On the local cohomology of secant varieties

2024/07/23 by Sebastián Olano, Debaditya Raychaudhury, Olano, Sebastian +1 · 2 citations
Mathematics · #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #Commutative Algebra and Its Applications #FOS: Mathematics #Tensor decomposition and applications

paper · pdf · doi:10.48550/arxiv.2407.16688

openalex publication_date 2024/07/23 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Given a sufficiently positive embedding X⊂ℙ^N of a smooth projective variety X, we consider its secant variety Σ that comes equipped with the embedding Σ⊂ℙ^N by its construction. In this article, we determine the local cohomological dimension \textrmlcd(ℙ^N,Σ) of this embedding, as well as the generation level of the Hodge filtration on the topmost non-vanishing local cohomology module H^q_Σ(O_ℙ^N), i.e., when q=\textrmlcd(ℙ^N,Σ). Additionally, we show that Σ has quotient singularities (in which case the equality \textrmlcd(ℙ^N,Σ)=\textrmcodim_ℙ^N(Σ) is known to hold) if and only if X≅ℙ¹. We also provide a complete classification of (X,L) for which Σ has (ℚ-)Gorentein singularities. As a consequence, we deduce that if Σ is a local complete intersection, then either X is isomorphic to ℙ¹, or an elliptic curve.

Cited by

• Partial Cohomologically Complete Intersections via Hodge Theory
• Local cohomology and singular cohomology of toric varieties via mixed Hodge modules

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