SPECTRUM OF LINE ARRANGEMENTS WITH MULTIPLICITIES: A COMBINATORIAL APPROACH VIA LATTICE POINT ENUMERATION

• 발행기관: 충청수학회
• 저자명: Youngho Yoon
• 간행물 정보: 『Journal of the Chungcheong Mathematical Society』Volume 38, No. 1, 9~24쪽, 전체 16쪽
• 주제분류: 자연과학 > 자연과학일반
• 파일형태: PDF
• 발행일자: 2025.02.28

국문 초록

We introduce a combinatorial method for computing the spectrum of singularities using lattice point enumeration in regions determined by the defining equation. For line arrangements with multiplicities, our approach reveals the interplay between global and local contributions to the spectrum through explicit counting of lattice points. A series of transformations of these regions preserves the spectral data while simplifying the counting process. We highlight the case of f(x, y) = xm1ym2 as an illustrative example, where the method reduces the problem to counting lattice points on two line segments and naturally explains the role of gcd(m1,m2). This work provides a new perspective on the spectra of line arrangements with multiplicities, with potential applications to general non-isolated singularities.

목차

1. Introduction

2. Preliminaries

3. Lattice Regions via Generating Vectors

4. Main Results

5. The Monomial Case

6. Conclusion and Future Work

References

키워드

Spectrum of singularity Line arrangement Lattice point enumeration Milnor fiber