Let $V$ be a complex projective variety and $L$ a nef line bundle on $V$ (i.e., $L$ is non-negative on every curve in $V$). Denote, as usual, $\deg\_LX = c\_1(L)^{\dim{X}}.[X]$ for $X$ a subvariety of $V$, considered as a prime cycle of dimension $\dim{X}$.


**Question.** *For subvarieties $X$ and $Y$ of $V$ with $\deg\_LX = \deg\_LY = 0$, does it follow that all positive-dimensional components $Z$ of $X \cap Y$ necessarily have $\deg\_LZ = 0$?* 



The kernel of a nef line bundle
Consider $V=\mathbb{P}^1\times \mathbb{P}^2$ with projections $p\_1\colon V \rightarrow \mathbb{P}^1 \text{ and } p\_2\colon V \rightarrow\mathbb{P}^2.$ Let $L = p\_1^\*(\mathcal{O}\_{\mathbb{P}^1}(1))$, let $X = p\_2^{-1}(\ell\_1)$ and $Y=p\_2^{-1}(\ell\_2)$ for two distinct lines $\ell\_1$ and $\ell\_2\subset \mathbb{P}^2$. Then $Z = X\cap Y \cong \mathbb{P}^1$ is a section of $p\_1$.


Finally we have $\deg\_L(X) = \deg\_L(Y) = 0$, but $\deg\_L(X\cap Y) = 1$.


