{"title":"Rank Zero Segre Integrals on Hilbert Schemes of Points on Surfaces","authors":"Yao Yuan","doi":"10.1093/imrn/rnae173","DOIUrl":null,"url":null,"abstract":"The generating function of the Segre integrals on Hilbert schemes of points on a surface $X$ can be determined by five universal series $A_{0}(z)$, $A_{1}(z)$, $A_{2}(z)$, $A_{3}(z)$, $A_{4}(z)$, due to the result of Ellingsrud–Göttsche–Lehn. These five series do not depend on the surface $X$ and depend on the element $\\alpha \\in K(X)$, to which the Segre integrals are associated, only through the rank. Marian–Oprea–Pandharipande have determined $A_{0}(z),A_{1}(z),A_{2}(z)$ for all ranks. For rank 0, it is easy to see $A_{4}(z)=1$. Marian–Oprea–Pandharipande also conjectured that $A_{3}(z)=A_{0}(z)A_{1}(z)$ for rank 0. We prove this conjecture by showing that when $X$ is the projective plan, the Segre integrals associated to the structure sheaf of a curve in the anti-canoncial class are all zero. Hence, the rank zero Segre integrals on the Hilbert schemes of points for all surfaces are determined.","PeriodicalId":14461,"journal":{"name":"International Mathematics Research Notices","volume":"20 1","pages":""},"PeriodicalIF":0.9000,"publicationDate":"2024-08-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"International Mathematics Research Notices","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1093/imrn/rnae173","RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 0
Abstract
The generating function of the Segre integrals on Hilbert schemes of points on a surface $X$ can be determined by five universal series $A_{0}(z)$, $A_{1}(z)$, $A_{2}(z)$, $A_{3}(z)$, $A_{4}(z)$, due to the result of Ellingsrud–Göttsche–Lehn. These five series do not depend on the surface $X$ and depend on the element $\alpha \in K(X)$, to which the Segre integrals are associated, only through the rank. Marian–Oprea–Pandharipande have determined $A_{0}(z),A_{1}(z),A_{2}(z)$ for all ranks. For rank 0, it is easy to see $A_{4}(z)=1$. Marian–Oprea–Pandharipande also conjectured that $A_{3}(z)=A_{0}(z)A_{1}(z)$ for rank 0. We prove this conjecture by showing that when $X$ is the projective plan, the Segre integrals associated to the structure sheaf of a curve in the anti-canoncial class are all zero. Hence, the rank zero Segre integrals on the Hilbert schemes of points for all surfaces are determined.
期刊介绍:
International Mathematics Research Notices provides very fast publication of research articles of high current interest in all areas of mathematics. All articles are fully refereed and are judged by their contribution to advancing the state of the science of mathematics.