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?
 A: $\DeclareMathOperator\NS{NS}$I don't have  know a reference off hand. So let me just give you a proof when
$\operatorname{char} k=0$. I expect it's true in general.
We can assume without loss of generality that $k=\mathbb{C}$.
Then via the exponential sequence and the Lefschetz $(1,1)$ theorem, $\NS(X)_\mathbb{Q}$
(your $\NS(X)$) can be identified the space of $(1,1)$ classes in $H^2(X,\mathbb{Q})$.
Given a nonzero $D\in \NS(X)_\mathbb{Q}$, there exists $D''\in H^{d-1,d-1}(X)\cap H^{2d-2}(X,\mathbb{Q})$ such that $D\cup D''\ne0$ by Poincare duality. Hard Lefschetz tells
us that $D'' = \Theta^{d-2}\cdot D'$ for some $D'\in \NS(X)_\mathbb{Q}$.

Here's a different argument which works over any algebraically closed field.
Let $Y\subset X$ be a surface given as the intersection of $d-2$ general hyperplanes with
respect to the embedding given by $N\Theta$, $N\gg 0$.
Claim: The restriction $\NS(X)_\mathbb{Q}\to \NS(Y)_\mathbb{Q}$ is injective.
Proof:
We can see, from the Kummer sequence, that $\NS(X)_\mathbb{Q}\subset \NS(X)_\mathbb{Q_\ell}$ embeds into $\ell$-adic cohomology $H^2(X_{\text{ét}},\mathbb{Q}_\ell)$. Therefore the claim follows from weak Lefschetz [Milne, Etale cohomology].
The pairing on $\NS(X)_\mathbb{Q}$ is a nonzero multiple of the restriction of the pairing on $Y$. The result now follows from the Hodge index theorem for surfaces [Hartshorne, Alg. Geom.].
A: S. Kleiman, Les théorèmes de finitude pour le foncteur de Picard, Exp. XIII of Séminaire de géométrie algébrique du Bois-Marie - 1966--1967 - Théorie des intersections et théorème de Riemann-Roch (SGA6) (P. Berthelot, A. Grothendieck, L. Illusie, eds), Lect. Notes in Math. 225, Springer, 1971, 616--666,
See Cor. 7.4 (i) p. 665.
