Consider the curve given by $z^{2g-2}y^2=\displaystyle\prod_{i=1}^{2g}(x-a_iz)$. This is a hyperelliptic curve and has genus $g-1$. At the same time it is a curve defined by an equation of $d=2g$ and hence by adjunction formula its genus should be ${d-1\choose 2}={2g-1 \choose 2}$, and $g-1\neq {2g-1 \choose 2}$. What am I missing here?
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1$\begingroup$ This is a fairly standard exercise on hyperelliptic curves, it would be good for you to work it out yourself. The problem lies with your application of the adjunction formula. $\endgroup$– Daniel LoughranCommented Dec 16, 2013 at 12:51
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1$\begingroup$ You mean the problem is that the curve is not smooth at [0:1:0]? $\endgroup$– Beginner2013Commented Dec 16, 2013 at 12:55
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1$\begingroup$ Exactly. The usual definition of a hyperelliptic curve is a smooth projective curve birational to a curve of the form $y^2 = f(x)$, where $f$ is separable. The equation you have written down is not smooth. To obtain the smooth projective model, one usually works in a weighted projective space. $\endgroup$– Daniel LoughranCommented Dec 16, 2013 at 13:05
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3$\begingroup$ Alternatively to working in a weighted projective space is to write your curve as two smooth affine curves that are glued together. (I think this is actually more standard than weighted projective spaces, but chacun a son gout.) Also, this question is likely to be closed, since it's really not research level, so it would be better on MathStackExchange. But don't feel that that's an insult or be discouraged, we all started by learning basic concepts. $\endgroup$– Joe SilvermanCommented Dec 16, 2013 at 13:56
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1$\begingroup$ You can also embed the curve in a quadric: mathoverflow.net/questions/79546 $\endgroup$– Jack HuizengaCommented Dec 16, 2013 at 15:10
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