Existence of log canonical closures

Christopher D. Hacon; Chenyang Xu

doi:10.1007/s00222-012-0409-0

Existence of log canonical closures

Christopher D. Hacon
, Chenyang Xu

Research output: Contribution to journal › Article › peer-review

118 Scopus citations

Abstract

Let f:X→U be a projective morphism of normal varieties and (X,Δ) a dlt pair. We prove that if there is an open set U⁰⊂U, such that (X,Δ)×_UU⁰ has a good minimal model over U⁰ and the images of all the non-klt centers intersect U⁰, then (X,Δ) has a good minimal model over U. As consequences we show the existence of log canonical compactifications for open log canonical pairs, and the fact that the moduli functor of stable schemes satisfies the valuative criterion for properness.

Original language	English (US)
Pages (from-to)	161-195
Number of pages	35
Journal	Inventiones Mathematicae
Volume	192
Issue number	1
DOIs	https://doi.org/10.1007/s00222-012-0409-0
State	Published - Apr 2013
Externally published	Yes

All Science Journal Classification (ASJC) codes

General Mathematics

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10.1007/s00222-012-0409-0

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title = "Existence of log canonical closures",

abstract = "Let f:X→U be a projective morphism of normal varieties and (X,Δ) a dlt pair. We prove that if there is an open set U0⊂U, such that (X,Δ)×UU0 has a good minimal model over U0 and the images of all the non-klt centers intersect U0, then (X,Δ) has a good minimal model over U. As consequences we show the existence of log canonical compactifications for open log canonical pairs, and the fact that the moduli functor of stable schemes satisfies the valuative criterion for properness.",

author = "Hacon, \{Christopher D.\} and Chenyang Xu",

note = "Funding Information: The first author was partially supported by NSF research grant no. 0757897, the second author was partially supported by NSF research grant no. 0969495. We are grateful to O. Fujino, J. Koll{\'a}r and J. MKernan for many useful comments and suggestions. We are also in debt to J. Koll{\'a}r for allowing us to use the materials of [] in Sect. 3. c ",

year = "2013",

month = apr,

doi = "10.1007/s00222-012-0409-0",

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journal = "Inventiones Mathematicae",

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TY - JOUR

T1 - Existence of log canonical closures

AU - Hacon, Christopher D.

AU - Xu, Chenyang

N1 - Funding Information: The first author was partially supported by NSF research grant no. 0757897, the second author was partially supported by NSF research grant no. 0969495. We are grateful to O. Fujino, J. Kollár and J. MKernan for many useful comments and suggestions. We are also in debt to J. Kollár for allowing us to use the materials of [] in Sect. 3. c

PY - 2013/4

Y1 - 2013/4

N2 - Let f:X→U be a projective morphism of normal varieties and (X,Δ) a dlt pair. We prove that if there is an open set U0⊂U, such that (X,Δ)×UU0 has a good minimal model over U0 and the images of all the non-klt centers intersect U0, then (X,Δ) has a good minimal model over U. As consequences we show the existence of log canonical compactifications for open log canonical pairs, and the fact that the moduli functor of stable schemes satisfies the valuative criterion for properness.

AB - Let f:X→U be a projective morphism of normal varieties and (X,Δ) a dlt pair. We prove that if there is an open set U0⊂U, such that (X,Δ)×UU0 has a good minimal model over U0 and the images of all the non-klt centers intersect U0, then (X,Δ) has a good minimal model over U. As consequences we show the existence of log canonical compactifications for open log canonical pairs, and the fact that the moduli functor of stable schemes satisfies the valuative criterion for properness.

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Existence of log canonical closures

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