Jarod Alper

Adequate moduli spaces and geometrically reductive group schemes

We introduce the notion of an adequate moduli space. The theory of adequate moduli spaces provides a framework for studying algebraic spaces which geometrically approximate algebraic stacks with reductive stabilizers in characteristic p. The denition of an adequate moduli space generalizes the existing notion of a good moduli space to characteristic p (and mixed characteristic). The most important examples of an adequate moduli space are: (1) the morphism from the quotient stack [X ss =G] of the semistable locus to the GIT quotient X ss ==G and (2) the morphism from an algebraic stack with nite inertia to the Keel{Mori coarse moduli space. It is shown that most of the fundamental properties of the GIT quotient X ss ==G follow from only the dening properties of an adequate moduli space. We provide applications of adequate moduli spaces to the structure of geometrically reductive and reductive group schemes. In particular, results of Seshadri and Waterhouse are generalized. The theory of adequate moduli spaces provides the possibility for intrinsic constructions of projective moduli spaces in characteristic p.

Finite Hilbert stability of (bi)canonical curves

We prove that a generic canonically or bicanonically embedded smooth curve has semistable mth Hilbert points for all m≥2. We also prove that a generic bicanonically embedded smooth curve has stable mth Hilbert points for all m≥3. In the canonical case, this is accomplished by proving finite Hilbert semistability of special singular curves with

A Lower Bound for the Determinantal Complexity of a Hypersurface

\mathbb{G}_{m}

On the choice of random directions for stochastic approximation algorithms

-action, namely the canonically embedded balanced ribbon and the canonically embedded balanced doubleA2k+1-curve. In the bicanonical case, we prove finite Hilbert stability of special hyperelliptic curves, namely Wiman curves. Finally, we give examples of canonically embedded smooth curves whose mth Hilbert points are non-semistable for low values of m, but become semistable past a definite threshold.

Associated forms and hypersurface singularities: The binary case

We prove that the determinantal complexity of a hypersurface of degree

Second flip in the Hassett–Keel program: existence of good moduli spaces

d > 2

Good Moduli Spaces for Artin Stacks

d>2 is bounded below by one more than the codimension of the singular locus, provided that this codimension is at least 5. As a result, we obtain that the determinantal complexity of the