$\DeclareMathOperator\NS{NS}\DeclareMathOperator\Pic{Pic}$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?



Non-degenerate pairing on Néron–Severi Group
$\DeclareMathOperator\NS{NS}$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 $D'\in \NS(X)\_\mathbb{Q}$.




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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.].


