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 $\alpha \in \operatorname{Cl}(X)$ an ample class. The moduli space is constructed as a quotient of the linear system $|\alpha|$ by $G = \operatorname{Aut}(X)$. Since the group G is...

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Bibliographic Details
Published in:Proceedings of the Edinburgh Mathematical Society Vol. 67; no. 2; pp. 577 - 616
Main Author: Bunnett, Dominic
Format: Journal Article
Language:English
Published: Cambridge, UK Cambridge University Press 01.05.2024
Subjects:
Algebra
Automorphisms
Hyperspaces
Theorems
Topological manifolds
14L24
14J70
geometric invariant theory
moduli theory
toric varieties
ISSN:0013-0915, 1464-3839
Online Access:Get full text
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Summary:We construct and study the moduli of stable hypersurfaces in toric orbifolds. Let X be a projective toric orbifold and $\alpha \in \operatorname{Cl}(X)$ an ample class. The moduli space is constructed as a quotient of the linear system $|\alpha|$ by $G = \operatorname{Aut}(X)$. 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.
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ISSN:0013-0915
1464-3839
DOI:10.1017/S0013091524000166