2026 Theses Doctoral

Geometry of the moduli space of curves on a hypersurface via the circle method

Hase-Liu, Matthew M.

We investigate the geometry of moduli spaces of curves on Fano hypersurfaces by passing to positive characteristic and using the circle method from analytic number theory. The first part of this thesis reinterprets the circle method geometrically to show that when a hypersurface's dimension is sufficiently large relative to its degree, the moduli space is irreducible and of the expected dimension. The second part, joint work with Jakob Glas, extends this analysis to the jet schemes of these spaces, proving that their singularities are terminal. Finally, the third part employs a strategy inspired by Mori's bend-and-break alongside these dimension bounds to confirm a prediction of geometric Manin's conjecture regarding the absence of accumulating maps. As a consequence, we establish a version of Poincaré duality for these moduli spaces within a specific range.