View source for Resolution of singularities

''desingularization''

The replacement of a singular [[Algebraic variety|algebraic variety]] by a birationally isomorphic non-singular variety. More precisely, a resolution of singularities of an algebraic variety <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081580/r0815801.png" /> over a ground field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081580/r0815802.png" /> is a proper birational morphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081580/r0815803.png" /> such that the variety <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081580/r0815804.png" /> is non-singular (smooth) (cf. [[Proper morphism|Proper morphism]]; [[Birational morphism|Birational morphism]]). A resolution of singularities of a scheme, of a complex-analytic space, etc. is defined analogously. The existence of a resolution of singularities enables one to reduce many questions to non-singular varieties, and in studying these one can use intersection theory and the apparatus of differential forms.

Usually a resolution of singularities is the result of successive application of monoidal transformations (cf. [[Monoidal transformation|Monoidal transformation]]). It is known that if the centre <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081580/r0815805.png" /> of a monoidal transformation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081580/r0815806.png" /> is admissible (that is, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081580/r0815807.png" /> is non-singular and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081580/r0815808.png" /> is a normal flat variety along <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081580/r0815809.png" />), then the numerical characteristics of the singularity of the variety (the multiplicity, the Hilbert function, etc.) are no worse than those of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081580/r08158010.png" />. The problem consists of choosing the centre of the blowing-up so that the singularities in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081580/r08158011.png" /> really are improved.

In the case of curves the problem of resolution of singularities essentially reduces to normalization. In the two-dimensional case the situation is more complicated. The existence of a resolution of singularities for any variety over a field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081580/r08158012.png" /> of characteristic zero has been proved. More precisely, for a reduced variety <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081580/r08158013.png" /> there exists a finite sequence of admissible monoidal transformations <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081580/r08158014.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081580/r08158015.png" />, with centres <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081580/r08158016.png" />, such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081580/r08158017.png" /> is contained in the set of singular points of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081580/r08158018.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081580/r08158019.png" /> is a non-singular variety. An analogous result is true for complex-analytic spaces. In positive characteristic the existence of a resolution of singularities has been established (1983) for dimensions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081580/r08158020.png" />.

The problem of resolution of singularities is closely connected with the problem of imbedded singularities, formulated as follows. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081580/r08158021.png" /> be imbedded in a non-singular algebraic variety <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081580/r08158022.png" />. Does there exist a proper mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081580/r08158023.png" />, with non-singular <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081580/r08158024.png" />, such that a) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081580/r08158025.png" /> induces an isomorphism from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081580/r08158026.png" /> onto <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081580/r08158027.png" />; and b) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081580/r08158028.png" /> is a divisor with normal crossings? (A divisor on a non-singular variety has normal crossings if it is given locally by an equation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081580/r08158029.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081580/r08158030.png" /> are part of a regular system of parameters on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081580/r08158031.png" />.)

The problem of imbedded singularities is a particular case of the problem of trivialization of a sheaf of ideals. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081580/r08158032.png" /> be a non-singular variety, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081580/r08158033.png" /> be a [[Coherent sheaf|coherent sheaf]] of ideals on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081580/r08158034.png" /> and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081580/r08158035.png" /> be a non-singular closed subvariety. The weak pre-image of the ideal <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081580/r08158036.png" /> under a blowing-up <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081580/r08158037.png" /> with centre in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081580/r08158038.png" /> is the sheaf of ideals

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081580/r08158039.png" /></td> </tr></table>

on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081580/r08158040.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081580/r08158041.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081580/r08158042.png" /> is the multiplicity of the ideal <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081580/r08158043.png" /> at a regular point of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081580/r08158044.png" />. Trivialization of a sheaf of ideals consists of finding a sequence of blowing-ups with non-singular centres for which the weak pre-image <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081580/r08158045.png" /> becomes the structure sheaf. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081580/r08158046.png" /> be a non-singular variety over a field of characteristic zero, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081580/r08158047.png" /> be a coherent sheaf of ideals over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081580/r08158048.png" /> and, in addition, let there be given a certain divisor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081580/r08158049.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081580/r08158050.png" /> with normal crossings. Then there exists a sequence of blowing-ups <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081580/r08158051.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081580/r08158052.png" />, with non-singular centres <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081580/r08158053.png" />, with the following properties: If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081580/r08158054.png" /> is defined as the weak pre-image of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081580/r08158055.png" /> under the blowing-up <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081580/r08158056.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081580/r08158057.png" /> is defined to be <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081580/r08158058.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081580/r08158059.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081580/r08158060.png" /> has only normal crossings (Hironaka's theorem). In addition, one may assume that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081580/r08158061.png" /> lies in the set of points of maximal multiplicity of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081580/r08158062.png" /> and has normal crossings with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081580/r08158063.png" />. For positive characteristic an analogous result is known only when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081580/r08158064.png" />.

Another problem of this type is the problem of eliminating points of indeterminacy of a rational transformation. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081580/r08158065.png" /> be a rational transformation of non-singular algebraic varieties. Does there exist a sequence of blowing-ups with non-singular centres

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081580/r08158066.png" /></td> </tr></table>

such that the induced transformation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081580/r08158067.png" /> is a morphism? This problem reduces to the problem of the existence of a trivialization of a sheaf of ideals, and the answer is affirmative if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081580/r08158068.png" /> or if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/r/r081/r081580/r08158069.png" />.

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  S.S. Abhyankar,   "Resolution of singularities of embedded algebraic surfaces" , Acad. Press  (1966)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  J. Lipman,   "Introduction to resolution of singularities"  R. Hartshorne (ed.) , ''Algebraic geometry (Arcata, 1974)'' , ''Proc. Symp. Pure Math.'' , '''29''' , Amer. Math. Soc.  (1975)  pp. 187–230</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  H. Hironaka,   "Resolution of singulariies of an algebraic variety over a field of characteristic zero I, II"  ''Ann. of Math.'' , '''79'''  (1964)  pp. 109–326</TD></TR></table>