The arithmetic genus of nonsingular curve C of degree d in PP^3 over an algebraically closed field is less than or equal to 1/2(d-1)(d-2). I must show it by comparing C with a suitable projection from a point into PP^2. How can I prove it?
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$\frac{1}{2}(d-1)(d-2)$ is the genus of a smooth plane curve of degree $d$. If you project from $P^3$ to $P^2$ off a point not contained in $C$ you can always get a plane curve of the same degree with at most nodes as singularities, which is birational to $C$ (hence has the same genus). Each node lowers the genus of the image curve by one w.r.t the formula you give. Hence you have the desired inequality. |
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