Geometry of Points Satisfying Cayley-Bacharach Conditions and Applications

2022/01/05 by Nicola Picoco, Picoco, Nicola · 2 citations
Mathematics · #Advanced Differential Equations and Dynamical Systems #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #FOS: Mathematics #Meromorphic and Entire Functions #math.AG

paper · pdf · doi:10.48550/arxiv.2201.01665

Minor changes

openalex publication_date 2022/01/05 · arxiv created 2022/01/18 · arxiv updated 2022/01/19 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

In this paper, we study the geometry of points in complex projective space that satisfy the Cayley-Bacharach condition with respect to the complete linear system of hypersurfaces of given degree. In particular, we improve a result by Lopez and Pirola and we show that, if k≥ 1 and Γ=\P₁,…,P_d\⊂ ℙⁿ is a set of distinct points satisfying the Cayley-Bacharach condition with respect to |O_ℙⁿ(k)|, with d≤ h(k-h+3)-1 and 3≤ h≤ 5, then Γ lies on a curve of degree h-1. Then we apply this result to the study of linear series on curves on smooth surfaces in ℙ³. Moreover, we discuss correspondences with null trace on smooth hypersurfaces of ℙⁿ and on codimension 2 complete intersections.

Cited by

• The Cayley-Bacharach property and the Levinson-Ullery conjecture
• Low degree subvarieties of universal hypersurfaces

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