## Advanced Studies in Pure Mathematics

### Varieties of lines on Fermat hypersurfaces

Tomohide Terasoma

#### 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 $F(X)$.

#### Article information

Dates
Revised: 1 April 2011
First available in Project Euclid: 24 November 2018

https://projecteuclid.org/ euclid.aspm/1543085018

Digital Object Identifier
doi:10.2969/aspm/06210459

Mathematical Reviews number (MathSciNet)
MR2933806

Zentralblatt MATH identifier
1263.14008

Keywords
Fano variety Fermat hypersurface

#### Citation

Terasoma, Tomohide. Varieties of lines on Fermat hypersurfaces. Arrangements of Hyperplanes — Sapporo 2009, 459--474, Mathematical Society of Japan, Tokyo, Japan, 2012. doi:10.2969/aspm/06210459. https://projecteuclid.org/euclid.aspm/1543085018