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Numerically Proportional

Algebraic Geometry
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Tsakanikas Nickos
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Numerically Proportional

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Post by Tsakanikas Nickos » Mon Feb 20, 2017 10:42 pm

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. \[ \exists \ \xi \in \mathbb{R} \smallsetminus {0} \, : \, L \equiv \xi M \]
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