We propose and study a new arithmetic invariant of non-hyperelliptic genus-4 curves: a canonical“quadratic” point on the Jacobian, defined by the two natural degree-2 maps to projective lines. Building on Xue’s result, that this point is generically non-torsion, we introduce a height-theoretic notion of bigness for families of curves in the moduli space of genus four curves, and give a criterion, via dimensions of modular quotients, for when it holds. We then exhibit two concrete 3- and 4-parameter families (the bi-involutions locus and a CM example) in which the canonical point is provably big, from which we deduce the finiteness of low-height curves and non-torsion at transcendental moduli. Our methods combine adelic Arakelov intersection theory with a generic Betti-rank argument.

In this thesis, which consists of two parts, we study some questions relating to the geometry of algebraic varieties in characteristic $p$, where $p$ is 0 or a prime number. The two parts may be read independently of one other; we give a separate introduction and background section for each one.

In the first part, we prove the Noether--Lefschetz theorem for divisor class groups of normal varieties in arbitrary characteristic. Our proof does not use any Hodge theory, monodromy, or cohomological arguments. The main ingredients are based on techniques and ideas that were available during M. Noether's lifetime. As an addendum to the first part, we give an alternative argument for showing the injectivity statement in Noether--Lefschetz using alterations, following ideas of Ravindra and Srinivas.

The second part is joint work with Joe Waldron and is in positive characteristic $p>0$. We study pathologies that can arise from the failure of Bertini's theorem, in particular geometric non-reducedness, and show a structural result that has applications to Fano varieties including Mori fiber spaces.