Garrity's book first asks the reader to compute the divisor associated to $y/z$ on $$V(y^2z-xz^2-x^3) \subset \Bbb{CP}^2.$$ A straightforward computation, which I have verified by looking it up in a draft of the book I found online that has answers, shows that
$$\DeclareMathOperator{div}{div} \div(y/z)=(0:0:1)+(1:0:i)+(1:0:-i)-(0:1:0)$$
which is a principal divisor with degree $1+1+1-1=2.$
A bit further down the page, Garrity then claims that all principal divisors have degree zero, because we are working with homogeneous rational functions, so the number of zeros (degree of numerator) is equal to the degree of the denominator (number of poles).
I find this plausible, but the computation I did above is an example of a divisor with nonzero degree.
What's going on here?