Exercise 111.56.1 (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 111.56.1 (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 111.56.2 (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 111.56.3. 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 111.56.4. 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 111.56.5 (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 111.56.6. In Situation 111.56.5 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 111.56.7. In Situation 111.56.5 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 111.56.8. Write down an $F$ as in Situation 111.56.5 with $d = 100$ such that the genus of $C$ is $0$.

Exercise 111.56.9. 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 111.56.10. Notation as in Situation 111.56.5. 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.

Your email address will not be published. Required fields are marked.

In your comment you can use Markdown and LaTeX style mathematics (enclose it like `$\pi$`

). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.

## Comments (0)