

\magnification=\magstep1
\def \bA {{\bf A}}
\def \bC {{\bf C}}
\def \bR {{\bf R}}
\def \bP {{\bf P}}
\def \bZ {{\bf Z}}
\def \bG {{\bf G}}
\def \bQ {{\bf Q}}
\def \bO {{\bf O}}
\def \bF {{\bf F}}
\def \bT {{\bf T}}
\def \bP {{\bf P}}


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

\bigskip


\centerline{ Problem Set 1A (Last revised \number\month/\number\day/\number\year)}
\bigskip


\item{1.}{Let $M_2(\bC)$ be the space of two by two matrices with
entries from $K$ considered as $\bA^4(\bC)$ via the 4 matrix entries as
coordinates.  Let $X$ be the set of matrices $B$ such that $B^2=B$.
Show that $X$ is a reducible algebraic variety.  
Find the irreducible components of $X$. Hint: Recall
the result proved in class  that over an infinite field the
similarity class of a square matrix is an irreducible set. }
\medskip


\item{2.}{Let $K$ be a field, $V$ an n-dimensional vector space over $K$
with basis $e_1,\ldots ,e_n$ and let $\wedge V$ be the exterior algebra (the K vector space with
basis $e_{i_1}\wedge \cdots \wedge e_{i_k}$ with $1 \leq
i_1<\cdots<i_k\leq n$ , $1 \leq k \leq n$ 
and algebra structure satisfying $w \wedge
w=0$.)  Let $\wedge^k V$ be the span of vectors
$e_{i_1}\wedge \cdots \wedge e_{i_k}$ with $1 \leq i_1<\cdots<i_k\leq
n$.  Show that the endomorphism of $V$  represented by a matrix $A$ induces a
endomorphism of $\wedge^k V$ and denote the matrix of this
endomorphism by $\wedge^k A$.}

\itemitem{a)}{Show that the matrix entries of $\wedge^k A$ are
polynomials
in the entries of$ $.A}

\itemitem{b)}{Discover the coefficients of the characteristic polynomial
of $A$ in terms of traces of $\wedge^k A$ by first computing on a 
convenient Zariski
dense set of matrices and using the irreducibility of $A^{n^2}(K)$
to verify your formula holds for all matrices.}

\medskip


\item{3.}{Let $F_n(x,y), F_{n-1}(x,y)$ be homogeneous polynomials
of degrees $n$ and $n-1$ respectively in $K[x,y]$ for a field K.
Show that the zero set of any irreducible polynomial of the form
$F(x,y)=F_n(x,y)+F_{n-1}(x,y)$ is a rational
variety.  Use this to parameterize the zero sets of $y^2-x^3, y^2-x^3-x^2$,
the Folium of Descartes $x^3+y^3-3xy$, the 5 leaved rose $(x^2+y^2)^3
-5x^4y+10x^2y^3-y^5$ (and more generally $r=\sin n\theta$ for odd n).}






\medskip

\item{4.}{Find a rational parameterization of  the lemniscate
$(x^2+y^2)^2=a^2(x^2-y^2)$. Hint: compute the intersection of
the lemniscate with the family of circles $x^2+y^2=t(x-y)$.}




\medskip
\item{5.}{Show that the zero set of $zy^2-x^2$ is rational.
(This affine cubic surface is called Whitney's umbrella).
More generally, any projective cubic surface F=0 containing a point P with
the gradient of  F vanishing at P and such that lines joining  P to
points on the surface do not all lie
wholly in the surface is rational. 
}



\itemitem{Proof:}
\medskip

\item{6.}{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{6.}{Let $V\subset \bA^n(K)$ and $W \subset \bA^m(K)$ be Zariski
closed subsets.  Show that the product set $V \times
W= \{(x_1,\ldots,x_n,y_1,\ldots, y_m)\in \bA^{n+m}|
(x_1,\ldots,x_n) \in V, (y_1,\ldots, y_m) \in W\}$ is Zariski closed in
$\bA^{m+n}$}.

\itemitem{a)}{Show that although as a set $\bA^n(K) \times \bA^m(K)$ is the
same as $A^{m+n}(K)$ there are subsets of $A^{m+n}(K)$ which are
closed in the Zariski topology but not in the product topology.}

\itemitem{b)}{Show that if $V,W$ as above are irreducible, then
$V \times W$ is irreducible. Hint: If the product is a union of closed
sets $Z_1,Z_2$ 
show that  the intersection of vertical sets $V \times \{w\}$ with
the closed set $Z_i$ in $\bA^{m+n}$ is wholly contained in some $Z_i$.
Show that the set $W_i$ of all $w \in W$ such that $V \times \{w\}$ 
is contained in $Z_i$ is closed.}

\itemitem{c)}{Use the preceeding to show that if $W \subset M_n(K)$ is an
irreducible set and $V$ is an irreducible set of invertible matrices
then the union of the conjugates of elements of $W$ by matrices in $V$
is an irreducible set.}




\itemitem{d)}{Show that the set of upper triangular matrices
of determinant 1 in $M_n(K)$ is an irreducible variety when $K$
is infinite.}


\itemitem{}{Show every matrix in $M_n(K)$ is similar to an upper
triangular matrix when $K$ is algebraically closed.  Show that the set
$SL(n,K)$ of matrices of determinant 1 with entries from an
algebraically closed field $K$ is an irreducible affine variety. (It
turns out that an algebraic group is irreducible if and only if it is
connected. Since in the representation theory of groups irreducible is
used to denote a simple module over the group ring, we generally refer
to a connected algebraic group intstead of an irreducible one.
So far we have seen that $GL(n,K), SL(n,k)$ are connected alebraic
groups.) }


\bye
