On the degree of a modular map

TL;DR

This paper investigates a rational map associated with a general cubic hypersurface in projective 4-space, calculating its degree as 2,074,320, revealing intricate geometric properties of lines on the hypersurface.

Contribution

It provides the first explicit computation of the degree of a natural rational map related to lines on a cubic hypersurface in $ ext{P}^4$.

Findings

The degree of the rational map is 2,074,320.

Six lines pass through a general point on the hypersurface.

The geometric configuration involves a rank 3 quadric cone with vertex at the point.

Abstract

Let $X$ be a general cubic hypersurface in $\mathbb P^4$. If $x\in X$ is a general point there are exactly six distinct lines in $X$ passing through $x$, that lie on the rank 3 quadric cone with vertex $x$ of lines that have intersection multiplicity at least 3 with $X$ in $x$. So there is a natural rational map $X\dasharrow \mathcal M_2$. In this paper we compute its degree to be 2074320.