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{Mathematics 535 \hfill Algebraic Geometry \hfill Fall 2016}

\bigskip


\centerline{ Problem Set 2 (Last revised \number\month/\number\day/\number\year)}
\bigskip

\item{0.}{(carryover from problem set 1) Let $R$ be a Boolean ring, that is $r^2=r$   for all $r \in
R$.

\itemitem{a)}{Show that $R$ is a commutative ring of  characteristic 2
and that all prime ideals are maximal.}

\itemitem{b)}{Show that spec($R$) is a compact Hausdorff space in
which every connected component is a point. (a so called  Stone
space)}

\itemitem{c)}{Show that if  X is compact Hausdorff and
totally disconnected then the ring $R$ of continuous functions from X
to
the field with two elements (considered as a topological space with
the discrete topology) is a Boolean algebra R.  Show that $X$ is
homeomorphic to spec($R$).}



\itemitem{d)}{Show that the constructions above yield an equivalence
of categories between Stone spaces with continuous maps, and
Boolean algebras with ring homomorphisms.
Since a homomorphism $R\rightarrow S$  yields by inverse image a map
$spec(S) \rightarrow spec(R)$ this relation is  a duality (introduced
by Marshall
 Stone in 1936).}



\medskip



\item{1.} {An aerial camera photographs a car traveling along a
  straight road on flat ground towards a junction. Before the junction
  there are two warning signs at distances of 4km and 2km from the
  junction. On the film, the signs are 1cm and 3cm from the junction
  and the car is 3/7cm from the junction. Use the cross ratio to
  determine how far the car is from the junction on the ground.}


\medskip




\item{2.} {(Harris 1.3) Show that any d points in projective space
  which are not contained in a line are the zero set of a collection
  of polynomials of degree d-1.}  

\medskip

\item{3.} {(Harris 1.5)Let $\Gamma$ be a collection of $d \leq kn$
  points in general position in projective $n$ space ($k \geq
  2$). Show that $\Gamma$ is the zero set of a collection of
  homogeneous polynomials of degrees at most $k$.}


\medskip
\item{4.} {(Harris 1.6) Let $P_1,\ldots,P_{n+2}$ and $Q_1,\ldots,Q_{n+2}$ be two ordered
sets
of $n+2$ points in general position in projective n--space. Show that
these varieties are projectively equivalent, and the equivalence can be chosen
to preserve the order.}

\bye
