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