In this thesis we study curves. In the first half, we study moduli spaces of curves and Gromov-Witten invariants, certain kinds of curves counts. We employ logarithmic geometry for this. Some major... Show moreIn this thesis we study curves. In the first half, we study moduli spaces of curves and Gromov-Witten invariants, certain kinds of curves counts. We employ logarithmic geometry for this. Some major results include the polynomiality of the double ramification cycle and recursive relations for the log double ramification cycle.In the second half we study rational points on curves, in particular Chabauty's method for finding the rational points and extensions of it. Major results include that the geometric (quadratic) Chabauty method is theoretically stronger than the original (quadratic) Chabauty method, and that local heights for quadratic Chabauty are explicitly computable. Show less
We define and study a natural system of tautological rings on the moduli spaces of marked curves at the level of differential forms. We show that certain 2-forms obtained from the natural normal... Show moreWe define and study a natural system of tautological rings on the moduli spaces of marked curves at the level of differential forms. We show that certain 2-forms obtained from the natural normal functions on these moduli spaces are tautological. Also we show that rings of tautological forms are always finite dimensional. Finally we characterize the Kawazumi–Zhang invariant as essentially the only smooth function on the moduli space of curves whose Levi form is a tautological form. Show less
