GTA Seminar : Bierstone -- Resolution Preserving Normal Crossings Singularities

Speaker: Prof. Edward Bierstone (Toronto)

http://www.math.toronto.edu/bierston/

Time: Tuesday, March 25, 12NOON--1PM. 

Room: Carslaw 535A. 

Lunch: seminar lunch is right after the talk.

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Title: Resolution Preserving Normal Crossings Singularities

Abstract: a normal crossings singularity means a transverse self-intersection.
Given a singular variety X (defined over the complex numbers, for example), 
can we find a proper mapping F from a variety Y to X such that Y has only normal 
crossings singularities, and F is an isomorphism over the open subset of X where 
the only singularities are normal crossings? This a fundamental question in 
birational geometry; for instance in the minimal model program. The answer 
depends on whether normal crossings is understood in an algebraic or more 
general local-analytic (formal) sense.

An illuminating example is the pinch point or Whitney umbrella 
X: z^2 + xy^2 = 0, which has general normal crossings singularities along the
nonzero x-axis. There is no proper birational mapping that eliminates the pinch 
point singularity at the origin without modifying normal crossings points.

So it makes sense to ask: can we find the smallest class of singularities S with
the following properties: (1) S includes all normal crossings singularities; (2) 
given X, there is a proper mapping F from Y to X such that Y has only singularities 
in S, and F is a isomorphism over the normal crossings locus of X? For surfaces X, 
it turns out that S comprises precisely normal crossings singularities and the pinch 
point. We can describe S completely also in dimension three, but the problem is open 
in higher dimension.

What is the higher-dimensional analogue of the pinch point? This is related to 
circulant matrices and plays an important part in the results above. Proofs of 
the results are based on the philosophy that the desingularization invariant 
can be used together with natural geometric information to compute local normal 
forms of singularities.

(Joint work with Sergio Da Silva, Pierre Lairez, Pierre Milman and Franklin Vera
Pacheco.)

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Seminar website:

http://www.maths.usyd.edu.au/u/SemConf/Geometry/