From resolution of singularities to equisingularities

Tzee-Char Kuo

Objects defined by polynomial, or analytic, equations can have singularities. For example, the cusp y^{2} − x^{3} = 0 has both partial derivatives vanish at 0. By a substitution x = X, y = XY, called a blow-up, the cusp is the image of a parabola, X = Y^{2}, which is smooth. We say the cusp has been desingularised by one blow-up. Similarly y^{2} = x^{5} is desingularised by two blow-ups.

Resolution of singularities amounts to desingularising an object by a succession of blow-ups.

When one wishes to classify singularities, one ought to begin with an equivalence relation. Consider, for example, the Whitney family: W_{t}(x, y) = xy(x + y)(x − ty), for t > 0. Should W_{t} and W_{t′} (t, t′ > 0) be declared equivalent? And in which sense?

The Equisingularity Problem is to search for a “nice and natural” equivalence relation between singularities.

In this lecture we briefly survey some recent developments on an intimate relationship between desingularisation and equisingularity. A substantial part of the lecture can be understood by students.