For which values of $d$ is the following possible: There exist a smooth hypersurface $X$ in $\mathbb{P}^3$ of degree $d$ with Picard number $d$, containing $d-1$ lines $l_1,...,l_{d-1}$ on the same plane such that the Neron-Severi group of $X$ is generated by the ample line bundle and the classes of these $d-1$ lines $l_i$?
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4$\begingroup$ Please explain what cases you know, allude to why you might care, what your expectation may be and why, etc. Questions asked in this fashion do not generally receive good responses. $\endgroup$– Jack HuizengaNov 29, 2013 at 5:52
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1$\begingroup$ What's the base field? Is it algebraically closed? $\endgroup$– Martin BrightNov 29, 2013 at 9:26
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$\begingroup$ @Bright: The base field in $\mathbb{C}$. $\endgroup$– Naga VenkataNov 29, 2013 at 14:36
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$\begingroup$ @Huizenga: The motivation comes from the study of Noether-Lefschetz locus which I unfortunately do not think is possible to explain in a paragraph. I am sorry. I would expect that for $d$ large enough this phenomenon happens. Again this is motivated by results on Noether-Lefschetz locus. $\endgroup$– Naga VenkataNov 29, 2013 at 14:42
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