Title: Dimension of the space of conics on Fano hypersurfaces,
Abstract: R. Beheshti showed that, for a smooth Fano hypersurface $X$ of degree $\leq
8$ over the complex number field $\mathbb{C}$, the dimension of the space of
lines lying in $X$ is equal to the expected dimension. We study the space of
conics on $X$. In this case, if $X$ contains some linear subvariety, then the
dimension of the space can be larger than the expected dimension. In this
paper, we show that, for a smooth Fano hypersurface $X$ of degree $\leq 6$ over
$\mathbb{C}$, and for an irreducible component $R$ of the space of conics lying
in $X$, if the $2$-plane spanned by a general conic of $R$ is not contained in
$X$, then the dimension of $R$ is equal to the expected dimension.