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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 $n\geq 1$).