Effective homology and periods of complex projective hypersurfaces

2023/06/08 by Pierre Lairez, Lairez, Pierre, Pierre Vanhove +2 · 4 citations
Computer Science · Mathematics · #14D05 #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #FOS: Computer and information sciences #FOS: Mathematics #Polynomial and algebraic computation #Primary 14Q15 #Secondary 32G20 #Symbolic Computation (cs.SC) #Topological and Geometric Data Analysis

paper · pdf · doi:10.48550/arxiv.2306.05263

openalex publication_date 2023/06/08 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We introduce a new algorithm for computing the periods of a smooth complex projective hypersurface. The algorithm intertwine with a new method for computing an explicit basis of the singular homology of the hypersurface. It is based on Picard-Lefschetz theory and relies on the computation of the monodromy action induced by a one-parameter family of hyperplane sections on the homology of a given section. We provide a SageMath implementation. For example, on a laptop, it makes it possible to compute the periods of a smooth complex quartic surface with hundreds of digits of precision in typically an hour.

Cited by

• Counting points on surfaces in polynomial time
• Galois Groups of Symmetric Cubic Surfaces
• Wall crossing structure from quantum phenomena to Feynman Integrals
• Periods of fibre products of elliptic surfaces and the Gamma conjecture

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