

\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 3 (Last revised \number\month/\number\day/\number\year)}
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\item{1.}{(Harris 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 $L_{\mu,\nu}$



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\item{2.} {(Harris 1.12) Show that any finite set of points on the twisted cubic is in
general position.}

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\item{3.} {(Harris 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.}



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\item{4.} {(Harris 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.}

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\item{5.}{(Harris 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
$P_i'$).  }



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\item{6.}{(Harris 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.}







\bye
