Electron-longitudinal acoustic phonon interaction: Difference between revisions

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In mathematics, '''canonical singularities''' appear as singularities of the [[canonical model]] of a [[projective variety]], and '''terminal singularities''' are special cases that appear as singularities of [[Minimal model program|minimal models]]. They were introduced by {{harvtxt|Reid|1980}}. Terminal singularities are important in the [[minimal model program]] because smooth minimal models do not always exist, and thus one must allow certain singularities, namely the terminal singularities.
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Suppose that ''Y'' is a normal variety such that its canonical class ''K''<sub>''Y''</sub> is '''Q'''-Cartier, and let ''f'':''X''→''Y'' be a resolution of the singularities of ''Y''.  
Then
:<math>\displaystyle K_X = f^*(K_Y)+\sum_i a_iE_i</math>
where the sum is over the irreducible exceptional divisors, and the ''a''<sub>''i''</sub> are rational numbers, called the discrepancies.  
   
Then the singularities of ''Y'' are called:
:'''terminal''' if  ''a''<sub>''i''</sub> > 0 for all ''i''
:'''canonical''' if  ''a''<sub>''i''</sub> &ge; 0 for all ''i''
:'''log terminal''' if  ''a''<sub>''i''</sub> > &minus;1 for all ''i''
:'''log canonical''' if  ''a''<sub>''i''</sub> &ge; &minus;1 for all ''i''.
 
==Properties==
The singularities of a projective variety ''V'' are canonical if the variety is [[normal variety|normal]], some power of the [[canonical line bundle]] of the non-singular part of ''V'' extends to a line bundle on ''V'', and ''V'' has the same [[plurigenera]] as any [[resolution of singularities|resolution]] of its singularities. ''V'' has canonical singularities if and only if it is a [[relative canonical model]].
 
The singularities of a projective variety ''V'' are terminal if the variety is [[normal variety|normal]], some power of the [[canonical line bundle]] of the non-singular part of ''V'' extends to a line bundle on ''V'', and ''V''  the pullback of any section of ''V''<sup>''m''</sup> vanishes along any codimension 1 component of the [[exceptional locus]] of a [[resolution of singularities|resolution]] of its singularities.
 
==Classification in small  dimensions==
Two dimensional terminal singularities are smooth.
If a variety has terminal singularities, then its singular points have codimension at least 3, and in particular in dimensions 1 and 2 all terminal singularities are smooth. In 3 dimensions they are isolated and were classified by {{harvtxt|Mori|1985}}.
 
Two dimensional canonical singularities are the same as [[du Val singularity|du Val singularities]], and are analytically isomorphic to quotients
of '''C'''<sup>2</sup> by finite subgroups of SL<sub>2</sub>('''C''').
 
Two dimensional log terminal singularities are  analytically isomorphic to quotients
of '''C'''<sup>2</sup> by finite subgroups of GL<sub>2</sub>('''C''').
 
Two dimensional log canonical singularities have  been classified by {{harvtxt|Kawamata|1988}}.
 
==Pairs==
More generally one can define these concepts for a pair (''X'',Δ) where Δ is a formal linear combination of prime divisors with rational coefficients. The pair is called
*'''terminal''' if Discrep(''X'',Δ) >0
*'''canonical''' if Discrep(''X'',Δ) ≥0
*'''klt''' (Kawamata log terminal) if Discrep(''X'',Δ) >&minus; 1 and |Δ|≤0
*'''plt''' (purely log terminal) if Discrep(''X'',Δ) >&minus; 1
*'''lc''' (log canonical) if Discrep(''X'',Δ) ≥&minus; 1.
 
==References==
*{{Citation | last1=Kollár | first1=János | title=Minimal models of algebraic threefolds: Mori's program | url=http://www.numdam.org/item?id=SB_1988-1989__31__303_0 | mr=1040578 | year=1989 | journal=Astérisque | issn=0303-1179 | issue=177 | pages=303–326}}
*{{Citation | authorlink1=Shigefumi Mori | last1=Mori | first1=Shigefumi | title=On 3-dimensional terminal singularities | url=http://projecteuclid.org/euclid.nmj/1118787793 | mr=792770 | year=1985 | journal=Nagoya Mathematical Journal | issn=0027-7630 | volume=98 | pages=43–66}}
*{{Citation | last1=Reid | first1=Miles | author1-link=Miles Reid | title=Journées de Géometrie Algébrique d'Angers, Juillet 1979/Algebraic Geometry, Angers, 1979 | publisher=Sijthoff & Noordhoff | location=Alphen aan den Rijn | mr=605348 | year=1980 | chapter=Canonical 3-folds | pages=273–310}}
*{{Citation | last1=Reid | first1=Miles | author1-link=Miles Reid | title=Algebraic geometry, Bowdoin, 1985 (Brunswick, Maine, 1985) | publisher=[[American Mathematical Society]] | location=Providence, R.I. | series=Proc. Sympos. Pure Math. | mr=927963 | year=1987 | volume=46 | chapter=Young person's guide to canonical singularities | pages=345–414}}
 
[[Category:Singularity theory]]
[[Category:Algebraic geometry]]

Latest revision as of 13:05, 7 January 2015

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