Genus one curves and Brauer-Severi varieties

  title={Genus one curves and Brauer-Severi varieties},
  author={Aise Johan de Jong and Wei Ho},
  journal={arXiv: Algebraic Geometry},
  • A. Jong, Wei Ho
  • Published 19 July 2012
  • Mathematics
  • arXiv: Algebraic Geometry
Let K be a field. Let A be a central simple algebra over K and let X be the associated Brauer-Severi variety over K. It has recently been asked if there exists a genus 1 curve C over K such that K(C) splits A. In other words, is there a genus one curve C over K with a morphism to X? In this short note, we explicitly construct such a curve in the case where X has dimension at most 4 (equivalently, when A has degree at most 5). 
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