Let $X$ be a nonsingular threefold of degree $d$ contained in $\mathbb{P}^4$ and let $P$ be a point on $X$. Is it possible to find a plane $H$ contained in $T_P (X)$ such that $X \cap H$ is not a union of $d$ lines?
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4$\begingroup$ I recommend that you search for "Eckardt point". There are points on some smooth hypersurfaces such that the corresponding tangent hyperplane section is a cone. Thus, every $2$-plane contained in the tangent hyperplane intersects the hypersurface in a union of lines. $\endgroup$– Jason StarrCommented Nov 21, 2018 at 12:35
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It seems that the case that you have in mind is $d=2$. In this case (and over an algebraically closed field $k$, char$k$ ≠ 2), you only have two singular curves of degree 2: a double line and two lines sharing a point. So in the case of quadric threefold, you will get two lines or a double line as soon as H contains P (I guess, you want this assumption).
On the contrary, for $d\ge3$ you will almost never get a union of lines by this procedure. Note that you can take a hyperplane section $X\cap H'$ first and then take the tangent space of $P\in X\cap H'$, so your question is essentially about lines on degree-$d$ surfaces. And there are no more than $d(11d-24)$ lines lying on a smooth surface of degree $d\ge 3$. This result goes back to Salmon's work in 1849 and, as for stardard textbooks, it appears, for example, as Proposition 11.9 in "3264" by D. Eisenbud and J. Harris, which also contains references for better estimates, since for $d>3$ the $d(11d-24)$ estimate is not sharp.
So, back to your question, for $d\ge 3$, you will get a union of line and something of degree $d-1$ only if you are lucky enough to pick $P$ on one of finite (and small) number of lines in $X\cap H'$. And it is really rare to get a union of $d$ lines, $P$ in this case is "Eckardt point" mentioned by Jason Starr in comments.
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$\begingroup$ I was wondering, what if I ask the same question for a plane in $T_P (X)$ that does not pass through $P$? In the spirit of the approach you have already mentioned, if one finds a 3-plane $H'$ with $P \not\in H'$ and $H'$ is not a tangent any point on the threefold, can we assert that $X \cap H' \cap T_P(X)$ will not be union of d lines? $\endgroup$– M. D.Commented Nov 22, 2018 at 12:22
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$\begingroup$ Again, by the same argument, using Bertini theorem, we can say that $X\cap H'\cap T_P(X)$ under these assuptions will be generally smooth, so for $d>1$ it will contain no lines. $\endgroup$– A KCommented Nov 22, 2018 at 23:02