Review: Introduction to Singularities and Deformations

Introduction to Singularities and Deformations
Introduction to Singularities and Deformations by G.-M. Greuel
My rating: 3 of 5 stars

What’s good?
The introductory chapters, which set up the algebro-geometric background and an introduction to singularity theory, are very nice, with a particularly nice treatment of the often rather mysterious criterion of flatness. In particular, the authors make good use of geometric examples to motivate and clarify the algebra.

The basic chapters on deformation theory, where the theory is treated in generality, are also quite nice, with a very elegant introduction to the concept of versality, though I felt more could have been made of the distinction between versal and universal deformations. The application of this theory to deformations of curve singularities is similarly elegant. The introduction of equisingular deformations was good but, a very little geometrical discussion was present, seemed rather arbitrary – the fact that it is just about the only natural concept of a singularity that remains singular under deformation somehow got lost in the algebra.

What’s not so good?
The whole discussion of singularity theory could have done with many more geometrical examples to motivate the often rather abstruse algebra. For example, Pusieux coefficients are a very powerful algebraic tool, but without a geometric model it’s quite hard to visualise what they actually mean.

There was also a tendency to vanish down rabbit-holes, culminating in the last 80 or so pages of the book, which are a disquisition on various more and more abstruse forms of equinormalisable deformation, so abstruse that most of the space was actually taken up with definitions, and the interesting bit (the deformation theory) was often an afterthought.

The appendices on Sheaves and algebra were good, but the appendix on abstract deformation theory was terrible: it presented abstract theory, not making it clear which were definitions and which deductions, and giving no motivation at all. This was unfortunate, given that a particularly poorly described bit of theory turned out to be fundamental to a critical part of the argument.

More seriously, the book’s great flaw is that it works entirely locally, so with germs of varieties and singularities and deformations. Now this allows considerable simplification, e.g. in that non-singular deformations are de facto trivial, but a huge amount is lost in not going over to the global picture. Even an aversion to cohomology should not limit one in that way. I feel that there should at least have been some explanation of the local – global relation for deformations, and how one computes global deformations from local, e.g. with an explanation of how locally trivial deformations become trivial.

Conclusion

A reasonably good pedagogical book, but not one to stand on its own. I would recommend reading at least Deformations of Singularities by Jan Stevens in addition.

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