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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| Vydáno v: | Proceedings of the Edinburgh Mathematical Society Ročník 67; číslo 2; s. 577 - 616 |
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| Médium: | Journal Article |
| Jazyk: | angličtina |
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Cambridge, UK
Cambridge University Press
01.05.2024
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| ISSN: | 0013-0915, 1464-3839 |
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| Abstract | 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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| AbstractList | 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. 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. |
| Author | Bunnett, Dominic |
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| Cites_doi | 10.1016/j.aim.2009.02.007 10.2307/2951828 10.4310/PAMQ.2007.v3.n1.a3 10.1007/s13366-011-0084-0 10.1112/S0010437X0400123X 10.1112/topo.12075 10.1017/CBO9780511615436 10.1007/BF01404578 10.1007/BF02684747 10.1215/S0012-7094-94-07509-1 10.1090/S1056-3911-2013-00651-7 10.1112/blms.12581 |
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| Snippet | 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... 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... 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... |
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