Reduced superschemes and the combinatorics of toric supervarieties

TL;DR

This work develops a framework for reduced superschemes and the combinatorics of toric supervarieties by extending classical notions of reduced/integral/normal rings to the super setting. The authors introduce decorated fans and the $HR_1$ regularity condition, proving a bijective correspondence between $HR_1$ toric supervarieties and decorated fans in the large-orbit regime, and they extend the construction to decorated polytopes for projective cases. They apply the framework to orbit closures inside the isomeric supergrassmannian, where decorated polytopes arise via a momentum-map–like image. Overall, the work provides a concrete, combinatorial toolkit for toric superspaces with potential applications in representation theory and supergeometry.

Abstract

We propose new definitions of integral, reduced, and normal superrings and superschemes to properly establish the notion of a supervariety. We generalize several results about classical reduced rings and varieties to the supergeometric setting, including an equivalence of categories between certain toric supervarieties and decorated polyhedral fans. These decorated fans are shown to encode important geometric information about the corresponding toric supervarieties. We then investigate some naturally-occurring toric supervarieties inside the isomeric supergrassmannian, which we show admits a nice description as a decorated polytope.

Figures (5)

• Figure 1: A real picture of $X$, with blue fuzz for the $\xi_1$ direction, red fuzz for the $\xi_2$ direction, and a purple "thick point" at $0 \in U_1$.
• Figure 2: Two different closed subschemes of $X$ as in Example \ref{['WeirdP1Example']}
• Figure 3: The decorated fan of an action of $Q(1)^2$ on $\mathbb{P}^{2|2}$
• Figure 4: Two different combinatorial descriptions of $\mathop{\mathrm{QGr}}\nolimits(1,2)$ as a toric supervariety for $Q(1)$.
• Figure 5: Decorated fans for a resolution of singularities $X' \to X$ as in Example \ref{['ex:SingularButEvenSmooth']}. The additional ray is generated by $(1,1,1)$ and decorated by the 0 subspace of $\mathfrak{t}_{\overline{1}}$.

Theorems & Definitions (112)

• Theorem 1
• Theorem 2
• Definition 2.1
• Definition 2.2
• Lemma 2.3
• Definition 2.4
• Lemma 2.5
• proof
• Lemma 2.6
• proof