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Zhenjian Wang

Publications and source records attributed to Zhenjian Wang.

12 recordsLinked to original sources

On Strong Lefschetz Property of 0-dimensional Complete Intersections and Associated Forms

We prove that a homogeneous 0-dimensional complete intersection satisfies the Strong Lefschetz Property (SLP) in degree 1 if and only if its associated form has nonzero Hessian. This result is essentially known in the literature, but our proof differs from previous ones. We then investigate properties of the associated forms and show that they can always be reconstructed from their partial derivatives, in particular from their Jacobian ideals. As an application of our method, we prove that every smooth homogeneous polynomial which is not of Sebastiani-Thom type is uniquely determined by any nontrivial graded component of its Jacobian ideal.

math.AG

Structure of Torus Fibration Under the First Betti Number Restriction

We study torus bundles with affine structure groups. First, we establish a rigidity result under constraints on the first Betti numbers: If $ \text{b}_{1}(M)-\text{b}_{1}(N)=\dim M-\dim N $ holds for a torus bundle $M$ with an affine structure group over a closed manifold $N$, then $M$ can be classified. Second, we obtain some necessary and sufficient conditions for the topological splitting of principal torus bundles. These results improve the understanding of the geometry of collapsing sequences under the first Betti number constraints, thereby extending the prior work by Huang-Wang.

math.DG

When Metrics Disagree: Automatic Similarity vs. LLM-as-a-Judge for Clinical Dialogue Evaluation

As Large Language Models (LLMs) are increasingly integrated into healthcare to address complex inquiries, ensuring their reliability remains a critical challenge. Recent studies have highlighted that generic LLMs often struggle in clinical contexts, occasionally producing misleading guidance. To mitigate these risks, this research focuses on the domain-specific adaptation of \textbf{Llama-2-7B} using the \textbf{Low-Rank Adaptation (LoRA)} technique. By injecting trainable low-rank matrices into the Transformer layers, we efficiently adapted the model using authentic patient-physician transcripts while preserving the foundational knowledge of the base model. Our objective was to enhance precision and contextual relevance in responding to medical queries by capturing the specialized nuances of clinical discourse. Due to the resource-intensive nature of large-scale human validation, the model's performance was evaluated through a dual-track framework: \textbf{Track A} utilized traditional lexical similarity metrics (e.g., BLEU, ROUGE), while \textbf{Track B} employed an "LLM-as-a-Judge" paradigm using GPT-4 for semantic assessment. Our results demonstrate that while the LoRA-enhanced model achieved significant improvements across all quantitative lexical dimensions, a profound disagreement surfaced in the GPT-4 evaluation, which marginally favored the baseline model's conversational flow. This metric divergence underscores a pivotal finding: traditional automated scores may not fully reflect clinical utility. Consequently, we propose that while automated metrics and LLM judges serve as valuable developmental proxies, rigorous validation by human medical experts remains an indispensable requirement for the safe deployment of LLMs in healthcare settings.

cs.CL

New proofs for technical results in "Infinitesimal invariants of mixed Hodge structures'' (arXiv:2406.17118v1)

Cubic forms $C$ are constructed in the work of R. Aguilar, M. Green and P. Griffiths to establish the generic global Torelli theorem for Fano-K3 pairs $(X,Y)$, where $X: F=0$ is a cubic threefold in $\mathbb{P}^4$ and $Y\in|-K_X|$ is an anticanonical smooth section of $X$ defined by a quadratic form $Q$. In this article, we prove the following two results, which were previously verified with the computer aid of Macaulay2: for a generic pair $(X,Y)$, (i) the cubic form $C$ is smooth; (2) $(J_{F,3}:Q)=0$, and thereby give a precise meaning of the word ``generic" in this context.

math.AG

Monotonic invariants under blowups

We prove that the numerical invariant $3μ-4τ$ of a reduced irreducible plane curve singularity germ is non-negative, non-decreasing under blowups and strictly increasing unless the curve is non-singular. This provides a new perspective to understand the question posed by A. Dimca and G.-M. Greuel. Moreover, our work can be put in the general framework of discovering monotonic invariants under blowups.

math.AG

Deformation of Milnor Algebras

We investigate deformations of Milnor algebras of smooth homogeneous polynomials, and prove in particular that any smooth degree $d$ homogeneous polynomial in $n+1$ variables that is not of Sebastiani-Thom type is determined by the degree $k$ homogeneous component of its Jacobian ideal for any $d-1\leq k\leq (n+1)(d-2)$. Our results generalize the previous result on the reconstruction of a homogeneous polynomial from its Jacobian ideal.

math.AG

On algebraic surfaces associated to line arrangements

For a line arrangement in the complex projective plane $\mathbb{P}^2$, we investigate the compactification $\overline{F}$ of the affine Milnor fiber in $\mathbb{P}^3$ and its minimal resolution $\widetilde{F}$. We compute the Chern numbers in terms of the combinatorics of the line arrangement, then we show that the minimal resolution is never a quotient of a ball; in addition, we also prove that $\widetilde{F}$ is of general type when the arrangement has only nodes or triple points as singularities.

math.AG

On tangential deformations of homogeneous polynomials

The Jacobian ideal provides the set of infinitesimally trivial deformations for a homogeneous polynomial, or for the corresponding complex projective hypersurface. In this article, we investigate whether the associated linear deformation is indeed trivial, and show that the answer is no in a general situation. We also give a characterization of tangentially smoothable hypersurfaces with isolated singularities. Our results have applications in the local study of variations of projective hypersurfaces, complementing the global versions given by J. Carlson and P. Griffiths, R. Donagi and the author, and in the study of isotrivial linear systems on the projective space, showing that a general divisor does not belong to an isotrivial linear system of positive dimension.

math.AG