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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.

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