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## Numerically Proportional

Algebraic Geometry
Tsakanikas Nickos
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### Numerically Proportional

Let $X$ be a smooth projective surface over $\mathbb{C}$ and let $H$ be an ample divisor on $X$. Let $L$ and $M$ be two $\mathbb{Q}$-divisors on $X$ which are not numerically trivial, and such that $L^2 = L \cdot M = M^2 = 0$Show that $L$ and $M$ are numerically proportional, i.e. $\exists \ \xi \in \mathbb{R} \smallsetminus {0} \, : \, L \equiv \xi M$