Next we shall suppose that $C_1$, $C_2$ are embedded nonsingular curves with $C_1\subset P^2$, $C_2\subset P^2$, respectively, and that $C_1$, $C_2$ has degree $d_1$, $d_2$, respectively, and that $d_1=d_2$, and furthermore that $C_1$ and $C_2$ are isomorphic as abstract nonsingular curves over an algebraically closed field of arbitrary characteristic $p$, $p$ being either a prime or $0$ where here $0$ denotes the additive identity of the ring $Z$ of integers which is easily seen to be the Grothendieck group of the additive monoid of natural numbers $\{0,1,\ldots\}$, or which can alternatively be viewed as the divisor class group of projective $n$-space (here we assume $n\geq 1$). \end