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

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\centerline{ Problem Set 9 (Last revised \number\month/\number\day/\number\year)}
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Let $K$ be an algebraically closed field unless otherwise noted.
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\item{1.}{(Harris 6.4){For any point $p \in \bP^3$ and any plane $H
    \subset \bP^3$ containing p let $\Sigma_{p,H} \subset \bG(1,3)$ be
    the set of lines in $\bP^3$ passing through p and lying in H.
    Show that under the Plucker imbedding $\bG(1,3) \rightarrow \bP^5$
    the subvariety $\Sigma_{p,H}$ is mapped to a line, and conversely
    every line in $\bP^5$ lying on $\bG(1,3)$ is of the form
    $\Sigma_{p,H}$ for some p and H.}

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\item{2.}{(Harris 6.5) For a point $p \in \bP^3$ let $\Sigma_p \subset
  \bG(1,3)$ be the set of lines in $\bP^3$ passing through p.  For any
  plane $H \subset \bP^3$ let $\sigma_H \subset \bP^3$ be the set of
  lines in $\bP^3$ contained in H.  Show that the Plucker embedding
  carries both $\Sigma_p, \Sigma_H$ into 2--planes in $\bP^5$.  Show
  that conversely any 2--plane $\Lambda\simeq \bP^2 \subset \bG(1,3)
  \subset \bP^5$ either equals $\Sigma_p$ for some p or $\Sigma_H$ for
  some H. }

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\item{3.}{(Harris 6.6) Let $\ell_1,\ell_2 \subset \bP^3$ be skew lines
  (that is nonintersecting lines).  Show that the set $Q\subset
  \bG(1,3)$ of lines in $\bP^3$ meeting both is the intersection of
  $\bG(1,3)$ with a three--plane $\bP^3\subset \bP^5$ and hence is a
  quadric surface. Deduce that $Q$ is isomorphic to
  $\bP^1\times\bP^1$.  Do the same problem for lines $\ell_1,\ell_2
  \subset \bP^3$ which meet.}

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\item{4.}{(Harris 6.20) Let $Q\subset \bP^3$ be the zero set of
  $Z_0Z_3-Z_1Z_2$.  Show that the Fano variety $F_1(Q)$ of all lines
  contained in Q is a union of two conic curves. Compare this with the
  parametric description of the Fano variety given in the discussion
  of the Segre map.}


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\item{5.}{(Harris 7.6) Show that if $\phi:X -- \to \bP^n$ is a
  rational map and $Z \subset X$ is a subvariety such that the
  restriction $\eta$ of $\phi$ to Z is defined , then the graph
  $\Gamma_\eta \subset Z \times \bP^n$ is contained in the set
  $\pi_1^{-1}(Z)=\Gamma_\phi\cap (Z\times\bP^n) \subset X \times
  \bP^n$ but may not equal it. Thus the proper transform of Z can be
  properly contained in the total transform of Z.}
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\item{6.}{(Harris 7.7) Show that the image of a rational normal curve
  $C \subset \bP^n$ under projection from a point $p \in C$ is a
  rational normal curve in $\bP^{n-1} $.  Is this true if we project
  from a point not in C?}







\bye



