

\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 2008}

\bigskip


\centerline{ Problem Set 1A (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 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).}



\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.}{Find the tangent space to the Whitney umbrella at a point
$(a,b,c)$.  What are the singular points of this variety?}

\itemitem{Proof:}
\medskip




\bye


%

