Inflectionary Invariants for Isolated Complete Intersection Curve Singularities (original) (raw)

2017, arXiv (Cornell University)

We investigate the role played by curve singularity germs in the enumeration of inflection points in families of curves acquiring singular members. Let N ≥ 2, and consider an isolated complete intersection curve singularity germ f : (C N , 0) → (C N−1 , 0). We define a numerical function m → AD m (2) (f) that naturally arises when counting m th-order weight-2 inflection points with ramification sequence (0,. .. , 0, 2) in a 1-parameter family of curves acquiring the singularity f = 0, and we compute AD m (2) (f) for several interesting families of pairs (f, m). In particular, for a node defined by f : (x, y) → xy, we prove that AD m (2) (xy) = m+1 4 , and we deduce as a corollary that AD m (2) (f) ≥ (mult 0 ∆ f) • m+1 4 for any f, where mult 0 ∆ f is the multiplicity of the discriminant ∆ f at the origin in the deformation space. Significantly, we prove that the function m → AD m (2) (f) − (mult 0 ∆ f) • m+1 4 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. CONTENTS 1. Introduction 1.1. Motivations 1.2. Overview of Main Results 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 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 4. Automatic Degeneracies of a Node 4.1. Finding a Basis of P m (xy) ∨ 4.2. The Weight-2 Case 4.3. The Weight-1 Case 4.4. Flecnodes as Limits of Inflection Points 5. Automatic Degeneracies of Higher-Order Singularities 5.1. The Case of Cusps 5.2. An Algorithm for Finding a Basis of P m (f) ∨ 5.3. Bounds on Automatic Degeneracies