This is just a feeling that I had and I am curious if it is totally wrong or true to some extent.


Let $X\subseteq \mathbb{P}^r$ be an integral hypersurface of degree $r-1$, which is not a cone. In this choice of numerology, we know that through every point of $X$ there is a line that is contained in $X$. 


Let $p\in X$ and assume that there is an $n-$dimensional family of lines in $X$ that pass through $p$. Is it true that if $n$ is large enough, it will force $p$ to be a singular point?


If so, do we have an estimate for that number $n$?



Does a moving family of lines through a fixed point produce a singularity?
To expand the comment of @potentially dense: the answer is $r-2$, and it does not depend on the degree of $X$. In general, suppose $X\subset \mathbb{P}^r$ is a hypersurface, and $p$ a smooth point of $X$. Lines passing through $p$ and contained in $T\_p(X)$ are parametrized by a $\mathbb{P}^{r-2}$; since $T\_p(X)\not\subset X$, we see that lines passing through $p$ and contained in $X$ form a family of dimension at most $r-3$. Now if (and only if) $p$ is a Eckart point, i.e. $X\cap T\_p(X)$ is a cone, the dimension is exactly $r-3$. And it is easy to construct a smooth hypersurface with such a point, for instance $X$ given by $X\_0^{d-1}X\_1+X\_1^{d}+\ldots+ X\_r^d=0$, with 
$p=(1,0,\ldots ,0)$.


