Explicit Computation of the Height of a Gross-Schoen Cycle
Arithmetic geometry concerns the number-theoretic properties of geometric objects defined by polynomials. Mathematicians are interested in the rational solutions to these geometric objects.
- Wang, R.
- 18 oktober 2022
- Thesis in Leiden Repository
However, it is usually very difficult to answer questions like this.A. Beilinson and S. Bloch conjectured a very general height theory in 1980s, which was used by B. Gross and R. Schoen in their study of the Gross-Schoen cycles. The height of canonical Gross-Schoen cycles is conjectured to be non-negative. This was verified when the curve is an elliptic or hyperelliptic curve, while very few are known in the non-hyperelliptic case.During my PhD study, I study the Beilinson-Bloch height of canonical Gross-Schoen cycles on curves with an emphasis on the genus 3 case (almost all genus 3 curves are non-hyperelliptic). I studied its unboundedness and singular properties, and did explicit computation for the height of the canonical Gross-Schoen cycle of a specific plane quartic curve.The method used in my thesis should be helpful for verifications.