A hyperplane \(M\) in \(\mathbb{P}^{*}(V)\) corresponds to a line \(m\) in \(V\). Show that \(\varphi_{V}^{-1}(M)\) is the divisor \(\operatorname{div}(f)+D\), where \(f\) spans the line \(m\). Show that \(\varphi_{V}(X)\) is not contained in any hyperplane of \(\mathbb{P}^{*}(V)\). (d) Conversely, if \(\varphi: X \rightarrow \mathbb{P}^{r}\) is any morphism whose image is not contained in any hyperplane, show that the divisors \(\varphi^{-1}(M)\) form a linear system on \(X .\) (Hint: If \(D=\) \(\varphi^{-1}\left(M_{0}\right)\), then \(\left.\varphi^{-1}(M)=\operatorname{div}\left(M / M_{0}\right)+D .\right)\) (e) If \(V=L(D)\) and \(\operatorname{deg}(D) \geq 2 g+1\), show that \(\varphi_{V}\) is one-to-one. (Hint: See Corollary 3 .) Linear systems are used to map curves to and embed curves in projective spaces.

Show that there are curves of every positive genus. (Hint: Consider affine plane curves \(y^{2} a(x)+b(x)=0\), where \(\operatorname{deg}(a)=g, \operatorname{deg}(b)=g+2\).)

(a) Use linear systems to reprove that every curve of genus 1 is birationally equivalent to a plane cubic. (b) Show that every curve of genus 2 is birationally equivalent to a plane curve of degree 4 with one double point. (Hint: Use a \(g_{4}^{3}\).)

Let \(f: X \rightarrow Y\) be a nonconstant (therefore surjective) morphism of projective nonsingular curves, corresponding to a homomorphism \(\tilde{f}\) of \(k(Y)\) into \(k(X)\). The integer \(n=[k(X): k(Y)]\) is called the degree of \(f .\) If \(P \in X, f(P)=Q\), let \(t \in \mathscr{O}_{Q}(Y)\) be a uniformizing parameter. The integer \(e(P)=\operatorname{ord}_{P}(t)\) is called the ramification index of \(f\) at \(P\). (a) For each \(Q \in Y\), show that \(\sum_{f(P)=Q} e(P) P\) is an effective divisor of degree \(n\) (see Proposition 4 of \(\$ 8.2\) ). (b) \((\operatorname{char}(k)=0)\) With \(t\) as above, show that \(\operatorname{ord}_{P}(d t)=\) \(e(P)-1 .\) (c) \((\operatorname{char}(k)=0)\) If \(g_{X}\) (resp. \(\left.g_{Y}\right)\) is the genus of \(X\) (resp. \(Y\) ), prove the Hurwitz Formula $$ 2 g_{X}-2=\left(2 g_{Y}-2\right) n+\sum_{P \in X}(e(P)-1) $$ (d) For all but a finite number of \(P \in X, e(P)=1\). The points \(P \in X\) (and \(f(P) \in Y\) ) where \(e(P)>1\) are called ramification points. If \(Y=\mathbb{P}^{1}\) and \(n>1\), show that there are always some ramification points. If \(k=\mathbb{C}\), a nonsingular projective curve has a natural structure of a one-dimensional compact complex analytic manifold, and hence a two-dimensional real analytic manifold. From the Hurwitz Formula (c) with \(Y=\mathbb{P}^{1}\) it is easy to prove that the genus defined here is the same as the topological genus (the number of "handles") of this manifold. (See Lang's "Algebraic Functions" or my "Algebraic Topology", Part X.)

Let \(P\) be a point on a nonsingular curve \(X\) of genus \(g .\) Let \(N_{r}=N_{r}(P)=l(r P) .\) (a) Show that \(1=N_{0} \leq N_{1} \leq \cdots \leq\) \(N_{2 g-1}=g .\) So there are exactly \(g\) numbers \(00 .\) (b) The following are equivalent: (i) \(P\) is a Weierstrass point; (ii) \(l(g P)>1 ;\) (iii) \(l(W-g P)>0\); (iv) There is a differential \(\omega\) on \(X\) with \(\operatorname{div}(\omega) \geq g P\) (c) If \(r\) and \(s\) are not gaps at \(P\), then \(r+s\) is not a gap at \(P\). (d) If 2 is not a gap at \(P\), the gap sequence is \((1,3, \ldots, 2 g-1)\). Such a Weierstrass point (if \(g>1\) ) is called hyperelliptic. The curve \(X\) has a hyperelliptic Weierstrass point if and only if there is a morphism \(f: X \rightarrow \mathbb{P}^{1}\) of degree \(2 .\) Such an \(X\) is called a hyperelliptic curve. (e) An integer \(n\) is a gap at \(P\) if and only if there is a differential of the first kind \(\omega\) with \(\operatorname{ord}(\omega)=n-1\).