Arithmetic geometry concerns the number-theoretic properties of geometric objects defined by polynomials. Mathematicians are interested in the rational solutions to these geometric objects. However... Show moreArithmetic geometry concerns the number-theoretic properties of geometric objects defined by polynomials. Mathematicians are interested in the rational solutions to these geometric objects. 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. Show less
This thesis discusses three aspects of Chinese tulou heritage management. Tulou are traditional fortified multifamily dwellings prevalent in southern Fujian. It first examines the tulou... Show moreThis thesis discusses three aspects of Chinese tulou heritage management. Tulou are traditional fortified multifamily dwellings prevalent in southern Fujian. It first examines the tulou interpretation prevailing in southern Fujian. Based on building studies, oral history, genealogies and interviews, it tries to reconstruct a relatively complete landscape biography to describe the essential episodes of built environment evolution in the Hekeng River Valley. This biography highlights the part non-tulou architecture and non-agricultural economic forms have played in the evolution of the settlement environment. It then discusses the regulations and laws which may have direct impact on not only the built heritage but also the life of local lineage society. It criticizes the harsh control of the local government over local people’s built heritage and their environment in the name of heritage preservation. It then moves on to examine the roles of different stakeholders in the heritage framework concerning the use of local people’s built heritage. It explores the possibility of reaching equilibrium among all the heritage players in the form of contracts, and offers some suggestions to the stakeholders getting involved in the tulou management issues. Show less
