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Let $X$ be a smooth projective variety over an algebraically closed field $k$. Let $NS(X)$ denote the group $[Pic(X)/Pic^a(X)]\otimes_{\mathbb{Z}}\mathbb{Q}$, where $Pic^a(X)$ 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.

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up vote 6 down vote accepted

I don't have know a reference off hand. So let me just give you a proof when $char\\, k=0$. 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 $D\cup D''\not=0$ 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 surface given as 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 $H^2(X_{et},\mathbb{Q}_\ell)$. Therefore the claim follows from weak Lefschetz [Milne, Etale cohomology]

The pairing on $NS(X)_\mathbb{Q}$ is nonzero multiple of the restriction of the pairing on $Y$. The result now follows from the Hodge index theorem for surfaces [Hartshorne, Alg. Geom.]

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