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On the moduli of hypersurfaces in toric orbifolds

Dominic Bunnett

Abstract

We construct and study the moduli of hypersurfaces in toric orbifolds. Let $X$ be a projective toric orbifold and $α\in Cl(X)$ 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 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.

On the moduli of hypersurfaces in toric orbifolds

Abstract

We construct and study the moduli of hypersurfaces in toric orbifolds. Let be a projective toric orbifold and an ample class. The moduli space is constructed as a quotient of the linear system by . Since the group is non-reductive in general, we use new techniques of non-reductive geometric invariant theory. Using the -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.

Paper Structure

This paper contains 15 sections, 33 theorems, 133 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Non-reductive geometric invariant theory
  3. Weight polytopes
  4. Automorphisms and toric varieties
  5. Toric orbifolds
  6. Graded automorphisms of the Cox ring
  7. Finiteness of the stabilisers
  8. The $A$-discriminant of a toric variety
  9. The $A$-discriminant
  10. Invariance of the $A$-discriminant
  11. Moduli of quasismooth hypersurfaces
  12. $\hat{U}$-stability for quasismooth hypersurfaces in weighted projective space
  13. Stability of quasismooth hypersurfaces in weighted projective space
  14. Explicit construction for $\mathbb P(1,\dots,1,r)$
  15. Stability of hypersurfaces in products of projective space

Key Result

Theorem 1

Let $X = \mathbb P(a_0, \dots , a_n) = \mathop{\mathrm{Proj}}\nolimits k[x_0, \dots , x_n]$ be a well-formed weighted projective space and let $d \geq \max\{a_0, \dots , a_n\} + 2$. Suppose that the $(\mathfrak C)$ condition holds for the action of $G=\mathop{\mathrm{Aut}}\nolimits(X)$ on $\mathcal{ In particular, there exists a geometric quotient $\mathcal{Y}^{\text{QS}}_d / G$ and hence a coarse

Figures (3)

  • Figure 1: $P_4$, the section polytope of $\mathcal{O}(4)$ in $W$ and $H$.
  • Figure 2: The weight diagram for $\hat{U} = \lambda_{g,N}(\mathbb G_m) \ltimes U$ for $N>>0$.
  • Figure 3: On the left is the section polytope $P$ of $\mathcal{O}(4)$. On the right is the Newton polytope of $\mathop{\mathrm{\textbf{V}}}\nolimits(f)$ where $f = xy^3 + x^2z+y^2z+z^2$.

Theorems & Definitions (94)

  • Theorem : Theorem \ref{['thm_main_NP_proof']}
  • Theorem : Corollary \ref{['invar_A_discrim']}
  • Theorem : Theorem \ref{['finite_stabilisers_thm']}
  • Definition 2.1
  • Definition 2.2
  • Remark 2.4
  • Lemma 2.5
  • Theorem 2.6
  • Definition 2.7
  • Definition 2.8
  • ...and 84 more