## Journal of Commutative Algebra

### Quaternion rational surfaces

#### 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 $μ$-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

#### 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

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