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LSpice
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Non-degenerate pairing on Néron–Severi Group

$\DeclareMathOperator\NS{NS}\DeclareMathOperator\Pic{Pic}$Let $X$ be a smooth projective variety over an algebraically closed field $k$. Let $\NS(X)$ denote the group $[\Pic(X)/\Pic^{\text a}(X)]\otimes_{\mathbb{Z}}\mathbb{Q}$, where $\Pic^{\text a}(X)$ denotes the divisors which are algebraically equivalent to 0. If $X$ is a surface, then the pairing on $\NS(X)$ given by $(D,D')\mapsto D\cdot D'$ is non-degenerate.

For higher dimensions, let $\Theta$ denote an ample divisor on $X$. Let $d$ be the dimension of $X$. Is it true that the pairing on $\NS(X)$ given by $(D,D')\mapsto D\cdot D'\cdot \Theta^{d-2}$ is non-degenerate?

Is there a good reference for the above result?

Rex
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