Let $X$ be a smooth degree $d$ ($d>5$) surface in $\mathbb{P}^3$. We now blow up $X$ at a point, embed it in some projective space, and and consider a projection of it into $\mathbb{P}^3$. The question is does this resulting surface have only ADE singularities? If not when is it the case? What is the degree of the final surface?
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5$\begingroup$ The phrasing could use a little work. Given that your blowup Y of X cannot be embedded in P^3 (the canonical bundle is of the wrong type) I assume you are asking for an embedding of Y in some projective space together with a projection from Y to P^3 which is birational onto its image and has only ADE singularities. Is this correct? $\endgroup$– Yusuf MustopaCommented Mar 5, 2013 at 1:44
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$\begingroup$ @Mustopa: Thank you. You are completely correct. $\endgroup$– Naga VenkataCommented Mar 5, 2013 at 6:59
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$\begingroup$ If you do not fix the embedding of Y into a projective space, the degree can be almost anything (with fixed d). $\endgroup$– Jérémy BlancCommented Mar 5, 2013 at 12:24
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