Dimension of the space of conics on Fano hypersurfaces

TL;DR

This paper investigates the dimension of the space of conics on smooth Fano hypersurfaces of degree up to 6, establishing conditions under which the dimension matches the expected value, especially when the spanned 2-plane is not contained in the hypersurface.

Contribution

It extends the understanding of conic spaces on Fano hypersurfaces by identifying when their dimension equals the expected dimension, particularly for degrees up to 6 and certain geometric conditions.

Findings

Dimension of the space of conics equals the expected dimension under specified conditions.

If the 2-plane spanned by a general conic is not contained in the hypersurface, the dimension matches the expected.

Results generalize known facts about lines to 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.