Degree of the 3-secant variety

TL;DR

This paper derives a formula for the degree of the 3-secant variety of a nonsingular projective variety using Segre classes and a generalized double point formula, addressing singularities via secant bundles.

Contribution

It introduces a new formula for the degree of the 3-secant variety based on Segre classes and extends double point formulas to handle singular cases.

Findings

Derived a formula for the degree of the 3-secant variety.

Reduced the problem to the 2-secant case using a generalized double point formula.

Handled singularities with secant bundles as birational models.

Abstract

In this paper, we present a formula for the degree of the 3-secant variety of a nonsingular projective variety embedded by a 5-very ample line bundle. The formula is provided in terms of Segre classes of the tangent bundle of a given variety. We use the generalized version of double point formula to reduce the calculation into the case of the 2-secant variety. Due to the singularity of the 2-secant variety, we use secant bundle as a nonsingular birational model and compute multiplications of desired algebraic cycles.