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\begin{center}\begin{large}\textbf{Math 137 - Problem Set 8\\
Due Friday, Apr 3}\end{large}\end{center}



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All rings are commutative, and $k$ is an algebraically closed field.

\begin{enumerate} 

\item Let $f \in k[x_1,\ldots, x_n]$ and $I = (f)$. Let $F \in k[x_1,\ldots, x_{n+1}]$ be the homogenization of $f$ with respect to $x_{n+1}$, and let $J$ be the homogenization of $I$ (i.e. the ideal generated by the homogenization of elements of $I$). Show that $J = (F).$

\item Let $V \subset \A^n$ be an affine algebraic set, and $I = I_a(V)$. Let $J \subset k[x_1, \ldots, x_{n+1}]$ be the homogenization of $I$. Define $\bar{V} \subset \P^n$ to be $V_p(J)$.

\begin{enumerate}

\item \textbf{(Extra credit)} Show that $J$ is radical.
\item Show that $\bar{V} \cap U_{n+1} = V$ (where we've identified $U_{n+1}$ and $\A^n$ in the natural way).

\item Show that $\bar{V}$ is equal to the Zariski closure of $V$ in $\P^n$. That is, $\bar{V}$ is the smallest projective algebraic set containing $V$.
\end{enumerate}

\item The \textit{\textbf{twisted cubic}} $C\subset \P^3$ is the image of the map $\phi : \P^1 \to \P^3$ defined $$\phi([s:t]) = [s^3 : s^2t: st^2: t^3].$$ 

\begin{enumerate}
\item Show that $C$ is the locus of points $[x:y:z:w]$ in $\P^3$ such that the rank of the matrix $$\begin{pmatrix}  x & y & z \\ y & z & w \end{pmatrix}$$ is 1. (Varieties described as the vanishing of minors of a matrix are called \textit{\textbf{determinantal varieties}}.)

\item Show that no two of the minors in part (a) determine $C$. That is, all three are necessary.

\end{enumerate}

%\item Recall that a \textit{\textbf{hypersurface}} of degree $d$ in $\P^n$ is an algebraic set of the form $V(F)$ where $F$ is a homogeneous polynomial of degree $d$ (with no multiple components). A \textit{\textbf{hyperplane}} is a hypersurface of degree one.

%\begin{enumerate}
%\item Let $P_1, \ldots, P_r \in \P^n$. Show there is a hyperplane in $\P^n$ that doesn't contain any of the $P_i$.
%\item Show that (a) generalizes to hypersurfaces of degree $d$, for any $d >0$.
%\end{enumerate}

\item Let $H = V(\sum a_i x_i)$ be a hyperplane in $\P^n$. 

\begin{enumerate}
\item Show that assigning $[a_1 : \ldots : a_{n+1}] \in \P^n$ to $H$ gives a one-to-one correspondence between hyperplanes in $\P^n$ and points in $\P^n$. This is called the \textbf{\textit{dual projective space}}. If $P \in \P^n$, let $P^*$ be the corresponding hyperplane; if $H$ is a hyperplane, let $H^*$ be the corresponding point. 

\item Show that $P \in H$ if and only if $H^* \in P^*$. 
\end{enumerate}

\item 
\begin{enumerate}
\item Let $P_1,P_2, P_3 \in \P^2$ be three distinct points not lying on a line. Show that if $Q_1, Q_2, Q_3 \in \P^2$ are distinct points not lying on a line, there is a projective change of coordinates $T: \P^2 \to \P^2$ such that $T(P_i) = Q_i$. (Hint: Show there's a projective change of coordinates sending them to $[0:0:1]$, $[0:1:0]$, and $[1:0:0]$.)

\item State and prove an analogous result for two sets of three lines in $\P^2$. (What conditions do you need to impose on the three lines?)
\end{enumerate}



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