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I'm looking for a formula or quick trick for computing the branch locus of a hypersurface under projection. My specific case:

Let $V\subset\mathbb{P}^3$ be a surface of degree $d$, $p\in\mathbb{P}^3$ (it's actually a node of $V$, if that's relevant), and I'm attempting to compute the branch locus of the projection $V\dashrightarrow \mathbb{P}^2$, which should be a plane curve.

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    Per$h$$a$ps this could be helpful: http://books.google.com/books?id=WUDrpDGHHEsC&pg=PA31&lpg=PA31&dq=computing+the+branch+locus+V+of+P^2&source=bl&ots=PTrmd4B3uF&sig=4JADnIAEff_R2YoKAFes$z$P_Gbc0&hl=e$n$&ei=XstETbPgGMGB8ga4htSZAg&sa=X&oi=book_result&ct=result&resnum=3&ved=0CCMQ6AEwAg#v=onepage&q=computing%20the%20branch%20locus%20V%20of%20P%5E2&f=false2011-01-30

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