On the moduli of hypersurfaces in toric orbifolds

We construct and study the moduli of stable hypersurfaces in toric orbifolds. Let X be a projective toric orbifold and α ∈ Cl(X) an ample class. The moduli space is constructed as a quotient of the linear system |α| by G = Aut(X) . Since the group G is non-reductive in general, we use new techniques of non-reductive geometric invariant theory. Using the A -discriminant of Gelfand, Kapranov and Zelevinsky, we prove semistability for quasismooth hypersurfaces of toric orbifolds. Further, we prove the existence of a quasi-projective moduli space of quasismooth hypersurfaces in a weighted projective space when the weighted projective space satisfies a certain condition. We also discuss how to proceed when this condition is not satisfied. We prove that the automorphism group of a quasismooth hypersurface of weighted projective space is finite excluding some low degrees.