

\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 2016}

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\centerline{ Problem Set 4 (Last revised \number\month/\number\day/\number\year)}
\bigskip


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





\medskip
\item{3.}{(Harris 2.2) Determine the ring of regular functions on $\bA^2
  -\{(0,0)\}$}

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\item{4.}{(Harris 2.3) Use 2.2 to show that $\bA^n -\{(0,0,\dots,0)\}$
is not isomorphic to an affine variety when $n\geq 2$. What is the
situation for $n=1$?}



\medskip

\item{5.}{(Harris 2.5) Show that the number of monomials of degree $d$ in $n+1$
  variables
is $n+d\choose n$.}


\bye


