Inflectionary Invariants for Isolated Complete Intersection Curve Singularities / Anand P. Patel and Ashvin A. Swaminathan.

Format:

Book

Author/Creator:

Patel, Anand P., author.

Swaminathan, Ashvin A., author.

Series:

Memoirs of the American Mathematical Society ; Volume 282.

Memoirs of the American Mathematical Society Series ; Volume 282

Language:

English

Subjects (All):

Intersection theory (Mathematics).

Invariants.

Deformations of singularities.

Curves.

Physical Description:

1 online resource (114 pages)

Edition:

First edition.

Place of Publication:

Providence, RI : American Mathematical Society, [2023]

Summary:

"We investigate the role played by curve singularity germs in the enumeration of inflection points in families of curves acquiring singular members. Let N [greater than or equal to] 2, and consider an isolated complete intersection curve singularity germ f : (CN, 0) [arrow] (CN[minus]1, 0). We define a numerical function m [x in a square] [arrow] ADm(2)(f) that naturally arises when counting mth-order weight-2 inflection points with ramification sequence (0, . . . , 0, 2) in a 1-parameter family of curves acquiring the singularity f [equals] 0, and we compute ADm(2)(f) for several interesting families of pairs (f,m). In particular, for a node defined by f : (x, y) [x in a square] [arrow] xy, we prove that ADm (2)(xy) [equals] [x in a square](m[plus]1)4 [x in a square], and we deduce as a corollary that ADm (2)(f) [greater than or equal to] (mult0 [delta symbol]f ) [x in a square](m[plus]1)4 [x in a square] for any f, where mult0 [delta symbol]f is the multiplicity of the discriminant [delta symbol]f at the origin in the deformation space. Significantly, we prove that the function m [x in a square] [arrow] ADm (2)(f)[minus](mult0 [delta symbol]f ) [x in a square](m[plus]1)4 [x in a square] is an analytic invariant measuring how much the singularity "counts as" an inflection point. We prove similar results for weight-2 inflection points with ramification sequence (0, . . . , 0, 1, 1) and for weight-1 inflection points, and we apply our results to solve a number of related enumerative problems"-- Provided by publisher.

Contents:

Cover

Title page

Acknowledgments

Chapter 1. Introduction

1.1. Motivations

1.2. Overview of Main Results

Chapter 2. Background Material

2.1. The Sheaves of Principal Parts

2.2. Families of Curves and Their Inflection Points

2.3. The Sheaves of Invincible Parts

Chapter 3. Defining Automatic Degeneracy

3.1. The Definition

3.2. Application to Counting Limiting Inflection Points

3.3. Relationship to Well-Known Invariants and Multiplicities

Chapter 4. Automatic Degeneracies of a Node

4.1. Finding a Basis of ^{ }( )^{∨}

4.2. The Weight-2 Case

4.3. The Weight-1 Case

4.4. Flecnodes as Limits of Inflection Points

Chapter 5. Automatic Degeneracies of Higher-Order Singularities

5.1. The Case of Cusps

5.2. An Algorithm for Finding a Basis of ^{ }( )^{∨}

5.3. Bounds on Automatic Degeneracies

Chapter 6. Examples of Computing Automatic Degeneracies

6.1. Conditions for Automatic Degeneracies to be Zero

6.2. An Explicit Basis of ^{ }( )^{∨} for 1≤ ≤4

6.3. Computation of _{(1,1)}²( )

6.4. Computation of ₍₂₎³( ^{ }- ^{ })

6.5. Computation of ₍₁₎²( ^{ }- ^{ })

6.6. Computation of ₍₂₎⁴( ²- ^{ })

6.7. Computation of ₍₁₎³( ²- ^{ })

6.8. Expected Values of Various Planar Automatic Degeneracies

6.9. An Example in the Non-Planar Case

Chapter 7. Other Enumerative Applications

7.1. Counting Hyperflexes in a Pencil of Plane Curves

7.2. Counting Septactic Points in a Pencil of Plane Curves

7.3. Calculating Classes of Weierstrass Divisors

Appendix A. Summary of Open Problems

Bibliography

Back Cover.

Notes:

Description based on print version record.

Includes bibliographical references.

Other Format:

Print version: Patel, Anand P. Inflectionary Invariants for Isolated Complete Intersection Curve Singularities

ISBN:

9781470473532

1470473534