arXiv subjects

Shizhuo Zhang

Publications and source records attributed to Shizhuo Zhang.

At least 19 recordsLinked to original sources

arXiv2026

Categorical reconstruction of del Pezzo surfaces: Hochschild--Serre algebras and spinor modifications

We prove that, for every smooth complex del Pezzo surface of degree at most four, the enhanced right orthogonal to the structure sheaf determines the surface up to isomorphism. In degrees one, two, and three, we recover the anticanonical equation from intrinsic pieces of the Hochschild-Serre algebra via graded matrix factorizations; in degree four, the relevant Serre diagonal recovers the orbifold canonical ring associated with the pencil of quadrics. We also give an alternative proof in degrees one and two using the Bertini and Geiser involutions, equivariant topological K-theory, and classical Torelli. Then, we study the Clifford component associated with a conic bundle structure over the projective line. In every degree at most four, we construct an abstract spinor bundle whose spinor modification produces another, generally non-isomorphic, del Pezzo surface. We show via Kuznetsov's modification theorem that the relevant base-linear equivalences are precisely those induced by spinor modifications.

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arXiv2026

A note on the noncommutative Hodge conjecture for graded matrix factorizations

Let $m\geq 2$ and $d\geq 7$. We consider homogeneous polynomials in $2m+2$ variables of the form $f=F_0(u_0,v_0)+\cdots+F_m(u_m,v_m)$, where the $F_i$ are independently very general squarefree binary forms of degree $d$. We prove the rational noncommutative Hodge conjecture for the dg category $\mathrm{MF}^{\mathrm{gr}}(f)$ of graded matrix factorizations. Its Hochschild homology is the direct sum of the scalar-invariant Jacobian sector and $d-1$ one-dimensional point sectors. Boundary--bulk images of explicit rank-one factorizations generate a lattice of rank $(d-1)^{m+1}$ in the identity sector, while grading shifts of the stabilized residue field generate all point sectors. A reduced-Burau calculation shows that the identity-sector lattice exhausts the rational Hodge classes at a very general parameter. Consequently, $\dim_{\mathbb{Q}}\operatorname{Hdg}\bigl(\mathrm{MF}^{\mathrm{gr}}(f),\mathbb{Q}\bigr)=(d-1)^{m+1}+d-1$, and the rational topological $K$-rank is $\bigl((d-1)^{2m+2}+d-1\bigr)/d+d-1$. Finally, applying the additivity of the noncommutative Hodge conjecture for semiorthogonal decompositions together with Orlov's decompositions for Fano, Calabi--Yau, and general-type hypersurfaces proves the rational Hodge conjecture for the associated smooth projective hypersurface.

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arXiv2026

EPW varieties as moduli spaces on ordinary GM surfaces and special GM threefolds

We show that the double dual EPW sextic associated with a strongly smooth Gushel-Mukai surface can be realized as a moduli space of semistable objects on its bounded derived category. Also, we observe that the double dual EPW surface associated with a special Gushel-Mukai threefold can be realized as a moduli space of semistable objects on its Kuznetsov component. Then we discuss extensions of our main results to double EPW sextics and double EPW surfaces and a refinement of a statement of Bayer and Perry about Gushel-Mukai threefolds with equivalent Kuznetsov components, under a mild assumption.

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arXiv2026

Categorical resolutions and birational geometry of nodal Gushel-Mukai varieties

An ordinary Gushel-Mukai variety with a single isolated node is the intersection of the Grassmannian $G(2, 5)$ with a nodal quadric and a linear space. We consider such intersections in dimension three, four and five. We describe a flop between the blowup of such a variety and a quadric fibration over $\mathbb{P}^2$: at the level of derived categories, this flop establishes an equivalence between the categorical resolution of the Kuznetsov component of the Gushel-Mukai variety and the derived category of modules on the even part of the Clifford algebra of the quadric fibration. As a first application, we extend a result of Kuznetsov and Perry to the nodal case, and we describe a subfamily of rational, nodal Gushel-Mukai fourfolds whose Kuznetsov components admit a categorical resolution of singularities by an actual K3 surface of degree two without a Brauer twist. This produces evidence for a version of Kuznetsov's rationality conjecture. We also describe the relation with Verra threefolds and fourfolds at the birational and categorical level. In particular, in the three-dimensional case, we investigate alternative birational models by hyperbolic equivalence and by Kuznetsov's spinor modifications. We show that the categorical resolution of the Kuznetsov component of a one-nodal Gushel-Mukai threefold determines its birational class, and we explicitly construct another birational model, which is again a conic fibration over $\mathbb{P}^2$ branched over a sextic, with the same Kuznetsov component up to autoequivalences.

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arXiv2026

Brill-Noether reconstruction of index one prime Fano threefolds

We show by a uniform argument that every index one prime Fano threefold $X$ of genus $g\geq 6$ can be reconstructed as a Brill-Noether locus inside a Bridgeland moduli space of stable objects in the Kuznetsov component $\mathcal{K}u(X)$. As an application, we verify Mukai's conjecture on the existence of dual embeddings of $X$. Moreover, we establish a refined categorical Torelli theorem for $X$ and classify autoequivalences of $\mathcal{K}u(X)$. We also give an alternative disproof of Kuznetsov's Fano threefold conjecture.

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arXiv2025

Infinitesimal Torelli problems for special Gushel-Mukai and related Fano threefolds: Hodge theoretical and categorical perspectives

We investigate infinitesimal Torelli problems for some of the Fano threefolds of the following two types: (a) those which can be described as zero loci of sections of vector bundles on Grassmannians (for instance, ordinary Gushel-Mukai threefolds), and (b) double covers of rigid Fano threefolds branched along a $K3$ surface (such as, special Gushel-Mukai threefolds). The differential of the period map for ordinary Gushel-Mukai threefolds has been studied by Debarre, Iliev and Manivel; in particular, it has a $2$-dimensional kernel. The main result of this paper is that the invariant part of the infinitesimal period map for a special Gushel-Mukai threefold is injective. We prove this result using a Hodge theoretical argument as well as a categorical method. Through similar approaches, we also study infinitesimal Torelli problems for prime Fano threefolds with genus $7$, $8$, $9$, $10$, $12$ (type (a)) and for special Verra threefolds (type (b)). Furthermore, a geometric description of the kernel of the differential of the period maps for Gushel-Mukai threefolds (and for prime Fano threefolds of genus $8$) is given via a Bridgeland moduli space in the Kuznetsov components.

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arXiv2025

Lagrangian families of Bridgeland moduli spaces from Gushel-Mukai fourfolds

Let $X$ be a very general Gushel-Mukai (GM) variety of dimension $n\geq 4$, and let $Y$ be a smooth hyperplane section. There are natural pull-back and push-forward functors between the semi-orthogonal components (known as the Kuznetsov components) of the derived categories of $X$ and $Y$. In this paper, we prove that the Bridgeland stability of objects is preserved by both pull-back and push-forward functors. We then explore various applications of this result, such as constructing an $8$-dimensional smooth family of Lagrangian subvarieties for each moduli space of stable objects in the Kuznetsov component of a general GM fourfold and proving the projectivity of the moduli spaces of semistable objects of any class in the Kuznetsov component of a general GM threefold, as conjectured by Perry, Pertusi, and Zhao.

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arXiv2025

Double EPW cubes from twisted cubics on Gushel-Mukai fourfolds

In this paper, we conduct the first systematic investigation of twisted cubics on Gushel-Mukai (GM) fourfolds. We then study the double EPW cube, a 6-dimensional hyperkähler manifold associated with a general GM fourfold $X$, through the Bridgeland moduli space, and show that it is the maximal rationally connected (MRC) quotient of the Hilbert scheme of twisted cubics on $X$. We also prove that a general double EPW cube admits a covering by Lagrangian subvarieties constructed from the Hilbert schemes of twisted cubics on GM threefolds, which provides a new example for a conjecture of O'Grady.

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arXiv2024

Three approaches to a categorical Torelli theorem for cubic threefolds of non-Eckardt type via the equivariant Kuznetsov components

Let $Y$ be a cubic threefold with a non-Eckardt type involution $τ$. Our first main result is that the $τ$-equivariant category of the Kuznetsov component $\mathcal{K}u_{\mathbb{Z}_2}(Y)$ determines the isomorphism class of $Y$ for general $(Y,τ)$. We shall prove this categorical Torelli theorem via three approaches: a noncommutative Hodge theoretical one (using a generalization of the intermediate Jacobian construction due to Alexander Perry), a Bridgeland moduli theoretical one (using equivariant stability conditions), and a Chow theoretical one (using some techniques in [kuznetsovnonclodedfield2021]).The remaining part of the paper is devoted to proving an equivariant infinitesimal categorical Torelli for non-Eckardt cubic threefolds $(Y,τ)$. To accomplish it, we prove a compatibility theorem on the algebra structures of the Hochschild cohomology of the bounded derived category $D^b(X)$ of a smooth projective variety $X$ and on the Hochschild cohomology of a semi-orthogonal component of $D^b(X)$. Another key ingredient is a generalization of a result in [macri2009infinitesima] which shows that the twisted Hochschild-Kostant-Rosenberg isomorphism is compatible with the actions on the Hochschild cohomology and on the singular cohomology induced by an automorphism of $X$. In appendix, we prove an equivariant categorical Torelli theorem for arbitrary cubic threefold with a geometric involution under a natural assumption.

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arXiv2024

Kuznetsov's Fano threefold conjecture via Hochschild-Serre algebra

Let $Y$ be a smooth quartic double solid regarded as a degree 4 hypersurface of the weighted projective space $\mathbb{P}(1,1,1,1,2)$. We study the multiplication of Hochschild-Serre algebra of its Kuznetsov component $\mathcal{K}u(Y)$, via matrix factorization. As an application, we give a new disproof of Kuznetsov's Fano threefold conjecture.

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arXiv2024

IVHS via Kuznetsov components and categorical Torelli theorems for weighted hypersurfaces

We study the categorical Torelli theorem for smooth (weighted) hypersurfaces in (weighted) projective spaces via the Hochschild--Serre algebra of its Kuznetsov component. In the first part of the paper, we show that a natural graded subalgebra of the Hochschild--Serre algebra of the Kuznetsov component of a degree $d$ weighted hypersurface in $\mathbb{P}(a_0,\ldots,a_n)$ reconstructs the graded subalgebra of the Jacobian ring generated by the degree $t:=\mathrm{gcd}(d,Σ_{i=0}^na_i)$ piece under mild assumptions. Using results of Donagi and Cox--Green, this gives a categorical Torelli theorem for most smooth hypersurfaces $Y$ of degree $d \le n$ in $\mathbb{P}^n$ such that $d$ does not divide $n+1$ (the exception being the cases of the form $(d,n) = (4, 4k + 2)$, for which a result of Voisin lets us deduce a generic categorical Torelli theorem when $k \ge 150$). Next, we show that the Jacobian ring of the Veronese double cone can be reconstructed from its graded subalgebra of even degree, thus proving a categorical Torelli theorem for the Veronese double cone. In the second part, we rebuild the infinitesimal Variation of Hodge structures of a series of (weighted) hypersurfaces from their Kuznetsov components via the Hochschild--Serre algebra. As a result, we prove categorical Torelli theorems for two classes of (weighted) hypersurfaces: $(1):$ Generalized Veronese double cone; $(2):$ Certain $k$-sheeted covering of $\mathbb{P}^n$, when they are generic. Then, we prove a refined categorical Torelli theorem for a Fano variety whose Kuznetsov component is a Calabi--Yau category of dimension $2m+1$. Finally, we prove the actual categorical Torelli theorem for generalized Veronese double cone and $k$-sheeted covering of $\mathbb{P}^n$.

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arXiv2024

Serre algebra, matrix factorization and categorical Torelli theorem for hypersurfaces

Let $X$ be a smooth Fano variety. We attach a bi-graded associative algebra $\mathrm{HS}(\mathcal{K}u(X))=\bigoplus_{i,j\in \mathbb{Z}} \mathrm{Hom}(\mathrm{Id},S_{\mathcal{K}u(X)}^{i}[j])$ to the Kuznetsov component $\mathcal{K}u(X)$ whenever it is defined. Then we construct a natural sub-algebra of $\mathrm{HS}(\mathcal{K}u(X))$ when $X$ is a Fano hypersurface and establish its relation with Jacobian ring $\mathrm{Jac}(X)$. As an application, we prove a categorical Torelli theorem for Fano hypersurface $X\subset\mathbb{P}^n(n\geq 2)$ of degree $d$ if $\mathrm{gcd}(n+1,d)=1.$ In addition, we give a new proof of the $[Pir22, Theorem 1.2]$ using a similar idea.

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arXiv2024

Categorical Torelli theorems for Gushel-Mukai threefolds

We show that a general ordinary Gushel-Mukai(GM) threefold $X$ is reconstructed from the Kuznetsov component $\mathcal{K}u(X)$ together with an extra data coming from tautological sub-bundle of Grassmannian $\mathrm{Gr}(2,5)$. We also prove that $\mathcal{K}u(X)$ determines birational isomorphism class of $X$, while $\mathcal{K}u(X')$ determines the isomorphism class of a general special GM threefold $X'$. As an application, we prove a conjecture of Kuznetsov-Perry in dimension three under a mild assumption. Finally, we use $\mathcal{K}u(X)$ to restate a conjecture of Debarre-Iliev-Manivel regarding fibers of the period map for ordinary GM threefolds.

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arXiv2024

Linear subspaces of the intersection of two quadrics via Kuznetsov component

Let $Q_i(i=1,2)$ be $2g$ dimensional quadrics in $\mathbb{P}^{2g+1}$ and let $Y$ be the smooth intersection $Q_1\cap Q_2$. We associate the linear subspace in $Y$ with vector bundles on the hyperelliptic curve $C$ of genus $g$ by the left adjoint functor of $Φ:D^b(C)\rightarrow D^b(Y)$. As an application, we give a different proof of the classification of line bundles and stable bundles of rank $2$ on hyperelliptic curves given by Desale and Ramanan. When $g=3$, we show that the projection functor induces a closed embedding $α:Y\rightarrow SU^s_C(4,h)$ into the moduli space of stable bundles on $C$ of rank $4$ of fixed determinant.

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arXiv2023

GraphPrompt: Graph-Based Prompt Templates for Biomedical Synonym Prediction

In the expansion of biomedical dataset, the same category may be labeled with different terms, thus being tedious and onerous to curate these terms. Therefore, automatically mapping synonymous terms onto the ontologies is desirable, which we name as biomedical synonym prediction task. Unlike biomedical concept normalization (BCN), no clues from context can be used to enhance synonym prediction, making it essential to extract graph features from ontology. We introduce an expert-curated dataset OBO-syn encompassing 70 different types of concepts and 2 million curated concept-term pairs for evaluating synonym prediction methods. We find BCN methods perform weakly on this task for not making full use of graph information. Therefore, we propose GraphPrompt, a prompt-based learning approach that creates prompt templates according to the graphs. GraphPrompt obtained 37.2\% and 28.5\% improvement on zero-shot and few-shot settings respectively, indicating the effectiveness of these graph-based prompt templates. We envision that our method GraphPrompt and OBO-syn dataset can be broadly applied to graph-based NLP tasks, and serve as the basis for analyzing diverse and accumulating biomedical data. All the data and codes are avalible at: https://github.com/HanwenXuTHU/GraphPrompt

cs.CL↗

arXiv2023

New perspectives on categorical Torelli theorems for del Pezzo threefolds

Let $Y_d$ be a del Pezzo threefold of Picard rank one and degree $d\geq 2$. In this paper, we apply two different viewpoints to study $Y_d$ via a particular admissible subcategory of its bounded derived category, called the Kuznetsov component: (i) Brill-Noether reconstruction. We show that $Y_d$ can be uniquely recovered as a Brill-Noether locus of Bridgeland stable objects in its Kuznetsov component. (ii) Exact equivalences. We prove that, up to composing with an explicit auto-equivalence, any Fourier-Mukai type equivalence of Kuznetsov components of two del Pezzo threefolds of degree $2\leq d\leq 4$ can be lifted to an equivalence of their bounded derived categories. As a result, we obtain a complete description of the group of Fourier-Mukai type auto-equivalences of the Kuznetsov component of $Y_d$. We also describe the group of Fourier-Mukai type auto-equivalences of Kuznetsov components of index one prime Fano threefolds $X_{2g-2}$ of genus $g=6$ and $8$. As an application, first we identify the group of automorphisms of $X_{14}$ and its associated $Y_3$. Then we give a new disproof of Kuznetsov's Fano threefold conjecture by assuming Gushel-Mukai threefolds are general. In an appendix, we classify instanton sheaves on quartic double solids, generalizing a result of Druel.

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arXiv2023

Making Large Language Models Better Reasoners with Step-Aware Verifier

Few-shot learning is a challenging task that requires language models to generalize from limited examples. Large language models like GPT-3 and PaLM have made impressive progress in this area, but they still face difficulties in reasoning tasks such as GSM8K, a benchmark for arithmetic problems. To improve their reasoning skills, previous work has proposed to guide the language model with prompts that elicit a series of reasoning steps before giving the final answer, achieving a significant improvement on GSM8K from 17.9% to 58.1% in problem-solving rate. In this paper, we present DIVERSE (Diverse Verifier on Reasoning Step), a novel approach that further enhances the reasoning capability of language models. DIVERSE has three main components: first, it generates diverse prompts to explore different reasoning paths for the same question; second, it uses a verifier to filter out incorrect answers based on a weighted voting scheme; and third, it verifies each reasoning step individually instead of the whole chain. We evaluate DIVERSE on the latest language model code-davinci-002 and show that it achieves new state-of-the-art results on six of eight reasoning benchmarks (e.g., GSM8K 74.4% to 83.2%).

cs.CL↗

arXiv2023

Conics on Gushel-Mukai fourfolds, EPW sextics and Bridgeland moduli spaces

We identify the double dual EPW sextic $\widetilde{Y}_{A^{\perp}}$ and the double EPW sextic $\widetilde{Y}_A$, associated with a very general Gushel-Mukai fourfold $X$, with the Bridgeland moduli spaces of stable objects of character $Λ_1$ and $Λ_2$ in the Kuznetsov component $\mathcal{K}u(X)$. This provides an affirmative answer to a question of Perry-Pertusi-Zhao. As an application, we prove a conjecture of Kuznetsov-Perry for very general Gushel-Mukai fourfolds.

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