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 ≥ 2 N \geq 2 , and consider an isolated complete intersection curve singularity germ f : ( C N , 0 ) → ( C N − 1 , 0 ) f \colon (\mathbb {C}^N,0) \to (\mathbb {C}^{N-1},0) . We define a numerical function m ↦ AD ( 2 ) m ⁡ ( f ) m \mapsto \operatorname {AD}_{(2)}^m(f) that naturally arises when counting m t h m^{\mathrm {th}} -order weight- 2 2 inflection points with ramification sequence ( 0 , … , 0 , 2 ) (0, \dots , 0, 2) in a 1 1 -parameter family of curves acquiring the singularity f = 0 f = 0 , and we compute AD ( 2 ) m ⁡ ( f ) \operatorname {AD}_{(2)}^m(f) for several interesting families of pairs ( f , m ) (f,m) . In particular, for a node defined by f : ( x , y ) ↦ x y f \colon (x,y) \mapsto xy , we prove that AD ( 2 ) m ⁡ ( x y ) = ( m + 1 4 ) , \operatorname {AD}_{(2)}^m(xy) = {{m+1} \choose 4}, and we deduce as a corollary that AD ( 2 ) m ⁡ ( f ) ≥ ( mult 0 ⁡ Δ f ) ⋅ ( m + 1 4 ) \operatorname {AD}_{(2)}^m(f) \geq (\operatorname {mult}_0 \Delta _f) \cdot {{m+1} \choose 4} for any f f , where mult 0 ⁡ Δ f \operatorname {mult}_0 \Delta _f is the multiplicity of the discriminant Δ f \Delta _f at the origin in the deformation space. Significantly, we prove that the function m ↦ AD ( 2 ) m ⁡ ( f ) − ( mult 0 ⁡ Δ f ) ⋅ ( m + 1 4 ) 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 prove similar results for weight- 2 2 inflection points with ramification sequence ( 0 , … , 0 , 1 , 1 ) (0, \dots , 0, 1,1) and for weight- 1 1 inflection points, and we apply our results to solve a number of related enumerative problems.

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