Varieties of lines on Fermat hypersurfaces

Abstract

Let $X$ be a hypersurface in the projective space $\mathbf{P}^{n+1}$ and $G(n + 1, 1)$ be the Grassmann variety $G(n+ 1, 1)$ of lines in $\mathbf{P}^{n+1}$. The subvariety $F(X)$ of $G(n+ 1, 1)$ consisting of lines contained in $X$ is called the Fano variety of $X$. We study a detailed structure of the Fano variety of the Fermat hypersurface $X$ of degree $d$ for $n \geq d$. More precisely, we show that a certain open subset $F^0 (X)$ of $F(X)$ has a fibration structure over a moduli space of marked pointed rational curves and that the fibers are complete intersections of Fermat hypersurfaces introduced in [T]. We also study singularities of $F(X)$.