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Suppose $X\subset\Bbb P^3$ is a smooth projective curve over an algebraically closed field. Define a multisecant to be a line $L$ which intersects $X$ in at least three distinct points. If $X$ has no multisecants, does this imply that there are no lines $L$ which meet $X$ in three points counted with multiplicity? (Specifically, does this imply that $X$ has no tangent line which intersects $X$ at another point?)

Background: a course I'm taking has used the "three distinct points" definition of multisecant and mentioned a result on exactly what space curves have multisecants without proving it. I want to prove it for myself, and I figured out how to prove it as long as I can assume "multisecant" means "secant line with sum of intersection multiplicities at least three". This would give me that any line intersects my curve in at most two points counted with multiplicity, but that's not actually the definition we have to work with. I was wondering if this more basic definition implies the one I want, but I'm a bit stuck in figuring this out. Intuitively, it seems like one ought to be able to "nudge" the tangent line a bit so that that the two points separate while maintaining a third intersection with the curve, but I do not know how to formalize this (and I'm worried that maybe there's some weird curve where it's not true).

Hank Scorpio
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  • The slightly more technical name for being "able to 'nudge' the tangent line" is a perturbation argument. Note that an intersection of line and curve with multiplicity one is conserved by a small continuous variation in the line's parameters (e.g. slope and intercept). There are some technical details to consider in working with a projective curve as opposed to an ordinary real planar curve. – hardmath Dec 17 '21 at 19:58
  • @hardmath Sure, if this were a plane curve, it's eay: the tangent line intersects the curve in at least three points counted with mulitplicity by assumption, which by Bezout means that the curve is of degree at least three, and therefore every line intersects it in at least three points counted with multiplicity. As a point with higher intersection multiplicity must contain the tangent line at the multiple intersection point, so by looking at the dual curve to see that one can always find a line with distinct intersection points. But idk how to make this work for space curves, hence my q. – Hank Scorpio Dec 17 '21 at 20:32
  • I'm not enough of a geometer to hope to fill in those details. But I suspect that searching with "perturbation argument" rather than "nudge" would be productive in finding good literature on space curves. – hardmath Dec 17 '21 at 20:42

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