Igor Krylov, Takuzo Okada, Erik Paemurru, Jihun Park
{"title":"2 n2-inequality for cA1 points and applications to birational rigidity","authors":"Igor Krylov, Takuzo Okada, Erik Paemurru, Jihun Park","doi":"10.1112/s0010437x24007164","DOIUrl":null,"url":null,"abstract":"<p>The <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240528083622301-0572:S0010437X24007164:S0010437X24007164_inline3.png\"><span data-mathjax-type=\"texmath\"><span>$4 n^2$</span></span></img></span></span>-inequality for smooth points plays an important role in the proofs of birational (super)rigidity. The main aim of this paper is to generalize such an inequality to terminal singular points of type <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240528083622301-0572:S0010437X24007164:S0010437X24007164_inline4.png\"><span data-mathjax-type=\"texmath\"><span>$cA_1$</span></span></img></span></span>, and obtain a <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240528083622301-0572:S0010437X24007164:S0010437X24007164_inline5.png\"><span data-mathjax-type=\"texmath\"><span>$2 n^2$</span></span></img></span></span>-inequality for <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240528083622301-0572:S0010437X24007164:S0010437X24007164_inline6.png\"><span data-mathjax-type=\"texmath\"><span>$cA_1$</span></span></img></span></span> points. As applications, we prove birational (super)rigidity of sextic double solids, many other prime Fano 3-fold weighted complete intersections, and del Pezzo fibrations of degree <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240528083622301-0572:S0010437X24007164:S0010437X24007164_inline7.png\"><span data-mathjax-type=\"texmath\"><span>$1$</span></span></img></span></span> over <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240528083622301-0572:S0010437X24007164:S0010437X24007164_inline8.png\"><span data-mathjax-type=\"texmath\"><span>$\\mathbb {P}^1$</span></span></img></span></span> satisfying the <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240528083622301-0572:S0010437X24007164:S0010437X24007164_inline9.png\"><span data-mathjax-type=\"texmath\"><span>$K^2$</span></span></img></span></span>-condition, all of which have at most terminal <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240528083622301-0572:S0010437X24007164:S0010437X24007164_inline10.png\"><span data-mathjax-type=\"texmath\"><span>$cA_1$</span></span></img></span></span> singularities and terminal quotient singularities. These give first examples of birationally (super)rigid Fano 3-folds and del Pezzo fibrations admitting a <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240528083622301-0572:S0010437X24007164:S0010437X24007164_inline11.png\"><span data-mathjax-type=\"texmath\"><span>$cA_1$</span></span></img></span></span> point which is not an ordinary double point.</p>","PeriodicalId":55232,"journal":{"name":"Compositio Mathematica","volume":null,"pages":null},"PeriodicalIF":1.3000,"publicationDate":"2024-05-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Compositio Mathematica","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1112/s0010437x24007164","RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 0
Abstract
The $4 n^2$-inequality for smooth points plays an important role in the proofs of birational (super)rigidity. The main aim of this paper is to generalize such an inequality to terminal singular points of type $cA_1$, and obtain a $2 n^2$-inequality for $cA_1$ points. As applications, we prove birational (super)rigidity of sextic double solids, many other prime Fano 3-fold weighted complete intersections, and del Pezzo fibrations of degree $1$ over $\mathbb {P}^1$ satisfying the $K^2$-condition, all of which have at most terminal $cA_1$ singularities and terminal quotient singularities. These give first examples of birationally (super)rigid Fano 3-folds and del Pezzo fibrations admitting a $cA_1$ point which is not an ordinary double point.
期刊介绍:
Compositio Mathematica is a prestigious, well-established journal publishing first-class research papers that traditionally focus on the mainstream of pure mathematics. Compositio Mathematica has a broad scope which includes the fields of algebra, number theory, topology, algebraic and differential geometry and global analysis. Papers on other topics are welcome if they are of broad interest. All contributions are required to meet high standards of quality and originality. The Journal has an international editorial board reflected in the journal content.