Inflectionary Invariants for Isolated Complete Intersection Curve Singularities

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 $\operatorname{AD}_{(2)}^m(f) \geq (\operatorname{mult}_0 Δ_f) \cdot {{m+1} \choose 4}$ for any $f$, where $\operatorname{mult}_0 Δ_f$ is the multiplicity of the discriminant $Δ_f$ at the origin in the deformation space. Furthermore, we show that the function $m \mapsto \operatorname{AD}_{(2)}^m(f) -(\operatorname{mult}_0 Δ_f) \cdot {{m+1} \choose 4}$ 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.

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