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For a [[Local ring|local ring]], this is the determination of a regular local ring birationally equivalent to it. For an irreducible algebraic variety (cf. [[Irreducible variety|Irreducible variety]]) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060230/l0602301.png" /> over a field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060230/l0602302.png" /> a resolving system is a family of irreducible projective varieties <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060230/l0602303.png" /> birationally equivalent to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060230/l0602304.png" /> (that is, such that the rational function fields <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060230/l0602305.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060230/l0602306.png" /> are isomorphic) and satisfying the following condition: For any valuation (place) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060230/l0602307.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060230/l0602308.png" /> there is a variety <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060230/l0602309.png" /> such that the centre <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060230/l06023010.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060230/l06023011.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060230/l06023012.png" /> is a non-singular point. The existence of a resolving system (the local uniformization theorem) was proved for arbitrary varieties over a field of characteristic zero (see [[#References|[1]]]), and also for two-dimensional varieties over any field and three-dimensional varieties over an algebraically closed field of characteristic other than 2, 3 or 5 (see [[#References|[2]]]). The existence of a resolving system for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060230/l06023013.png" /> consisting of a single variety implies resolution of the singularities of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060230/l06023014.png" /> and can be obtained from the local uniformization theorem in dimension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060230/l06023015.png" />. In the general case the local uniformization theorem implies the existence of a finite resolving system (see [[#References|[3]]]).
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For a [[Local ring|local ring]], this is the determination of a regular local ring birationally equivalent to it. For an irreducible algebraic variety (cf. [[Irreducible variety|Irreducible variety]]) $V$ over a field $k$ a resolving system is a family of irreducible projective varieties $\{V_\alpha\}$ birationally equivalent to $V$ (that is, such that the rational function fields $k(V_\alpha)$ and $k(V)$ are isomorphic) and satisfying the following condition: For any valuation (place) $v$ of $k(V)$ there is a variety $V'\in\{V_\alpha\}$ such that the centre $P'$ of $v$ on $V'$ is a non-singular point. The existence of a resolving system (the local uniformization theorem) was proved for arbitrary varieties over a field of characteristic zero (see [[#References|[1]]]), and also for two-dimensional varieties over any field and three-dimensional varieties over an algebraically closed field of characteristic other than 2, 3 or 5 (see [[#References|[2]]]). The existence of a resolving system for $V$ consisting of a single variety implies resolution of the singularities of $V$ and can be obtained from the local uniformization theorem in dimension $\leq3$. In the general case the local uniformization theorem implies the existence of a finite resolving system (see [[#References|[3]]]).
  
 
====References====
 
====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> O. Zariski,   "Local uniformization on algebraic varieties" ''Ann. of Math. (2)'' , '''41''' (1940) pp. 852–896</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> S.S. Abhyankar,   "Resolution of singularities of arithmetic surfaces" , Acad. Press (1966)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top"> W.V.D. Hodge,   D. Pedoe,   "Methods of algebraic geometry" , '''3''' , Cambridge Univ. Press (1954)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top"> O. Zariski,   P. Samuel,   "Commutative algebra" , '''2''' , Springer (1975)</TD></TR></table>
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<table><TR><TD valign="top">[1]</TD> <TD valign="top"> O. Zariski, "Local uniformization on algebraic varieties" ''Ann. of Math. (2)'' , '''41''' (1940) pp. 852–896 {{MR|0002864}} {{ZBL|0025.21601}} {{ZBL|66.1327.02}} </TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> S.S. Abhyankar, "Resolution of singularities of arithmetic surfaces" , Acad. Press (1966)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top"> W.V.D. Hodge, D. Pedoe, "Methods of algebraic geometry" , '''3''' , Cambridge Univ. Press (1954) {{MR|0061846}} {{ZBL|0055.38705}} </TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top"> O. Zariski, P. Samuel, "Commutative algebra" , '''2''' , Springer (1975) {{MR|0389876}} {{MR|0384768}} {{ZBL|0313.13001}} </TD></TR></table>
  
  
  
 
====Comments====
 
====Comments====
The resolution of singularities for algebraic varieties of arbitrary dimension over an algebraically closed field of characteristic zero has been achieved by H. Hironaka in 1964 [[#References|[a1]]]. Over algebraically closed fields of characteristic <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060230/l06023016.png" /> resolution of singularities for varieties of dimension 2, and for varieties of dimension 3 provided <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060230/l06023017.png" />, has been proved by S.S. Abhyankar [[#References|[a2]]].
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The resolution of singularities for algebraic varieties of arbitrary dimension over an algebraically closed field of characteristic zero has been achieved by H. Hironaka in 1964 [[#References|[a1]]]. Over algebraically closed fields of characteristic $p>0$ resolution of singularities for varieties of dimension 2, and for varieties of dimension 3 provided $p>5$, has been proved by S.S. Abhyankar [[#References|[a2]]].
  
 
For (local) uniformization in analytic geometry and in the theory of functions of a complex variable (Riemann surfaces) cf. [[Uniformization|Uniformization]].
 
For (local) uniformization in analytic geometry and in the theory of functions of a complex variable (Riemann surfaces) cf. [[Uniformization|Uniformization]].
  
 
====References====
 
====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> H. Hironaka,   "Resolution of singularities of an algebraic variety over a field of characteristic zero" ''Ann. of Math.'' , '''79''' (1964) pp. 109–326</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> S.S. Abhyankar,   "Resolution of singularities of arithmetic surfaces" , Harper &amp; Row (1965)</TD></TR></table>
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<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> H. Hironaka, "Resolution of singularities of an algebraic variety over a field of characteristic zero" ''Ann. of Math.'' , '''79''' (1964) pp. 109–326 {{MR|0199184}} {{ZBL|0122.38603}} </TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> S.S. Abhyankar, "Resolution of singularities of arithmetic surfaces" , Harper &amp; Row (1965) {{MR|200272}} {{ZBL|}} </TD></TR></table>

Latest revision as of 14:25, 1 May 2014

For a local ring, this is the determination of a regular local ring birationally equivalent to it. For an irreducible algebraic variety (cf. Irreducible variety) $V$ over a field $k$ a resolving system is a family of irreducible projective varieties $\{V_\alpha\}$ birationally equivalent to $V$ (that is, such that the rational function fields $k(V_\alpha)$ and $k(V)$ are isomorphic) and satisfying the following condition: For any valuation (place) $v$ of $k(V)$ there is a variety $V'\in\{V_\alpha\}$ such that the centre $P'$ of $v$ on $V'$ is a non-singular point. The existence of a resolving system (the local uniformization theorem) was proved for arbitrary varieties over a field of characteristic zero (see [1]), and also for two-dimensional varieties over any field and three-dimensional varieties over an algebraically closed field of characteristic other than 2, 3 or 5 (see [2]). The existence of a resolving system for $V$ consisting of a single variety implies resolution of the singularities of $V$ and can be obtained from the local uniformization theorem in dimension $\leq3$. In the general case the local uniformization theorem implies the existence of a finite resolving system (see [3]).

References

[1] O. Zariski, "Local uniformization on algebraic varieties" Ann. of Math. (2) , 41 (1940) pp. 852–896 MR0002864 Zbl 0025.21601 Zbl 66.1327.02
[2] S.S. Abhyankar, "Resolution of singularities of arithmetic surfaces" , Acad. Press (1966)
[3] W.V.D. Hodge, D. Pedoe, "Methods of algebraic geometry" , 3 , Cambridge Univ. Press (1954) MR0061846 Zbl 0055.38705
[4] O. Zariski, P. Samuel, "Commutative algebra" , 2 , Springer (1975) MR0389876 MR0384768 Zbl 0313.13001


Comments

The resolution of singularities for algebraic varieties of arbitrary dimension over an algebraically closed field of characteristic zero has been achieved by H. Hironaka in 1964 [a1]. Over algebraically closed fields of characteristic $p>0$ resolution of singularities for varieties of dimension 2, and for varieties of dimension 3 provided $p>5$, has been proved by S.S. Abhyankar [a2].

For (local) uniformization in analytic geometry and in the theory of functions of a complex variable (Riemann surfaces) cf. Uniformization.

References

[a1] H. Hironaka, "Resolution of singularities of an algebraic variety over a field of characteristic zero" Ann. of Math. , 79 (1964) pp. 109–326 MR0199184 Zbl 0122.38603
[a2] S.S. Abhyankar, "Resolution of singularities of arithmetic surfaces" , Harper & Row (1965) MR200272
How to Cite This Entry:
Local uniformization. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Local_uniformization&oldid=16581
This article was adapted from an original article by V.I. Danilov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article