On the moduli of hypersurfaces in toric orbifolds

Abstract

We construct and study the moduli of hypersurfaces in toric orbifolds. Let $X$ be a projective toric orbifold and $\alpha \in Cl(X)$ an ample class. The moduli space is constructed as a quotient of the linear system $|\alpha |$ 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 we prove semistability for certain toric orbifolds. Further, we show that quasismooth hypersurfaces in a weighted projective space are stable 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.

27 pages; (v2 results expanded to more cases, intro improved)

References in corpus (1)

• On K-stability of some del Pezzo surfaces of Fano index 2

Cited by in corpus (1)

• A remarkable class of elliptic surfaces of amplitude 1 in weighted projective space