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\textbf{18.782 Introduction to Arithmetic Geometry\hspace{118pt}Fall~2013}\\\vspace{4pt}
\textbf{Problem Set \#10\hspace{237pt}Due: 12/3/2013}
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These problems are related to the material covered in Lectures 21-22.
I have made every effort to proof-read them, but some errors may remain.
The first person to spot each error will receive 1-5 points of extra credit.

The problem set is due by the start of class on 12/3/2013 and should be submitted electronically as a pdf-file e-mailed to  \url{drew@math.mit.edu}.
You can use the latex source for this problem set as a template for writing up your solutions;
be sure to include your name in your solutions and to identify collaborators and any sources not listed in the syllabus.

Recall that we have defined a \emph{curve} as a smooth projective variety of dimension one (and varieties are defined to be irreducible algebraic sets).

\subsection*{Problem 1. Bezout's theorem (50 points)}

In this problem $k$ is an algebraically closed field.\\
\noindent
A curve in $\P^2$ is called a \emph{plane curve}.\footnote{Plane curves are not usually required to be smooth or irreducible, but ours are.}

\begin{enumerate}
\item[\textbf{(a)}] Prove that every plane curve $X/k$ is a hypersurface, meaning that its ideal $I(X)$ is of the form $(f)$, where $f$ is  a homogeneous polynomial in $k[x,y,z]$.  Then show that every generator for $I(X)$ has the same degree.
\end{enumerate}
\noindent
The \emph{degree} of $X$ (denoted $\deg X$) is the degree of any generator for its homogeneous ideal.

\begin{enumerate}
\item[\textbf{(b)}] Let $F/k$ be a function field, let $P$ be a place of $F$, and let $f\in \O_P$.
Prove that the ring $\O_P/(f)$ is a $k$-vector space of dimension $\ord_P(f)$.
\end{enumerate}

Given a nonconstant homogeneous polynomial $g\in k[x,y,z]$ that is relatively prime to~$f$, we can represent $g$ as an element of the local ring $\O_{X,P}$ of functions in $X$ that are regular at~$P$ by picking a homogeneous polynomial $h$ that does not vanish at $P$ and representing~$g$ as $g/h$ reduced modulo $I(X)$, an element of $k(X)$.  Note that in terms of computing $\ord_P(g)$ it makes no difference which $h$ we pick, $\ord_P(g)$ will always be equal to the order of vanishing of $g$ at $P$, a nonnegative integer.
We then define the \emph{divisor} of $g$ in $\Div_k X$ to be
\[
\div_X g = \sum \ord_P(g)P.
\]
Note that $\div_X g$ is \emph{not} a principal divisor.\footnote{By varying $h$ locally we eliminate the poles that would be present if we fixed a global choice for $h$.}
Indeed, $\deg\div_X g$ is never zero.

\begin{enumerate}
\item[\textbf{(c)}] Prove that $\deg \div_X g$ depends only on $\deg g$ (i.e. $\deg \div_X g = \deg \div_X h$ whenever $g$ and $h$ have the same degree and are both relatively prime to $f$).  Then prove that $\deg \div_X g$ is a linear function of $\deg g$.
\end{enumerate}
\noindent
Now suppose that $g$ is irreducible and nonsingular, so it defines a plane curve $Y/k$.

\begin{enumerate}
\item[\textbf{(d)}] Prove that $\deg \div_Y f = \deg \div_X g$.
\end{enumerate}

\begin{definition}
Let $f$ and $g$ be two nonconstant homogeneous polynomials in $k[x,y,z]$ with no common factor, and let $P$ be a point in $\P^2$.
The \emph{intersection number} of $f$ and $g$ at $P$ is
\[
I_P(f,g) := \dim_k \O_{\P^2,P}/(f,g)
\]
Here $\O_{\P^2,P}$ denotes the ring of functions in $k(\P^2)$ that are regular at $P$, and $f$ and $g$ are represented as elements of this ring by choosing homogeneous denominators of appropriate degree that do not vanish at $P$, exactly as described above.
\end{definition}
As above, let $X/k$ and $Y/k$ denote plane curves defined by relatively prime homogeneous polynomials $f$ and $g$, and let $I(f,g)=\sum_P I_P(f,g)$.

\begin{enumerate}
\item[\textbf{(e)}] Prove that $I(f,g)$ is equal to $\deg \div_X g=\deg\div_Y f$.
\item[\textbf{(f)}] Prove \textbf{Bezout's Theorem} for plane curves:
\[
I(f,g)=\deg f\deg g.
\]
\end{enumerate}
In fact Bezout's theorem holds even when $f$ and  $g$ are not necessarily irreducible and nonsingular, but you need not prove this.
It should be clear that $f$ and $g$ do not need to be irreducible; just factor them and apply the theorem to all pairs of factors.
You proof should also handle cases where just one of $f$ or $g$ is singular; it takes a bit more work to handle the case where both $f$ and $g$ are singular and intersect at a common singularity.
The assumption that $k=\kbar$ is necessary, in general, but the inequality $I(f,g)\le\deg f\deg g$ always holds.

\subsection*{Problem 2. Derivations and differentials (50 points)}

A \emph{derivation} on a function field $F/k$ is a $k$-linear map $\delta\colon F\to F$ such that
\[
\delta(fg)=\delta(f)g+f\delta(g).
\]
for all $f,g\in F$.

\begin{enumerate}
\item[\textbf{(a)}] Prove that the following hold for any derivation $\delta$ on $F/k$:
\begin{enumerate}[(i)]
\item $\delta(c)=0$ for all $c\in k$.
\item $\delta(f^n)=nf^{n-1}\delta(f)$ for all $f\in F^\times$ and $n\in \Z$.
\item If $k$ has positive characteristic $p$ then $\delta(f^p)=0$ for all $f\in F$.
\item $\delta(f/g)=(\delta(f)g-f\delta(g))/g^2$ for all $f,g\in F$ with $g\ne 0$.
\end{enumerate}
\end{enumerate}
\noindent
To simplify matters, we henceforth assume that $k$ has characteristic zero.\footnote{For those who are interested, the key thing that changes in characteristic $p>0$ is that everywhere we require an element $x$ to be transcendental we need to additionally require it to be a \emph{separating element}, which means that $F/k(x)$ is a separable extension.}

The simplest example of a derivation is in the case where $F=k(x)$ is the rational function field and $\delta\colon F\to F$ is the map defined by $\delta(f)=\partial f/\partial x$.  We want to generalize this example to arbitrary function fields.

Let $x$ be a transcendental element of $F/k$.
Any $y\in F$ is then algebraic over $k(x)$ and has a minimal polynomial $\lambda\in k(x)[T]$.
After clearing denominators we can assume that $\lambda\in k[x,T]$.  We now formally define
\[
\frac{\partial y}{\partial x} := -\frac{\partial \lambda/\partial x}{\partial \lambda/\partial T}(y) \in k(x,y)\subseteq F
\]
and let the map $\delta_x\colon F\to F$ send $y$ to $\partial y/\partial x$.

One can show (but you are not asked to do this) that $\delta_x$ is a derivation on $F/k$.
Note that we get a derivation $\delta_x$ for each transcendental $x$ in $F$.  Now let $\Der_F$ be the set of all deriviations on $F/k$.

\begin{enumerate}
\item[\textbf{(b)}] Let $x$ be a transcendental element of $F/k$. Prove that for any $\delta_1,\delta_2\in \Der_F$ we have $\delta_1(x)=\delta_2(x)\Rightarrow \delta_1=\delta_2$.
Conclude that $\delta_x$ is the unique $\delta\in \Der_F$ for which $\delta(x)=1$.
\item[\textbf{(c)}] Prove the following:
\begin{enumerate}[(i)]
\item For all $\delta_1,\delta_2\in \Der_F$ the map $(\delta_1+\delta_2)\colon F\to F$ defined by $f\mapsto \delta_1(f)+\delta_2(f)$ is a derivation (hence an element of $\Der_F$).
\item For all $f\in F$ and $\delta\in \Der_F$ the map $(f\delta)\colon F\to F$ defined by $g\mapsto f\delta(g)$ is a derivation (hence an element of $\Der_F$).
\item Every $\delta\in \Der_F$ satisfies $\delta=\delta(x)\delta_x$ (in particular, the \emph{chain rule} $\delta_y=\delta_y(x)\delta_x$ holds for any transcendental $x,y\in F/k$).
\end{enumerate}
\end{enumerate}
It follows that we may view $\Der_F$ as one-dimensional $F$-vector space with any $\delta_x$ as a basis vector.
But rather than fixing a particular basis vector; instead, let us define a relation on the set $S$ of pairs $(u,x)$ with $u,x\in F$ and $x$ transcendental over $k$:
\begin{equation}
(u,x)\sim (v,y)\Longleftrightarrow v=u\delta_y(x).
\end{equation}
\begin{enumerate}
\item[\textbf{(d)}] Prove that $\sim$ is an equivalence relation on $S$.
\end{enumerate}

For each transcendental element $x\in F/k$, let the symbol $dx$ denote the equivalence class of $(1,x)$, and for $u\in F$ define $udx$ to be the equivalence class of $(u,x)$; we call $dx$ a \emph{differential}.
It follows from part (iii) of (d) that every derivation $\delta$ can be uniquely represented as $\delta=udx$ for some $u\in F$, but now we have the freedom to change representations; we may also write $\delta=vdy$ for any transcendental element $y$, where $v=u\delta_y(x)=u\partial x/\partial y$.

\begin{enumerate}
\item[\textbf{(e)}] Prove that $d(x+y)=dx+dy$ and $d(xy)=xdy+ydx$ for all transcendental $x,y\in F/k$.
\end{enumerate}
Let us now extend our differential notation to elements of $F$ that are not transcendental over $k$.
Recall that $k$ is algebraically closed in $F$, so we only need to consider elements of $k$.

\begin{enumerate}
\item[\textbf{(f)}] Prove that defining $da=0$ for all $a\in k$ ensures that (e) holds for all $x,y\in F$, and that no other choice does.
\end{enumerate}

Now momentarily forget everything above and just define $\Delta_F$ to the $F$-vector space generated by the set of formal symbols $\{dx: x\in F\}$, subject to the relations
\[
\textrm{(1) } d(x+y)=dx+dy,\qquad \textrm{(2) } d(xy)=xdy+ydx,\qquad\textrm{(3) } da=0\text{ for } a\in k.
\]
Note that $x$ and $y$ denote elements of $F$ (functions), not free variables, so $\Delta_F$ reflects the structure of $F$ and will be different for different function fields.

\begin{enumerate}
\item[\textbf{(g)}] Prove that $\dim_F\Delta_F=1$, and that any $dx$ with $x\not\in k$ is a basis.
\end{enumerate}

The set $\Delta=\Delta_F$ is often used as an alternative to the set of Weil differentials $\Omega$.
They are both one-dimensional $F$-vector spaces, hence isomorphic (as $F$-vector spaces).
But in order to be useful, we need to associate divisors to differentials in $\Delta$, as we did for $\Omega$.
\smallskip

For any differential $\omega\in \Delta$ and any place $P$, we may pick a uniformizer $t$ for $P$ and write $\omega = wdt$ for some function $w\in F$ that depends on our choice of $t$; note that $t$ is necessarily transcendental over $k$, since it is a uniformizer.
We then define $\ord_P(\omega):= \ord_P(w)$, and the divisor of $\omega$ is then given by
\[
\div \omega := \sum_P \ord_P(\omega)P.
\]
As in Problem 1, the value $\ord_P(\omega)$ does not depend on the choice of the uniformizer $t$.

\begin{enumerate}
\item[\textbf{(h)}] Prove that $\div udv = \div u + \div dv$ for any $u,v\in F$.
Conclude that the set of nonzero differentials in $\Delta$ constitutes a linear equivalence class of divisors.
\item[\textbf{(i)}] Let $F=k(t)$ be the rational function field.
Compute $\div dt$ and prove that it is a canonical divisor.
Conclude that a divisor $D\in \Div_k C$ is canonical if and only if $D=\div df$ for some transcendental $f\in F$.
\end{enumerate}

\noindent
Part (i) holds for arbitrary curves, but you are not asked to prove this.
It follows that the space of differentials $\Delta$ plays the same role as the space of Weil differentials $\Omega$, and it has the virtue of making explicit computations much easier.

\begin{enumerate}
\item[\textbf{(j)}] Prove that the curve $x^2+y^2+z^2$ over $\Q$ has genus $0$ (even though it is not isomorphic to $\P^1$ because it has no rational points) by explicitly computing a canonical divisor.
\end{enumerate}

\subsection*{Problem 3. Survey}
Complete the following survey by rating each problem on a scale of 1 to~10 according to how interesting you found the problem (1 = ``mind-numbing," 10 = ``mind-blowing"), and how difficult you found the problem (1 = ``trivial," 10 = ``brutal").  Also estimate the amount of time you spent on each problem.

\begin{center}
\begin{tabular}{l|r|r|r|}
& Interest & Difficulty & Time Spent\\\hline
Problem 1 & & & \\\hline
Problem 2 & & & \\\hline
\end{tabular}
\end{center}
\noindent
Please rate each of the following lectures that you attended, according to the quality of the material (1=``useless", 10=``fascinating"), the quality of the presentation (1=``epic fail", 10=``perfection"), the pace (1=``way too slow", 10=``way too fast"), and the novelty of the material (1=``old hat", 10=``all new").

\begin{center}
\begin{tabular}{l|l|r|r|r|r|}
Date & Lecture Topic & Material & Presentation & Pace & Novelty\\\hline
11/19 & Riemann's inequality & & & & \\\hline 
11/21 & The Riemann-Roch theorem & & & &\\\hline 
\end{tabular}
\end{center}

\noindent
Feel free to record any additional comments you have on the problem sets or lectures; in particular, how you think they might be improved.

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