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\begin{center} {\large \textbf{Math 526 -- Homework  9}} \smallskip \\ 
Due 8:30am on Thursday, April 12, 2018\ \\ \ \\
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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} 
Given a subset $S\subset \P^n(k)$, define the cone over $S$ to be 
\[{\rm cone}(S)  \ = \  \{ (\lambda x_0, \hdots, \lambda x_n) \ \text{ such that } \lambda \in k \text{ and }  [x_0: \hdots, x_n]\in S\} \ \subseteq \A^{n+1}(k). \]
Let $X\subset \P^n(k)$ be a projective variety and let $\phi:\P^n(k)\dashrightarrow \P^m(k)$ be a rational map given 
by $\phi(x) = [\phi_0(x): \hdots : \phi_m(x)]$ where $\phi_0(x), \hdots, \phi_m(x)$ are homogeneous of degree $d$. \\
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Prove that if $k$ is algebraically closed, then ${\rm cone}(\phi(X)) = \phi({\rm cone}(X))$. 
\end{prob}

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\begin{prob} 
Hassett 10.3
\end{prob}

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\begin{prob} 
Hassett 10.4
\end{prob}

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\begin{prob} 
Hassett 10.7
\end{prob}



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