Singularities of the moduli space of rational curves on a lowdegree hypersurface in projective space
Singularities of the moduli space of rational curves on a lowdegree hypersurface in projective space

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^{n1}. 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(d1)2^{d1}. 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 betterbehaved 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.