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I have just started learning Riemann surfaces and I am using the book by Rick Miranda: Algebraic curves and Riemann Surfaces. #F in section 1.3 asks to determine the genus of the curve in $\mathbb{P}^3$ defined by the two equations $x_0x_3=2x_1x_2$ and $x_0^2 + x_1^2 +x_2^2 +x_3^2 = 0$. #G also has a similar question in which he asks to determine the genus of the twisted cubic. Please explain how to approach this type of question.

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    I posted the same question in mathoverflow. You may check the various answers given there. http://mathoverflow.net/questions/55312/problem-in-rick-miranda-finding-genus-of-a-projective-curve2011-02-16
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    See my answer [link](http://math.stackexchange.com/questions/31371/how-to-compute-genus/56262#56262). It is a formula that you can prove with sheaf cohomology. It appears also in Exercise I.7.2(b) of Hartshorne, but I don't know an elementary proof.2011-08-08
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    I remember Miranda's book is not difficult. So maybe you can read the sections again and try to do it for yourself.2011-11-20
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    In my copy of Miranda, the equation is $x_0 x_3 = x_1 x_2$, not $x_0 x_3 = 2 x_1 x_2$. I have therefore changed the statement of the problem, under the assumption that the $2$ is a typo.2011-11-21
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    Having considered the problem a bit more, it seems that the 2 is required for the equations to define a smooth complete intersection curve. For example, without the 2, there is a critical point at $[1:i:-i:1]$. Presumably this is an error in my copy of Miranda, which was fixed in a later printing. I have therefore restored the 2 in the statement of the problem.2011-11-21

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