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Relations in the Sarkisov program

Part of: Birational geometry

Published online by Cambridge University Press:  28 August 2013

Anne-Sophie Kaloghiros
Anne-Sophie Kaloghiros*
Affiliation:
Department of Mathematics, Imperial College London, 180 Queen’s Gate, London SW7 2AZ, UK email a.kaloghiros@imperial.ac.uk
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Abstract

The Sarkisov program studies birational maps between varieties that are end products of the Minimal Model Program (MMP) on nonsingular uniruled varieties. If $X$ and $Y$ are terminal $ \mathbb{Q} $-factorial projective varieties endowed with a structure of Mori fibre space, a birational map $f: X\dashrightarrow Y$ is the composition of a finite number of elementary Sarkisov links. This decomposition is in general not unique: two such define a relation in the Sarkisov program. I define elementary relations, and show they generate relations in the Sarkisov program. Roughly speaking, elementary relations are the relations among the end products of suitable relative MMPs of $Z$ over $W$ with $\rho (Z/ W)= 3$.

Keywords

Sarkisov programMinimal Model Program

MSC classification

Secondary: 14E30: Minimal model program (Mori theory, extremal rays) 14E05: Rational and birational maps
Type
Research Article
Information
Compositio Mathematica , Volume 149 , Issue 10 , October 2013 , pp. 1685 - 1709
Copyright
© The Author(s) 2013 

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