On linear sections of a nonsingular threefold 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? 
 A: 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.
