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\begin{center} {\large \textbf{Math 526 -- Homework  8}} \smallskip \\ 
Due 8:30am on Thursday, April 5, 2018\ \\ \ \\
 \end{center}

Please indicate any sources you used for a given problem on the solution to that problem. 
For example, if you worked with another student to get the solution to a problem, please indicate who.
You are welcome to work together in small groups (2-4 people), but please try the problems on your own first. 
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\begin{prob} 
Let $k$ be an algebraically closed field and let $J\subseteq k[x_0, \hdots, x_n]$ be a homogeneous ideal. 
Show that $\J(\X(J))$ equals $\sqrt{J}:\langle x_0, \hdots, x_n\rangle$. \\ \ \\
\end{prob}

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\begin{prob} 
Let $p = [1:1:1:1] \in \P^3$ and consider two different representations of the map given by projection from $p$:
\begin{align*}
&\pi_p:\P^3(\R) \dashrightarrow \P^2(\R) \ \  &\text{ given by }  \ \ &\pi_p([x:y:z:w]) = [x-w:y-w:z-w]\\
&\tilde{\pi}_p:\P^3(\R) \dashrightarrow \P^2(\R) \ \ &\text{ given by } \ \ &\tilde{\pi}_p([x:y:z:w]) = [x-2y+z:y - 2z + w:z- 2w +x]
\end{align*}
Find a projective linear transformation $\phi:\P^2(\R)\rightarrow \P^2(\R)$ (given by a $3\times 3$ matrix) 
for which maps $\phi\circ\pi_p$ and $\tilde{\pi}_p$ are equal.
\end{prob}

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\begin{prob} 
Hassett 9.7
\end{prob}

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\begin{prob} 
Hassett 10.5 (abc) 
\end{prob}

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