Suppose $X$ is a smooth projective surface of general type of degree $d$ in $\mathbb{P}^n$. Is it possible to find a curve of degree $k$ which intersects $X$ in $dk$ points ?
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$\begingroup$ Certainly if your surface is contained in some $\mathbb{P}^3\subset \mathbb{P}^n$. $\endgroup$– abxApr 1, 2021 at 4:20

$\begingroup$ I am sorry. I meant for a nondegenerated surface in $\mathbb{P}^n$. $\endgroup$– LAPRASApr 1, 2021 at 4:41

$\begingroup$ You are asking for some $k$ or for all $k$? $\endgroup$– Francesco PolizziApr 1, 2021 at 6:49

$\begingroup$ I am asking for all $k$. But is it true for some $k$? If yes, then can we say something about $k$? $\endgroup$– LAPRASApr 1, 2021 at 7:10
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1 Answer
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The answer is no for all $k$, in fact, there are already counterexamples for $k=1$.
For instance, there exist smooth, nondegenerate surfaces $X \subset \mathbb{P}^6$ such that every line intersect them in at most two points, see
S. Di Rocco, K. Ranestad: On surfaces in $\mathbb{P}^{6}$ with no trisecant lines, Ark. Mat. 38, No. 2 (2000), 231261 . ZBL1035.14011.


$\begingroup$ I do not know off the top of my head. I should think about it. $\endgroup$ Apr 1, 2021 at 7:54