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\begin{document}
\pagestyle{plain}
\title{Some amusing problems related to algebraic geometry}
\author{Ravi Vakil}
\date{September 9, 1999.}
\maketitle

{\parskip=12pt % closing bracket is just before the bibliography 

Here are some problems to pique your interest; we will solve (or
mostly solve) them during the semester.  Some are deep, and others are
just fun.  You won't actually need algebraic geometry to
solve a fair number of these problems, but if you solve them (or make
progress), you'll secretly pick up a lot of algebro-geometrical
insight. 

{\bf 1.} {\em Infinite Chomp.} The game of Chomp is played as follows.
Fix non-negative integers $m$ and $n$.   Cookies are placed in a
rectangular array at the points $(x,y)$ where $0 \leq x \leq m$ and $0
\leq y \leq n$ ($x$, $y$ are integers).  The cookie at (0,0) is poisoned.  
Two players alternate moving; a move involves picking a cookie, and
eating it and every cookie above and to the right of it.  The player
who dies loses.  (There's a neat proof that if $m$ and $n$ aren't both
zero, the first player has a winning strategy, that doesn't reveal
what that strategy is.)  Clearly the game ends in a finite number of terms.

Infinite Chomp is the same, except cookies are placed at $(x,y)$ where
$x$ and $y$ run through all non-negative integers.  Prove that the
game is guaranteed to end in a finite number of terms.  Generalize
this to where cookies are placed on $(\zed^+)^n$ for any $n$.
   
(This is secretly related to the Hilbert Basis Theorem.)

{\bf 2.}  Suppose $f_1(x_1, \dots, x_n)=0$, \dots, $f_r(x_1, \dots, x_n)=0$ is a
system of $r$ polynomial equations in $n$ unknowns, with integral
coefficients, and suppose this system has a finite number of complex
solutions.  Show that each solution is algebraic, i.e. if $(x_1,
\dots, x_n)$ is a solution, then  $x_i \in \overline{Q}$ for all $i$.

{\bf 3.} {\em Pascal's theorem.}  If a hexagon is inscribed in an irreducible conic,
then the opposite sides meet in collinear points.

{\bf 4.} {\em Pappas' theorem.} Let $L_1$ and $L_2$ be two lines.  Let $p_1$,
$p_2$, and $p_3$ be distinct points of $L_1$, and let $q_1$, $q_2$,
and $q_3$ be distinct points of $L_2$ (none lying on $L_1 \cap L_2$).
Let $L_{ij}$ be the line between $P_i$ and $Q_j$.  For each $i$, $j$,
$k$ with $\{ i, j, k \} = \{ 1, 2, 3 \}$, let $R_k = L_{ij} \cap L_{ji}$.
Then $R_1$, $R_2$, and $R_3$ are collinear.

{\bf 5.} We'll see that the integers $\zed$ should really be thought of as 
some weird curve.

{\bf 6.} {\em Group law on an elliptic curve.} We'll define the group
law on an elliptic curve, and see that it is associative.  (We'll
first define elliptic curves, which secretly come up in the next three
problems.)

{\bf 7.} Find all rational solutions to $y^2 = a x^3 + b x^2$ where $a, b \in \Q$.

{\bf 8.}  Prove that if $p(t)$ and $q(t)$ are polynomials (with complex coefficients)
such that $p(t)^2 = q(t)^3 + 1$, then $p(t)$  and $q(t)$ are both constant.

{\bf 9.} {\em Poncelet's theorem.}  Let $C$ and $D$ be two ellipses, with
$C$ contained in $D$.  Pick a point $p_0$ on $C$, and draw a tangent
to $D$ from $p_0$, which meets $C$ again at some other point $p_1$.
Repeat this process (picking the ``other'' tangent from $p_1$).
Suppose after $n$ repetitions, you return to your starting point, i.e.
$p_n=p_0$.  Prove that this is true no matter where you start.

{\bf 10.}  If $f(x)$ is a polynomial with integer coefficients, of degree $d=2g+1$ or $2g+2$, 
square-free mod $p$, then
$$
\left| \sum_{i=0}^{p-1} \left( \frac {f(i)} p \right) \right| \leq 2 g \sqrt{p}+1.
$$

(The ``fraction'' denotes the symbol for quadratic residues.)
If $g >1$, the proof uses the Weil conjectures for curves.
(In specific cases, an explicit formula can be given for the
left-hand side.)

{\bf 11.} The degree of the discriminant $b^2-4ac$ of the quadratic $a
x^2 + bx +c$ is 2.  Show that the degree of the discriminant of a
degree $d$ polynomial is $2(d-1)$.

{\bf 12.} Fix a positive integer $d$, and let $\si_1$, \dots, $\si_n$
be $n$ transpositions in $S_d$ (i.e.  they swap two elements of $\{ 1,
\dots, d \}$).  Suppose they generate $S_d$, and $$\si_1 \dots \si_n =
e,$$ where $e$ is the identity.  Show that $n \geq 2(d-1)$. 


} % end of parskip; it started just before the introduction
\end{document}
