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?

weightedprojective space. $\endgroup$ – Daniel Loughran Dec 16 '13 at 13:05