Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

On computing local monodromy and the numerical local irreducible decomposition

Parker B. Edwards, Jonathan D. Hauenstein

TL;DR

Addressing the local structure of a holomorphic germ $oldsymbol{V}$ at a point $x^*$, the paper develops a theory of local monodromy actions for generic linear projections and uses them to recover the local irreducible decomposition via numerical witness sets. It shows that local monodromy actions arise as sub-actions of global monodromy and can be extended beyond tiny neighborhoods through analytic continuation, enabling a practical algorithm with guarantees. The authors implement the method in open-source software and demonstrate its effectiveness on cones, Whitney umbrellas, Brieskorn-type hypersurfaces, and a four-bar linkage, reporting local/global fiber and branch degrees and the corresponding monodromy groups. This work provides a robust numerical framework for local singularity analysis with broad implications for computational algebraic geometry and related fields.

Abstract

Similarly to the global case, the local structure of a holomorphic subvariety at a given point is described by its local irreducible decomposition. Following the paradigm of numerical algebraic geometry, an algebraic subvariety at a point is represented by a numerical local irreducible decomposition comprised of a local witness set for each local irreducible component. The key requirement for obtaining a numerical local irreducible decomposition is to compute the local monodromy action of a generic linear projection at the given point, which is always well-defined on any small enough neighborhood. We characterize some of the behavior of local monodromy action of linear projection maps under analytic continuation, allowing computations to be performed beyond a local neighborhood. With this characterization, we present an algorithm to compute the local monodromy action and corresponding numerical local irreducible decomposition for algebraic varieties. The results are illustrated using several examples facilitated by an implementation in an open source software package.

On computing local monodromy and the numerical local irreducible decomposition

TL;DR

Addressing the local structure of a holomorphic germ at a point , the paper develops a theory of local monodromy actions for generic linear projections and uses them to recover the local irreducible decomposition via numerical witness sets. It shows that local monodromy actions arise as sub-actions of global monodromy and can be extended beyond tiny neighborhoods through analytic continuation, enabling a practical algorithm with guarantees. The authors implement the method in open-source software and demonstrate its effectiveness on cones, Whitney umbrellas, Brieskorn-type hypersurfaces, and a four-bar linkage, reporting local/global fiber and branch degrees and the corresponding monodromy groups. This work provides a robust numerical framework for local singularity analysis with broad implications for computational algebraic geometry and related fields.

Abstract

Similarly to the global case, the local structure of a holomorphic subvariety at a given point is described by its local irreducible decomposition. Following the paradigm of numerical algebraic geometry, an algebraic subvariety at a point is represented by a numerical local irreducible decomposition comprised of a local witness set for each local irreducible component. The key requirement for obtaining a numerical local irreducible decomposition is to compute the local monodromy action of a generic linear projection at the given point, which is always well-defined on any small enough neighborhood. We characterize some of the behavior of local monodromy action of linear projection maps under analytic continuation, allowing computations to be performed beyond a local neighborhood. With this characterization, we present an algorithm to compute the local monodromy action and corresponding numerical local irreducible decomposition for algebraic varieties. The results are illustrated using several examples facilitated by an implementation in an open source software package.

Paper Structure

This paper contains 12 sections, 15 theorems, 8 equations, 7 figures, 1 table, 1 algorithm.

Table of Contents

  1. Introduction
  2. Illustrating example
  3. Background
  4. Local irreducible decomposition and local parameterization
  5. Numerical algebraic geometry and homotopy continuation
  6. Hyperplane sections and fundamental groups of complements
  7. Local monodromy actions and groups
  8. Monodromy functors
  9. Local monodromy actions of holomorphic subvarieties
  10. Computing local monodromy actions
  11. Examples
  12. Conclusion

Key Result

Theorem 1

If $\mathbf{V}$ is a reduced germ of a holomorphic subvariety of $\mathbb{C}^N$ with pure dimension $d$, there is a Zariski open set of linear projections $\mathbb{C}^N\to\mathbb{C}^d$ where, if ${\widetilde{\pi}}$ is a member, the projection map germ $\boldsymbol{{\widetilde{\pi}}}\vert_\mathbf{V}:

Figures (7)

  • Figure 1: Intersecting the cone (blue) with a general slice (green) yields an irreducible curve (cyan), while intersecting with a general slice through the origin (red) yields two lines (black).
  • Figure 2: Intersection of the cone (blue) with line (red) yields two points (black) which are the start points of two paths (yellow) that limit to the origin (green).
  • Figure 3: (a) Critical points (magenta) with respect to ${\widetilde{\pi}}$ on the cone (blue); (b) critical locus (magenta) in the image of ${\widetilde{\pi}}$ intersected with a line (black) passing through $\gamma_1$ (cyan) yielding two points (blue).
  • Figure 4: (a) Illustration of a basic loop starting at $\gamma_1$ and only encircling $p_1$ once counterclockwise; (b) pictorial illustration of monodromy action interchanging ${\widetilde{s}}_1$ and ${\widetilde{s}}_2$.
  • Figure 5: Illustrating first translation homotopy in \ref{['lem:generators']}: (a) loop $\ell_i^{t_1}$ (black, orange arrow), $\ell_i^{t_2}$ (pink, dashed), path $p_i\vert_{[t_1,t_2]}$ (red), and path $\gamma$ (bottom, blue arrow); (b) loop $\ell_i^{t_1}$ has been translated to $\ell_i^{t_1}+c_t$ for some $t\in [t_1,t_2]$; (c) loop ends translation at $\ell_i^{t_1}+c_{\gamma(t_2)-\gamma(t_1)}$.
  • ...and 2 more figures

Theorems & Definitions (48)

  • Theorem
  • Lemma 2.1
  • proof
  • Remark 2.2
  • Definition 2.3: From BHS16
  • Definition 2.4: BHS16
  • Example 2.5
  • Lemma 2.6
  • proof
  • Remark 2.7
  • ...and 38 more