## Journal of Commutative Algebra

- J. Commut. Algebra
- Volume 12, Number 2 (2020), 237-261.

### Quaternion rational surfaces

Jerome William Hoffman, Xiaohong Jia, and Haohao Wang

#### Abstract

A quaternion rational surface is a rational surface generated by two rational space curves via quaternion multiplication. In general, the structure of the graded minimal free resolution of a rational surface is unknown. The goal of this paper is to construct the graded minimal free resolution of a quaternion rational surface generated by two rational space curves. We will provide the explicit formulas for the maps of these graded minimal free resolutions. The approach we take is to utilize the information of the $\mu $-bases of the generating rational curves, and create the generating sets for the first and second syzygy modules in the graded minimal free resolutions. In addition, we show that the ideal generated by the first syzygy module expressed in terms of moving planes is exactly the same as the ideal generated by the parametrization in the affine ring.

#### Article information

**Source**

J. Commut. Algebra, Volume 12, Number 2 (2020), 237-261.

**Dates**

Received: 9 March 2017

Revised: 20 August 2017

Accepted: 30 August 2017

First available in Project Euclid: 2 June 2020

**Permanent link to this document**

https://projecteuclid.org/euclid.jca/1591063221

**Digital Object Identifier**

doi:10.1216/jca.2020.12.237

**Mathematical Reviews number (MathSciNet)**

MR4105546

**Zentralblatt MATH identifier**

07211337

**Subjects**

Primary: 14Q05: Curves

Secondary: 14Q10: Surfaces, hypersurfaces 13D02: Syzygies, resolutions, complexes

**Keywords**

syzygy graded minimal free resolution implicit equations

#### Citation

Hoffman, Jerome William; Jia, Xiaohong; Wang, Haohao. Quaternion rational surfaces. J. Commut. Algebra 12 (2020), no. 2, 237--261. doi:10.1216/jca.2020.12.237. https://projecteuclid.org/euclid.jca/1591063221