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K-moduli of Fano 3-folds can have embedded points

Andrea Petracci

TL;DR

This work provides an explicit obstructed K-polystable toric Fano $3$-fold $X$ whose $K$-moduli stack $\mathcal{M}^{\mathrm{Kss}}_{3}$ and moduli space $M^{\mathrm{Kps}}_{3}$ have embedded points at $[X]$. It analyzes $X$ via toric polytopes $P$ and $Q$, computes the hull $A$ of the $\mathbb{Q}$-Gorenstein deformation, and then determines the invariant quotient $A^G$ under the automorphism group, yielding $A^G \cong \mathbb{C} \llbracket u_1, \dots, u_6 \rrbracket / (u_1^2, u_2^2, u_1 u_2, u_1 u_3, u_2 u_3)$; this nontrivial quotient confirms embedded points in the moduli. The paper also connects deformation-theoretic data to mirror-symmetry by constructing a Laurent polynomial $f$ with polytope $P$ and discussing a conjectural mirror $X'$ with period matching $\pi_f$, though a full Gromov–Witten calculation is not carried out. Overall, it demonstrates that K-moduli for $n=3$ can exhibit intricate non-reduced structures and relates these phenomena to both invariant theory and mirror symmetry.

Abstract

We exhibit an example of obstructed K-polystable Fano 3-fold $X$ such that the K-moduli stack of K-semistable Fano varieties and the K-moduli space of K-polystable Fano varieties have an embedded point at $[X]$.

K-moduli of Fano 3-folds can have embedded points

TL;DR

This work provides an explicit obstructed K-polystable toric Fano -fold whose -moduli stack and moduli space have embedded points at . It analyzes via toric polytopes and , computes the hull of the -Gorenstein deformation, and then determines the invariant quotient under the automorphism group, yielding ; this nontrivial quotient confirms embedded points in the moduli. The paper also connects deformation-theoretic data to mirror-symmetry by constructing a Laurent polynomial with polytope and discussing a conjectural mirror with period matching , though a full Gromov–Witten calculation is not carried out. Overall, it demonstrates that K-moduli for can exhibit intricate non-reduced structures and relates these phenomena to both invariant theory and mirror symmetry.

Abstract

We exhibit an example of obstructed K-polystable Fano 3-fold such that the K-moduli stack of K-semistable Fano varieties and the K-moduli space of K-polystable Fano varieties have an embedded point at .

Paper Structure

This paper contains 10 sections, 2 theorems, 21 equations, 1 figure.

Table of Contents

  1. Introduction
  2. The polytopes $P$ and $Q$
  3. The variety $X$
  4. Some computations
  5. Cotangent sheaves and groups
  6. The sheaf $\mathscr{T}^{\mathrm{qG}, 1}_{X}$
  7. The hull of $\mathrm{Def}^{\mathrm{qG}}_{X}$
  8. The action the automorphism group
  9. Conclusion
  10. Mirror symmetry

Key Result

Theorem 1.1

There exists a K-polystable toric Fano $3$-fold $X$ with canonical singularities, such that the miniversal ring of the stack $\mathcal{M}^\mathrm{Kss}_{3}$ at the point corresponding to $X$ is and the completion of the structure sheaf of the scheme $M^\mathrm{Kps}_{3}$ at the point corresponding to $X$ is

Figures (1)

  • Figure 1: The polytope $P$

Theorems & Definitions (2)

  • Theorem 1.1
  • Corollary 1.2