# Surface Intersection on $\mathbb{P}^3$

I just can't figure out how to solve this problem.

"Let $S=Z(xy-zw)\subset \mathbb{P}^3$ and $l \subset S$ a line. Show that does not exist any surface $S'\subset \mathbb{P}^3$ such that $S\cap S'=l$."

Thank you so much for the help

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Think about degrees. –  Andrew Oct 21 '12 at 21:06
Such a surface would have some positive degree $d$. What's the degree of $S$? So what would that make the degree of $S \cap S'$? This produces your contradiction. (At least, provided we are interpreting the equation $S \cap S' = l$ scheme-theoretically.)