If you have an explicit equation for a (-1) curve on a surface, and an explicit equation for the surface, so that the surface comes as a blowup with the (-1) curve being the exceptional divisor, is there some formula for writing down the equations of the blow-down map (like the formula for the blowup)?
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Additional details might help. By "equation for the surface" and "formula for the blow-up" do you mean you have/want (i) homogeneous polynomials (in ambient coordinates) cutting out your surface, (ii) an implicit equation (in local coordinates) for a $(-1)$-curve, (iii) a parametric description of the blow-down map (in local coordinates), or something else? – Andrew D. Hwang Jul 21 '13 at 18:16
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Let's say that I have the homogeneous equations of the surface in projective space, and also the homogeneous equations of the curve. I would like whatever you can give me about the blow-down. – Tony Jul 21 '13 at 18:30