2025/07/11 by Dawei Chen, Fei Yu, Chen, Dawei +1
Mathematics · #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #Algebraic structures and combinatorial models #Commutative Algebra and Its Applications #FOS: Mathematics #Geometric Topology (math.GT)
paper · pdf · doi:10.48550/arxiv.2507.09078
openalex publication_date 2025/07/11 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Given a holomorphic differential on a smooth complex algebraic curve, we associate to it a Gorenstein curve singularity with \mathbb Gm-action via a test configuration. This construction decomposes the strata of holomorphic differentials with prescribed orders of zeros into negatively graded miniversal deformation spaces of such singularities. Additionally, it provides a natural description for the singular curves that appear in the boundary of the miniversal deformation spaces. Our approach leads to a number of applications. We classify the unique Gorenstein singularity with \mathbb Gm-action for each nonvarying stratum of holomorphic differentials and study when these nonvarying strata can be compactified by weighted projective spaces. Moreover, extending the classical results about ADE singularities, we establish the K(π,1)-property for non-hypersurface complete intersection singularities of type U7, U8, U9, and Sk. We also study singularities with bounded α-invariants in the log minimal model program for \mathcal Mg and utilize them to bound the slopes of effective divisors in \mathcal Mg. Finally, we show that the loci of subcanonical points with fixed semigroups have trivial tautological rings and provide a criterion to determine whether they are affine varieties.