Shiraishi–Mori Construction

• Shiraishi–Mori construction is a dual-framework approach that creates non-generic, structured objects through algebraic geometric compactifications and operator-theoretic scar embedding.
• In algebraic geometry, it completes affine spaces into Mori fiber spaces using pencil constructions and birational modifications that yield families of Fano hypersurfaces with specific rigidity properties.
• In quantum many-body theory, it designs exact non-thermal eigenstates via local projectors and Zeeman terms, providing a systematic pathway to engineer and classify ferromagnetic scar towers.

The Shiraishi–Mori construction encompasses two distinct but foundational schemes: an algebraic geometric method for completing affine spaces into Mori fiber spaces, and an operator-theoretic framework for embedding quantum many-body scar (QMBS) subspaces within non-integrable quantum spin Hamiltonians. In both contexts, it systematically constructs non-generic, highly structured objects—either geometric compactifications or exact, non-thermal eigenstates—in settings dominated by genericity and ergodicity. Contemporary developments have exhaustively classified such constructions in both algebraic and quantum settings, with each direction independently named after Shiraishi and Mori. This article presents detailed expositions of both threads, their structural mechanisms, and their long-range consequences.

1. Algebraic Geometric Construction: Completions of Affine Spaces

The classical Shiraishi–Mori construction, as generalized by later authors, yields completions of affine space $\mathbb{A}^n$ into total spaces $X$ admitting Mori fiber space structures $f\colon X \to \mathbb{P}^1$ whose generic fibers exhibit prescribed (often non-rational, birationally rigid, or non-stably rational) features. The construction proceeds as follows (Dubouloz et al., 2020):

• One starts with $X = \mathbb{P}^n$ (homogeneous coordinates $[x_0:\cdots:x_n]$), a hyperplane $H = \{x_0 = 0\}$, and an integral hypersurface $F \subset \mathbb{P}^n$ of degree $d$ with $2 \le d \le n$, such that $F \cap H$ is irreducible and smooth on $F$.
• The associated pencil $$L = \langle F, dH \rangle \subset | \mathcal{O}_{\mathbb{P}^n}(d)|$$ yields a rational fibration

$$\tilde X = \{ (x,[u:v]) \in \mathbb{P}^n \times \mathbb{P}^1 \mid v \cdot F(x) + u \cdot x_0^d = 0 \}$$

with projection $f^0: \tilde X \to \mathbb{P}^1$.

• After one small resolution, or alternatively blowing up the base locus $S = F \cap H \subset \mathbb{P}^n$ and running a relative minimal model program (MMP), one produces a $\mathbb{Q}$-factorial terminal variety $X$ with $f: X \to \mathbb{P}^1$ a Mori fiber space. The key geometric properties are:
  • $X$ contains $\mathbb{A}^n \cong \mathbb{P}^n \setminus H$ as an open affine chart.
  • $-K_X$ is $f$-ample and the relative Picard number $\rho(X/\mathbb{P}^1)=1$.
  • For $t \neq \infty$, $X_t \cong \{F - t x_0^d=0\} \subset \mathbb{P}^n$ is a smooth Fano hypersurface of degree $d$.

This technique systematically produces families of affine completions with prescribed fiber type, including cases with non-rational, birationally rigid, or non-stably rational general fibers, thereby recovering—among other cases—the classical Shiraishi–Mori examples for $(n,d)=(2,2)$ and $(n,d)=(3,2)$.

2. Explicit Model and Birational Modifications

The completed space $X$ is explicitly described as the bi-projective hypersurface

$$X = \{ v F(x) + u x_0^d = 0 \} \subset \mathbb{P}^n_{[x]} \times \mathbb{P}^1_{[u:v]}$$

with structure morphism $f: X \to \mathbb{P}^1$. The chart $\{ x_0 \ne 0, u \ne 0 \}$ defines an open embedding of $\mathbb{A}^n$ into $X$ via normalization $u=1$ and $x_i' = x_i/x_0$.

The process involves:

• Blowing up the base locus $S = F \cap H\subset \mathbb{P}^n$ to obtain $\hat{\mathbb{P}}^n$;
• The pencil becomes a morphism $\hat f: \hat{\mathbb{P}}^n \to \mathbb{P}^1$ given by $|\tau^* \mathcal{O}_{\mathbb{P}^n}(dH) - E|$;
• $\hat{\mathbb{P}}^n$ is $\mathbb{Q}$-factorial and terminal for $d \le n$, and final contraction of the strict transform of $H$ yields $X$ with $\rho(X/\mathbb{P}^1)=1$.

This geometric framework allows $X$ to serve as a Mori fiber space over $\mathbb{P}^1$, and for generic $t$, $X_t$ is a smooth degree $d$ hypersurface with properties including non-rationality, birational rigidity, or failure of stable rationality depending on $(n,d)$, as established by classical theorems in higher-dimensional geometry (Dubouloz et al., 2020).

3. Quantum Many-Body Scar Construction: Operator-Theoretic Framework

Independently, the Shiraishi–Mori construction appeared in quantum many-body theory as an operator-theoretic method for engineering non-thermal scar eigenstates within an otherwise thermalizing many-body spectrum. The framework, as generalized and exhaustively classified in (Omiya, 16 Jan 2026), proceeds as follows:

• On a spin-system Hilbert space $\mathcal{H} = \bigotimes_{x \in \Lambda} h_x$, select a scar subspace $\mathcal{S} \subset \mathcal{H}$, a family of local projectors $\{P_j\}_j$, and decompose the Hamiltonian as $H = H_Z + A$.
• The annihilator $A = \sum_j A_j$ with $A_j = h_j P_j$ or $P_j h_j' P_j$, and $P_j|\text{scar}\rangle=0$ for all $j$, ensures $A|\text{scar}\rangle=0$.
• The Zeeman term $H_Z$ acts only within $\mathcal{S}$ (typically $H_Z = \omega \sum_x S^z_x$), generating an exactly solvable scar spectrum—typically a tower of equidistant states—embedded in the full many-body spectrum.

This architectural principle isolates a robust set of scar eigenstates and guarantees their persistence under deformations that preserve the annihilator structure.

4. Exhaustiveness for Ferromagnetic Many-Body Scars

The foundational result presented in (Omiya, 16 Jan 2026) establishes that any local Hamiltonian $H$ harboring a complete tower of “magnon-type” ferromagnetic scars must admit a decomposition precisely of the Shiraishi–Mori form (possibly generalized). Specifically, Theorem 5.2 asserts:

• If $H$ is local and every totally symmetric weight-basis state is an exact eigenstate, then

$$H = \sum_{x \in \Lambda} h^{(1)}_{[x]} P_x + \sum_{\langle x,y\rangle} h^{(2)}_{[xy]} P_{xy} + \sum_{x \in \Lambda} h_x^{\rm Zeeman}$$

where $P_x$, $P_{xy}$ are on-site and two-site local projectors (each annihilating all scars) and the Zeeman term is a sum of Cartan generators. Equivalently, $H = H_A + H_Z$, $H_A|\text{scar}\rangle=0$, $H_Z|\text{scar}\rangle = E|\text{scar}\rangle$.

This result establishes the SM construction as not merely sufficient, but necessary (up to mild generalizations) for the existence of a full ferromagnetic (symmetric) scar tower (Omiya, 16 Jan 2026).

5. Mechanisms of Projector Decomposition and Examples

Derivation of the SM form follows a sequence of algebraic and locality-based reductions:

• The Hamiltonian is split into its permutation-symmetric and -asymmetric components, with the symmetric part $H_Z$ preserving the scar subspace and the asymmetric part $H_A$ annihilating it.
• Lemma 4.1 guarantees that any global annihilator of the symmetric sector can be decomposed into a sum of local projectors (on-site and nearest-neighbor), each individually annihilating the scar subspace.
• The locality of $H$ and the structure of the symmetric sector enforce that $H_Z$ must be a sum (or polynomial) in on-site Cartan operators, i.e., strictly a Zeeman term.

Prominent examples include:

• The spin-1 XY model (see Sec. 2, (Omiya, 16 Jan 2026)), which, after a momentum twist, organizes as a sum of two-site projectors and on-site Zeeman terms, each projector killing the scar manifold.
• The PXP model, where stable spin-wave scar towers can be rewritten in terms of forbidden configuration projectors and a uniform Zeeman tilt.
• Dzyaloshinskii–Moriya (DM)–type perturbations recast as local interactions that separately annihilate the symmetric sector.

6. Implications, Limitations, and Extensions

Within the operator-theoretic domain, the SM construction is provably exhaustive for the case of a complete, totally symmetric (magnon-type) scar tower. This restricts the full algebraic structure of the Hamiltonian and precludes exotic mechanisms for scar formation in such settings (Omiya, 16 Jan 2026).

Known limitations include:

• The construction mandates a complete symmetric sector; partial or “skipped weight” scar towers (e.g., “even Dicke” states or AKLT-type scars) may require alternative, possibly subgroup-specific projector architectures.
• While the existence of local projectors is guaranteed, the strict finiteness of their spatial range in all cases remains an open question; strict nearest-neighbor decompositions are proven, but generalization to arbitrarily long-range tails is conjectural.
• Approximate scars, as in realistic experimental architectures, suggest a near-annihilating “approximate SM” structure whose deviation controls the timescale of quantum revivals.

Potential generalizations include extensions to SU(2)-embedded scar towers and to higher-rank symmetry sectors, broadening the applicability of the exhaustion theorem.

7. Synthesis and Comparative Perspective

The Shiraishi–Mori construction, in both geometric and operator-algebraic instantiations, leverages networked projectors or birational modifications to realize exceptional subspaces or fibers within generic, often ergodic, environments. In algebraic geometry, the pencil construction with rational or arbitrarily rigid fibers systematizes compactifications of affine space with explicit control on the generic fiber type (Dubouloz et al., 2020). In quantum many-body theory, the SM operator decomposition is shown to encompass and exhaust all known mechanisms for realizing exact ferromagnetic scar towers (Omiya, 16 Jan 2026). This unifying principle not only demarcates the boundaries of such constructions but also offers a platform for systematic exploration of non-generic phenomena in both fields.