Singularities of the moduli space of rational curves on a low-degree hypersurface in projective space

Singularities of the moduli space of rational curves on a low-degree hypersurface in projective space

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Will Sawin, Columbia University
Fine Hall 322

We study the moduli space of smooth genus zero curves of fixed degree on a smooth, degree d hypersurface in P^{n-1}. In particular, we study its singular locus. We give a lower bound for the codimension of the singular locus that increases linearly with the degree of the curve, as long as n> 3(d-1)2^{d-1}. We prove a similar bound for the locus of rational curves that aren't very free. Thus, the geometry of the moduli space, and of a typical rational curve on it, becomes better-behaved as the degree of the curve increases. The proof relies on a reduction to a finite field counting problem that is ultimately solved using techniques from analytic number theory, and is joint work with Tim Browning.