# Inflectionary Invariants for Isolated Complete Intersection Curve Singularities

## TL;DR

This paper introduces and computes new numerical invariants for isolated complete intersection curve singularities, linking them to inflection point counts in families of curves and providing tools for enumerative geometry.

## Contribution

It defines the invariant _{(2)}^m(f) for singularities and computes it for specific cases, connecting it to the multiplicity of the discriminant and inflection point enumeration.

## Key findings

- _{(2)}^m(xy) = {{m+1}  4} for a node.
- _{(2)}^m(f)   ( ext{mult}_0 \u2206_f)  {{m+1}  4} for general f.
- The difference _{(2)}^m(f) - ( ext{mult}_0 \u2206_f)  {{m+1}  4} is an analytic invariant.

## Abstract

We investigate the role played by curve singularity germs in the enumeration of inflection points in families of curves acquiring singular members. Let $N \geq 2$, and consider an isolated complete intersection curve singularity germ $f \colon (\mathbb{C}^N,0) \to (\mathbb{C}^{N-1},0)$. We introduce a numerical function $m \mapsto \operatorname{AD}_{(2)}^m(f)$ that arises as an error term when counting $m^{\mathrm{th}}$-order weight-$2$ inflection points with ramification sequence $(0, \dots, 0, 2)$ in a $1$-parameter family of curves acquiring the singularity $f = 0$, and we compute $\operatorname{AD}_{(2)}^m(f)$ for various $(f,m)$. Particularly, for a node defined by $f \colon (x,y) \mapsto xy$, we prove that $\operatorname{AD}_{(2)}^m(xy) = {{m+1} \choose 4},$ and we deduce as a corollary that $\operatorname{AD}_{(2)}^m(f) \geq (\operatorname{mult}_0 \Delta_f) \cdot {{m+1} \choose 4}$ for any $f$, where $\operatorname{mult}_0 \Delta_f$ is the multiplicity of the discriminant $\Delta_f$ at the origin in the deformation space. Furthermore, we show that the function $m \mapsto \operatorname{AD}_{(2)}^m(f) -(\operatorname{mult}_0 \Delta_f) \cdot {{m+1} \choose 4}$ is an analytic invariant measuring how much the singularity "counts as" an inflection point. We obtain similar results for weight-$2$ inflection points with ramification sequence $(0, \dots, 0, 1,1)$ and for weight-$1$ inflection points, and we apply our results to solve various related enumerative problems.