

\magnification=\magstep1
\def \bA {{\bf A}}
\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 2002}

\bigskip


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


\item{1.} {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 $F(x,y)=F_n(x,y)+F_{n-1}(x,y)$ 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 $r=sin n\theta$).}

\itemitem{Proof:}{}


\medskip

\item{2.}{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 $x^2+y^2=t(x-y)$.}

\itemitem{Proof:}{}


\medskip
\item{3.}{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{4.} {An aerial camera photographs a car travelling 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. How far is the car from the junction on the ground?}
\itemitem{Proof:}{}


\medskip




\item{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.}
\itemitem{Proof:}{}


\medskip

\item{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 $k$.}



\itemitem{Proof:}{}



\medskip
\item{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 equivalenc can be chosen
to preserve the order.}
\itemitem{Proof:} {
We show that $P_1,\ldots,P_{n+2}$ is equivalent to
$E_0,\ldots,E_n,E$ where $E_i$ is given by the line determined by the
standard basis vector $[x_0,\ldots,x_n],x_j=0 {\rm\ if\ } j \not = i, x_i=1$
and $E=[1,1,1,\ldots,1]$. Since projective equivalence is an equivalence
relation this will establish the result.  Since the points $P_i$ are in general position the lines given by the first $n+1$ points form a basis of the $n+1$ dimensional
vector space $K^{n+1}$. Thus the collection $\{P_i\}$ is projectively equivalent to 
$E_0,\ldots ,E_n,P$ for some $P$ via the linear
transformation taking the basis formed by the first $n+1$ $P_i$ to the
standard basis. Since projective transformations preserve independence of points, the point $P$ is independent form any choice of $n$ of the points $E_i$.
Hence if $P=[\lambda_0,\ldots,\lambda_n]$ all $\lambda_i$ are nonzero.
The diagonal matrix with $1/\lambda_i$ down the diagonal will transform
the lines given by $E_0,\ldots ,E_n,P$ to the lines given by $E_0,\ldots ,E_n,E$.}

 

\medskip
\item{1.11}{For $i=0,1,2$ let  $F_i$ be the determinant of the 2 by 2 matrix obtained
by removing the $3-i$ column from 
$$\pmatrix{ Z_0& Z_1& Z_2\cr Z_1& Z_2& Z_3}$$  
Show that the intersection of two distinct quadric surfaces $Q_i$ which are the zero set of $F_i$ 
is the union of a line and the twisted cubic.  More generally, for
$\lambda=[\lambda_0,\lambda_1,\lambda_2]$ a point in the projective plane
let $Q_\lambda$ be the zero set of $\lambda_0F_0+\lambda_1F_1+\lambda_2F_2$.
Show that for $\mu,\nu$ distinct the intersection of $Q_\mu,Q_\nu$ is
the union of the twisted cubic and a line $L_{\mu,\nu}$

\itemitem{Proof:} {}

\medskip
\item{1.12} {Show that any finite set of points on the twisted cubic is in
general position.}
\itemitem{Proof:}{ }
\medskip

\item{1.13} Let $p_i$ be 7 distinct points on a twisted cubic C.
Show that the zero locus of all quadratic polynomials which
vanish at all points $p_i$ is  precisely C.
\itemitem{Proof:}{}


\medskip
\item{1.15} Let $p_i$ be kd+1 distinct points on a rational normal curve 
C in
$\bP^d$.
Show that any polynomial F of degree k vanishing on the $p_i$ also 
vanishes on C. Show that exercise 1.5 is sharp.}

\itemitem{Proof:}{}






\medskip
\item{1.19}{Define the cross ratio of  4 ordered points
$P_1,P_2,P_3,P_4$ in 
general position in $\bP^1$ 
to be the image of $P_4$ under the projective transformation
sending $P_i$ to $E_i$ for $i=1,2,3$.
Show that $P_1,P_2,P_3,P_4$ are projectively equivalent (as an ordered set)
to $P_1',P_2',P_3',P_4'$
if and only if the cross ratio of the sets agrees. Generalize this to show that
n+3 ordered points $P_1,P_2,P_3,\ldots,P_{n+3}$ in general position in
$\bP^n$ are 
projectively equivalent to $P_1',P_2',P_3',\ldots,P_{n+3}'$ if and only if
the cross ratio of $Q_1,Q_2,Q_3,Q_i$ equals that of 
$Q_1',Q_2',Q_3',Q_i'$ for $i=4,\ldots,n+3$ where the points $Q_i$ are
the preimage of the points $P_i$ under the map from $\bP^1$ given by
the
unique rational normal curve through the $P_i$ (similarly for $P_i'$).  }

\itemitem{Proof:}{}




\medskip
\item{1.21}{Let P be a point in the projective plane and let $W_P$ be the
hyperplane in $\bP^5$ defined by the condition that the quadratic polynomial
$\lambda_0Z_0^2+\lambda_1Z_1^2+\lambda_2Z_2^2+\lambda_3Z_0Z_1+\lambda_4Z_0Z_2+\lambda_5Z_1Z_2$
vanishes at P.  If $P_i$ are 5 points in $\bP^2$ with no 4 collinear show
that the hyperplanes $W_{P_i}$ intersect in a single point.  Show that this means
that there is a unique plane conic curve through the 5 points.}

\itemitem{Proof:}{}




\medskip
\item{1.23}{Show that the construction on page 13 works, and reconcile the
statement in the end of the top paragraph on page 14 with 1.21}

\itemitem{Proof:} {}



\medskip

\item{1.24}{Let $\Lambda_i$ be d linear spaces of dimension d-2 in    
$\bP^d$. Show that the family of hyperplanes in $\bP^d$ containing a  
fixed $\Lambda_i$ is parameterized by $\lambda \in \bP^1$. Choose such
parameterizations of hyperplanes $H_i(\lambda)$ such that              
$H_1(\lambda)\cap \ldots \cap H_d(\lambda)$ is a point $p(\lambda)$.   
Show that the set of all $p(\lambda)$ is a rational normal curve.}

\itemitem{Proof:}{}


\medskip

\item{1.25}{Take $\Lambda_i=\{[z_0,z_1,\ldots,z_d] | z_{i-1}=z_i$ for 
$i=1,\ldots,d\}$ and $H_i([\lambda_0,\lambda_1])=                       
Z(\lambda_0z_{i-1}+\lambda_1z_{i})$ in the previous problem. Show    
that the intersection of all $H_i(\lambda)$ is the set of points       
$[Z_0,\ldots,Z_d]$ such that $$\pmatrix{Z_0& Z_1& Z_2&  .& .& .&
Z_{d-2}& Z_{d-1} \cr Z_1& Z_2&  Z_3& .& .& .&.& Z_d}$$             
has rank 1. }

\itemitem{Proof:}{}



\medskip

\item{1.27}{Show that the images of the maps $\mu,\nu :\bP^1 \to \bP^2$
given by $\mu[x_0,x_1]=[x_0^3,x_0x_1^2,x_1^3]$ and                    
$\nu[x_0,x_1]=[x_0^3,x_0x_1^2-x_0^3,x_1^3-x_0^2x_1]$ are algebraic    
varieties.}

\itemitem{Proof:}{}

\medskip
\item{1.28}{Let $\nu:\bP^1\to \bP^2$ be given by three homogeneous
cubic polynomials. Show that if the polynomials have no common zero,
then the image is a hypersurface which is the zero set of a cubic
polynomial.}


\itemitem{Proof:}{}



\medskip

\item{1.29}{Let $\nu_{\alpha,\beta}:\bP^1\to \bP^3$ be given by
$\nu_{\alpha,\beta}([x_0,x_1])=[x_0^4-\beta x_0^3x_1,x_0^3x_1-\beta x_0^2x_1^2,
\alpha x_0^2x_1^2-x_0x_1^3,\alpha x_0x_1^3-x_1^4]$.  Show that the image
of this map is a projective variety which is the zero locus of
one quadratic and two cubic polynomials.}



\itemitem{Proof:} {}



\bye


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