Singularities, Secants, and Syzygies

In this thesis, we explore three aspects of singularities and syzygies in algebraic geometry. First, we study singularities in positive characteristic. In particular, we prove an inversion of adjunction result for the F -signature, a numerical invariant which measures the asymptotic number of Frobenius splittings of a singularity. Next, we turn our attention to syzygies of secant varieties of curves. We prove that the Boij-S¨oderberg decomposition of the secant variety of a curve C embedded by a degree d line bundle L becomes pure as d → ∞. Finally, we analyze the singularities of nested Hilbert schemes of points on surfaces. We show that S[n,n+1,n+2] is klt with local complete intersection singularities. We conclude this section with a discussion of the applications of nested Hilbert schemes to secant varieties of surfaces. In addition to these original results, we have included extensive background material on singularities and syzygies of varieties. We also discuss questions for further research in this area.