Varieties of dimension $n$ with an $n$-parameter family of lines

Question

Let $X$ be a projective variety of dimension $n$ with an $n$-parameter family of lines. Is $X\cong \mathbb{P}^n$? I know this is true for $n=2$, but I'm curious about generalizations.

Answer 1

There are other examples, for example hypersurfaces $X_d\subset \mathbb{P}^{n+1}$ of dimension $n$ and low degree $d$ relative to $n$.

It suffices to show we can find $d$ so there is a 1-parameter family of lines through any point $p\in X_d$. The lines through $p$ in $\mathbb{P}^{n+1}$ are parameterized by a $\mathbb{P}^{n}$. You can show (by expanding out the equation defining $X$ around $p$) that the family of lines through $p$ in $X_d$ is dimension at least $n-d$.

Therefore, it suffices to have $d\leq n-1$.

More generally, if $L\subset X\subset \mathbb{P}^N$ and the canonical divisor $K_X$ restricts to something of high enough degree on $L$ (I think $\geq 3$), then we can wiggle the line in $X$ enough to get an $n$-dimensional family. Somebody who knows more about such varieties (and this story) should tell me more.