TITLE: The kernel of a nef line bundle
QUESTION [6 upvotes]: 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 $\deg_LZ = 0$?

REPLY [7 votes]: 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 $\deg_L(X\cap Y) = 1$.