unused proofs etc.tex

%It is well known that this line bundle is ample, We see ampleness by the an argument in \cite{}.
%\begin{lemma}
%For sufficiently large \(d\), \(L_d\) is ample
%\end{lemma}
%\begin{proof}
%Since \(L\) is ample then, for some \(d>0\), \(L^{\otimes d} \otimes \mathcal{J}_V\) is globally generated. Consider the linear system \(\Sigma \), defined as the image of the natural map \(H^0(L^{\otimes d} \otimes \mathcal{J}_V) \to H^0(L^{\otimes d})\), which has base locus \(V\). Let \(\varphi: X \to \PP^N\) be the associated rational map. We may identify \(W\) as the graph of \(\varphi\) in \(X \times \PP^N\). Then
%\[
%p_2^* \O(1) |_W = f^*L \otimes \O(-E),
%\]
%and so, for \(d\) large enough such that \(L^{\otimes d}\) is very ample, we have:
%\[
%L_{d+1} = f^*L^{\otimes d+1} \otimes \O(-E) \cong L^{\otimes d} \otimes p_2^* \O(1) |_W \cong p_1^* L^{\otimes d} \otimes f^* \O(1) |_W.
%\]
%\end{proof}

\include{preamble}
\begin{document}

\begin{figure}[H]
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