2017/05/24 by Anand Patel, Patel, Anand, Ashvin Swaminathan +1
Computer Science · Engineering · Mathematics · #14B05 #14C17 #14C20 #14C21 #14H20 #14N15 #Advanced Numerical Analysis Techniques #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #FOS: Mathematics #Polynomial and algebraic computation
paper · pdf · doi:10.48550/arxiv.1705.08761
openalex publication_date 2017/05/24 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 \colon (ℂN,0) → (ℂN-1,0). We introduce a numerical function m ↦ AD(2)m(f) that arises as an error term 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 = 0, and we compute AD(2)m(f) for various (f,m). Particularly, for a node defined by f \colon (x,y) ↦ xy, we prove that AD(2)m(xy) = m+1 \choose 4, and we deduce as a corollary that AD(2)m(f) ≥ (mult0 Δf) ⋅ m+1 \choose 4 for any f, where mult0 Δf is the multiplicity of the discriminant Δf at the origin in the deformation space. Furthermore, we show that the function m ↦ AD(2)m(f) -(mult0 Δf) ⋅ 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, …, 0, 1,1) and for weight-1 inflection points, and we apply our results to solve various related enumerative problems.