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Inflectionary Invariants for Isolated Complete Intersection Curve Singularities
About this Title
Anand P. Patel and Ashvin A. Swaminathan
Publication: Memoirs of the American Mathematical Society
Publication Year:
2023; Volume 282, Number 1397
ISBNs: 978-1-4704-6157-7 (print); 978-1-4704-7353-2 (online)
DOI: https://doi.org/10.1090/memo/1397
Published electronically: January 3, 2022
Keywords: Deformations of curve singularities,
inflection points,
sheaves of principal parts,
linear systems,
ramification theory,
Weierstrass points,
determinantal varieties,
intersection theory
Table of Contents
Chapters
- Acknowledgments
- 1. Introduction
- 2. Background Material
- 3. Defining Automatic Degeneracy
- 4. Automatic Degeneracies of a Node
- 5. Automatic Degeneracies of Higher-Order Singularities
- 6. Examples of Computing Automatic Degeneracies
- 7. Other Enumerative Applications
- A. Summary of Open Problems
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 define a numerical function $m \mapsto \operatorname {AD}_{(2)}^m(f)$ that naturally arises 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 several interesting families of pairs $(f,m)$. In particular, 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. Significantly, we prove 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 âcounts asâ an inflection point. We prove 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 a number of related enumerative problems.- E. Ballico and L. Gatto, Weierstrass points on singular curves, Rend. Sem. Mat. Univ. Politec. Torino 55 (1997), no. 2, 145â170 (1998). MR 1680491
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