Lines on the secant cubic hypersurfaces of Severi varieties

TL;DR

This paper investigates the geometry of lines on secant cubic hypersurfaces of Severi varieties, establishing a Torelli-type theorem that characterizes secant cubics via their Fano varieties of lines.

Contribution

It provides a geometric description of lines on secant cubics and proves a Torelli theorem linking the hypersurface to its Fano variety of lines.

Findings

Description of the Fano variety of lines on secant cubics

Verification of the Torelli theorem for secant cubics

Characterization of secant cubics via their lines

Abstract

The secant varieties of Severi varieties provide special examples of (singular) cubic hypersurfaces. An interesting question asks when a given cubic hypersurface is projectively equivalent to a secant cubic hypersurface. Inspired by the ''geometric'' Torelli theorem for smooth cubic hypersurfaces due to F. Charles, we study the geometry of lines on secant cubics and describe the Fano variety of lines. Then we verify the ''geometric'' Torelli theorem for the case of secant cubics. Namely, a cubic hypersurface is isomorphic to a secant cubic if and only if their Fano varieties of lines are isomorphic.