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Degree of the 3-secant variety

Doyoung Choi

TL;DR

This work derives a computable degree formula for the 3-secant variety $\sigma_3(X)$ of a nonsingular projective variety $X$ embedded by a $5$-very ample line bundle, expressing the degree in terms of Segre classes of the tangent bundle $T_X$. The approach reduces the problem to the 2-secant case using a generalized double point formula, and resolves singularities of the 2-secant variety via the secant bundle, a nonsingular birational model over a Hilbert scheme. The main result provides a concrete formula $deg(\sigma_3(X))=\frac{1}{3}\left(\frac{d_X d_Y}{2}-A\right)$ with $d_Y=deg(\sigma_2(X))$ and $A=\sum_{k\ge0}{3n+2\choose k}\,deg\,s_k(C_{\Delta(X)}(X\times\sigma_2(X)))$, enabling computation in terms of intrinsic invariants, notably the Segre classes of $T_X$. The paper also demonstrates explicit curve and surface cases, illustrating the practical applicability of the formula to concrete geometries.

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.

Degree of the 3-secant variety

TL;DR

This work derives a computable degree formula for the 3-secant variety of a nonsingular projective variety embedded by a -very ample line bundle, expressing the degree in terms of Segre classes of the tangent bundle . The approach reduces the problem to the 2-secant case using a generalized double point formula, and resolves singularities of the 2-secant variety via the secant bundle, a nonsingular birational model over a Hilbert scheme. The main result provides a concrete formula with and , enabling computation in terms of intrinsic invariants, notably the Segre classes of . The paper also demonstrates explicit curve and surface cases, illustrating the practical applicability of the formula to concrete geometries.

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.
Paper Structure (5 sections, 9 theorems, 74 equations)

This paper contains 5 sections, 9 theorems, 74 equations.

Table of Contents

  1. Introduction
  2. Generalized double point formula
  3. The secant bundle
  4. Main theorem
  5. Examples

Key Result

Theorem 1.1

Let $V \subset \mathbb{P}^N$ be an equi-dimensional closed subscheme with $\mathcal{L}:= \mathcal{O}_V(1)$. Let $\sigma_0 , \cdots , \sigma_d$ be global sections of $\mathcal{L}$. Let $v^i(\underline{\sigma}, V)$ be the $v$-cycle in the Vogel's intersection theory and $\sigma : V \dashrightarrow \ma

Theorems & Definitions (24)

  • Theorem 1.1: joins1999
  • Theorem 1.2
  • Remark 2.1
  • Definition 2.2: fulton2013
  • Definition 2.3: joins1999
  • Definition 2.4
  • Theorem 2.5: joins1999
  • Definition 3.1: cf. BS88 and catanese90
  • Remark 3.2
  • Proposition 3.3
  • ...and 14 more