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 ?

  • $\begingroup$ Certainly if your surface is contained in some $\mathbb{P}^3\subset \mathbb{P}^n$. $\endgroup$
    – abx
    Apr 1, 2021 at 4:20
  • $\begingroup$ I am sorry. I meant for a non-degenerated surface in $\mathbb{P}^n$. $\endgroup$
    – LAPRAS
    Apr 1, 2021 at 4:41
  • $\begingroup$ You are asking for some $k$ or for all $k$? $\endgroup$ Apr 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 $k$? $\endgroup$
    – LAPRAS
    Apr 1, 2021 at 7:10

1 Answer 1


The answer is no for all $k$, in fact, there are already counterexamples for $k=1$.

For instance, there exist smooth, non-degenerate 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), 231-261 . ZBL1035.14011.

  • $\begingroup$ Is it true for some $k$ ? $\endgroup$
    – LAPRAS
    Apr 1, 2021 at 7:51
  • $\begingroup$ I do not know off the top of my head. I should think about it. $\endgroup$ Apr 1, 2021 at 7:54

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