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28 March 2024
 
  » arxiv » math.AG/0412519

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A study of the Hilbert-Mumford criterion for the stability of projective varieties
J. Ross ; R. P. Thomas ;
Date 29 Dec 2004
Subject Algebraic Geometry; Differential Geometry MSC-class: 14L24 | math.AG math.DG
AbstractWe make a systematic study of the Hilbert-Mumford criterion for different notions of stability for polarised algebraic varieties $(X,L)$; in particular for K- and Chow stability. For each type of stability this leads to a concept of slope $mu$ for varieties and their subschemes; if $(X,L)$ is semistable then $mu(Z)lemu(X)$ for all $Zsubset X$. We give examples such as curves, canonical models and Calabi-Yaus. We prove various foundational technical results towards understanding the converse, leading to partial results; in particular this gives a geometric (rather than combinatorial) proof of the stability of smooth curves.
Source arXiv, math.AG/0412519
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