Order of Vanishing on Projective Variety

Question

I know for a curve $C$ in affine space, we can define a the order of vanishing at a smooth point $p \in C $by noting that $\mathcal{O}_p$, the local ring at $p$, is a DVR with maximal ideal $\mathfrak{m}_p$ and letting $\nu_p(f)$ be the largest $d$ such that $f \in \mathfrak{m}_p^d$. I was wondering if there is a similar notion for curves in projective space, i.e. how can we define valuation or orders of vanishing at a smooth point like the order of vanishing of $z/x$ at $[0:0:1]$ on $V(y^2z-x^3)$, or more generally any function in the fraction field? Thank you

Answer 1

Choose any affine chart containing the point $p \in C$ at which you want to compute these things. Then you just compute in the affine chart. It's an exercise to show that the result is independent of the chosen chart.