Pierre Lairez, Eric Pichon-Pharabod, Pierre Vanhove
{"title":"Effective homology and periods of complex projective hypersurfaces","authors":"Pierre Lairez, Eric Pichon-Pharabod, Pierre Vanhove","doi":"10.1090/mcom/3947","DOIUrl":null,"url":null,"abstract":"<p>We introduce a new algorithm for computing the periods of a smooth complex projective hypersurface. The algorithm intertwines 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.</p> <p>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.</p>","PeriodicalId":18456,"journal":{"name":"Mathematics of Computation","volume":"64 1","pages":""},"PeriodicalIF":2.1000,"publicationDate":"2024-02-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Mathematics of Computation","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1090/mcom/3947","RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS, APPLIED","Score":null,"Total":0}
引用次数: 0
Abstract
We introduce a new algorithm for computing the periods of a smooth complex projective hypersurface. The algorithm intertwines 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.
期刊介绍：
All articles submitted to this journal are peer-reviewed. The AMS has a single blind peer-review process in which the reviewers know who the authors of the manuscript are, but the authors do not have access to the information on who the peer reviewers are.
This journal is devoted to research articles of the highest quality in computational mathematics. Areas covered include numerical analysis, computational discrete mathematics, including number theory, algebra and combinatorics, and related fields such as stochastic numerical methods. Articles must be of significant computational interest and contain original and substantial mathematical analysis or development of computational methodology.
京公网安备 11010802042870号
京ICP备2023020795号-1
分享
求助内容：
应助结果提醒方式：
扫码关注我们
