Exercise 109.56.1 (Definitions). Provide definitions of the italicized concepts.

a

*radical ideal*of a ring,a

*finite type*ring homomorphism,a

*differential a la Weil*,a

*scheme*.

These were the questions in the final exam of a course on Commutative Algebra, in the Fall of 2013 at Columbia University.

Exercise 109.56.1 (Definitions). Provide definitions of the italicized concepts.

a

*radical ideal*of a ring,a

*finite type*ring homomorphism,a

*differential a la Weil*,a

*scheme*.

Exercise 109.56.2 (Results). State something formally equivalent to the fact discussed in the course.

result on hilbert polynomials of graded modules.

dimension of a Noetherian local ring $(R, \mathfrak m)$ and $\bigoplus _{n \geq 0} \mathfrak m^ n/\mathfrak m^{n + 1}$.

Riemann-Roch.

Clifford's theorem.

Chevalley's theorem.

Exercise 109.56.3. Let $A \to B$ be a ring map. Let $S \subset A$ be a multiplicative subset. Assume that $A \to B$ is of finite type and $S^{-1}A \to S^{-1}B$ is surjective. Show that there exists an $f \in S$ such that $A_ f \to B_ f$ is surjective.

Exercise 109.56.4. Give an example of an injective local homomorphism $A \to B$ of local rings, such that $\mathop{\mathrm{Spec}}(B) \to \mathop{\mathrm{Spec}}(A)$ is not surjective.

Situation 109.56.5 (Notation plane curve). Let $k$ be an algebraically closed field. Let $F(X_0, X_1, X_2) \in k[X_0, X_1, X_2]$ be an irreducible polynomial homogeneous of degree $d$. We let

\[ D = V(F) \subset \mathbf{P}^2 \]

be the projective plane curve given by the vanishing of $F$. Set $x = X_1/X_0$ and $y = X_2/X_0$ and $f(x, y) = X_0^{-d}F(X_0, X_1, X_2) = F(1, x, y)$. We denote $K$ the fraction field of the domain $k[x, y]/(f)$. We let $C$ be the abstract curve corresponding to $K$. Recall (from the lectures) that there is a surjective map $C \to D$ which is bijective over the nonsingular locus of $D$ and an isomorphism if $D$ is nonsingular. Set $f_ x = \partial f/\partial x$ and $f_ y = \partial f/\partial y$. Finally, we denote $\omega = \text{d}x/f_ y = - \text{d}y/f_ x$ the element of $\Omega _{K/k}$ discussed in the lectures. Denote $K_ C$ the divisor of zeros and poles of $\omega $.

Exercise 109.56.6. In Situation 109.56.5 assume $d \geq 3$ and that the curve $D$ has exactly one singular point, namely $P = (1 : 0 : 0)$. Assume further that we have the expansion

\[ f(x, y) = xy + h.o.t \]

around $P = (0, 0)$. Then $C$ has two points $v$ and $w$ lying over $P$ characterized by

\[ v(x) = 1, v(y) > 1 \quad \text{and}\quad w(x) > 1, w(y) = 1 \]

Show that the element $\omega = \text{d}x/f_ y = - \text{d}y/f_ x$ of $\Omega _{K/k}$ has a first order pole at both $v$ and $w$. (The behaviour of $\omega $ at nonsingular points is as discussed in the lectures.)

In the lectures we have shown that $\omega $ vanishes to order $d - 3$ at the divisor $X_0 = 0$ pulled back to $C$ under the map $C \to D$. Combined with the information of (1) what is the degree of the divisor of zeros and poles of $\omega $ on $C$?

What is the genus of the curve $C$?

Exercise 109.56.7. In Situation 109.56.5 assume $d = 5$ and that the curve $C = D$ is nonsingular. In the lectures we have shown that the genus of $C$ is $6$ and that the linear system $K_ C$ is given by

\[ L(K_ C) = \{ h\omega \mid h \in k[x, y],\ \deg (h) \leq 2\} \]

where $\deg $ indicates total degree^{1}. Let $P_1, P_2, P_3, P_4, P_5 \in D$ be pairwise distinct points lying in the affine open $X_0 \not= 0$. We denote $\sum P_ i = P_1 + P_2 + P_3 + P_4 + P_5$ the corresponding divisor of $C$.

Describe $L(K_ C - \sum P_ i)$ in terms of polynomials.

What are the possibilities for $l(\sum P_ i)$?

Exercise 109.56.8. Write down an $F$ as in Situation 109.56.5 with $d = 100$ such that the genus of $C$ is $0$.

Exercise 109.56.9. Let $k$ be an algebraically closed field. Let $K/k$ be finitely generated field extension of transcendence degree $1$. Let $C$ be the abstract curve corresponding to $K$. Let $V \subset K$ be a $g^ r_ d$ and let $\Phi : C \to \mathbf{P}^ r$ be the corresponding morphism. Show that the image of $C$ is contained in a quadric^{2} if $V$ is a complete linear system and $d$ is large enough relative to the genus of $C$. (Extra credit: good bound on the degree needed.)

Exercise 109.56.10. Notation as in Situation 109.56.5. Let $U \subset \mathbf{P}^2_ k$ be the open subscheme whose complement is $D$. Describe the $k$-algebra $A = \mathcal{O}_{\mathbf{P}^2_ k}(U)$. Give an upper bound for the number of generators of $A$ as a $k$-algebra.

[1] We get $\leq 2$ because $d - 3 = 5 - 3 = 2$.

[2] A quadric is a degree $2$ hypersurface, i.e., the zero set in $\mathbf{P}^ r$ of a degree $2$ homogeneous polynomial.

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