Resolution preserving normal crossings singularitiesProf. Edward Bierstone, University of Toronto
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.)