## Advanced Studies in Pure Mathematics

- Adv. Stud. Pure Math.
- Algebraic Varieties and Automorphism Groups, K. Masuda, T. Kishimoto, H. Kojima, M. Miyanishi and M. Zaidenberg, eds. (Tokyo: Mathematical Society of Japan, 2017), 163 - 182

### Cusp singularities and quasi-polyhedral sets

#### Abstract

We study cusp singularities from the viewpoint of locally polyhedral sets and reflection groups. Following the definition of quasi-polyhedral sets by Grünbaum, we consider a special kind of rational quasi-polyhedral sets with group action and refer to the relation with cusp singularities. We give also some examples of such quasi-polyhedral sets by using discrete groups generated by reflections.

#### Article information

**Dates**

Received: 8 April 2015

Revised: 25 September 2015

First available in Project Euclid:
21 September 2018

**Permanent link to this document**

https://projecteuclid.org/
euclid.aspm/1537498708

**Digital Object Identifier**

doi:10.2969/aspm/07510163

**Mathematical Reviews number (MathSciNet)**

MR3793365

**Zentralblatt MATH identifier**

1396.14046

**Subjects**

Primary: 14M25: Toric varieties, Newton polyhedra [See also 52B20]

Secondary: 52B20: Lattice polytopes (including relations with commutative algebra and algebraic geometry) [See also 06A11, 13F20, 13Hxx] 20F55: Reflection and Coxeter groups [See also 22E40, 51F15]

**Keywords**

Toric variety cusp singularity lattice polytope reflection group

#### Citation

Ishida, Masanori. Cusp singularities and quasi-polyhedral sets. Algebraic Varieties and Automorphism Groups, 163--182, Mathematical Society of Japan, Tokyo, Japan, 2017. doi:10.2969/aspm/07510163. https://projecteuclid.org/euclid.aspm/1537498708