Singular plane sections and the conics in the Fermat quintic threefold

We present explicit equations for the space of conics in the Fermat quintic threefold $X$, working within the space of plane sections of $X$ with two singular marked points. This space of two-pointed singular plane sections has a birational morphism to the space of bitangent lines to the Fermat quintic threefold, which in its turn is birational to a 625‑to‑1 cover of $\mathbb{P}^4$. We illustrate the use of the resulting equations in identifying special cases of one-dimensional families of conics in $X$.

Keywords

quintic threefold, conics, Hilbert scheme

2010 Mathematics Subject Classification

Primary 14J30. Secondary 14J32.

Full Text (PDF format)

Received 7 September 2021

Received revised 5 April 2022

Accepted 5 April 2022

Published 25 October 2022