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I heard once about one open problem. That was about existing a hypersurface of a small degree (5? or 6?) passing through some number (5? 6?) of 3-fold points and 2-fold lines (3 lines?).

It was said that experimentally we can always find such a hypersurface but there is no proof, and counting expected dimension shows that there should be none of them

Do you know exact numbers in this problem, or some research around?

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  • $\begingroup$ Should it be of dimension 2? Does this question related with 3-secant varieties? $\endgroup$
    – IBazhov
    Commented Feb 13, 2015 at 12:54
  • $\begingroup$ 1) I don't remember. 2) No, I guess. That is about existence of a variety, how would this be related to secant varieties? $\endgroup$
    – Nikita Kalinin
    Commented Feb 14, 2015 at 10:13
  • $\begingroup$ Such kind of varieties can arise in the following context. Let $X\subset \mathbb P^4$ be a surface. Let $S_3$ be a corresponding 3-secant variety and $I_3$ be incidence variety. Due to Severy, the degree $\delta$ of $I_3\to S_3$ is the number apparent triple points of $X$, i.e., the number of triple points of hypersurface $\pi(X)$, where $\pi:\mathbb P^4\to\mathbb P^3$ a projection from a point. $\endgroup$
    – IBazhov
    Commented Feb 14, 2015 at 14:20
  • $\begingroup$ I invite you to look into mathnet.or.kr/mathnet/kms_tex/973054.pdf for introduction and into sciencedirect.com/science/article/pii/S0001870804003482 for developments. See also mathsci.kaist.ac.kr/asarc/kwak/paper/smooth.pdf and dm.unipi.it/~depoi/pubblicazioni/pietro.pdf. $\endgroup$
    – IBazhov
    Commented Feb 14, 2015 at 14:20
  • $\begingroup$ the question is opposite. One has no intention to study a (particulary defined) variety which are singular (by coincidence). One wants to prove that for a given set of singularities (some number of triple points and double lines) there is no a hypersurface. These are quite different questions. One concerns (particular, though numerous)examples, the other concerns about existence. $\endgroup$
    – Nikita Kalinin
    Commented Feb 14, 2015 at 21:51

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