2017/08/31 by Meng Chen, Chen Jiang
Mathematics · #Algebraic Geometry and Number Theory #Combinatorics #Divisor (algebraic geometry) #Fano plane #Geometric Analysis and Curvature Flows #Geometry #Geometry and complex manifolds #Gravitational singularity #Mathematical analysis #Mathematics #Pure mathematics #math.AG
paper · pdf · doi:10.5802/aif.3367
published as Annales de l'Institut Fourier, Volume 70 (2020) no. 6, pp. 2473-2542 · 63 pages, comments are welcome; v2: final version, to appear in Annales de l'Institut Fourier
arxiv created 2019/11/28 · openalex publication_date 2021/04/15 · openalex created_date 2021/04/26 · arxiv updated 2021/12/24 · openalex updated_date 2026/08/06
By a canonical (resp. terminal) weak <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>ℚ</mml:mi> </mml:math> -Fano <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mn>3</mml:mn> </mml:math> -fold we mean a normal projective one with at worst canonical (resp. terminal) singularities on which the anti-canonical divisor is <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>ℚ</mml:mi> </mml:math> -Cartier, nef and big. For a canonical weak <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>ℚ</mml:mi> </mml:math> -Fano <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mn>3</mml:mn> </mml:math> -fold <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>V</mml:mi> </mml:math> , we show that there exists a terminal weak <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>ℚ</mml:mi> </mml:math> -Fano <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mn>3</mml:mn> </mml:math> -fold <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>X</mml:mi> </mml:math> , being birational to <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>V</mml:mi> </mml:math> , such that the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>m</mml:mi> </mml:math> -th anti-canonical map defined by <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo>|</mml:mo> <mml:mo>-</mml:mo> <mml:mi>m</mml:mi> <mml:msub> <mml:mi>K</mml:mi> <mml:mi>X</mml:mi> </mml:msub> <mml:mo>|</mml:mo> </mml:mrow> </mml:math> is birational for all <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>m</mml:mi> <mml:mo>≥</mml:mo> <mml:mn>52</mml:mn> </mml:mrow> </mml:math> . As an intermediate result, we show that for any <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>K</mml:mi> </mml:math> -Mori fiber space <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>Y</mml:mi> </mml:math> of a canonical weak <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>ℚ</mml:mi> </mml:math> -Fano <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mn>3</mml:mn> </mml:math> -fold, the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>m</mml:mi> </mml:math> -th anti-canonical map defined by <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo>|</mml:mo> <mml:mo>-</mml:mo> <mml:mi>m</mml:mi> <mml:msub> <mml:mi>K</mml:mi> <mml:mi>Y</mml:mi> </mml:msub> <mml:mo>|</mml:mo> </mml:mrow> </mml:math> is birational for all <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>m</mml:mi> <mml:mo>≥</mml:mo> <mml:mn>52</mml:mn> </mml:mrow> </mml:math> .