Background on Milnor fibers:
Suppose a smooth complex plane curve acquires an isolated singularity f(x,y) = 0 with Milnor number µ = dim.C[[x,y]]/(∂f/∂x, ∂f/∂y), and r local branches. Then we know from the theory of plane curve singularities, [Milnor’s Singular Points of Complex Hypersurfaces p.45], that a small spherical neighborhood of the singular point meets a nearby smooth curve C(t) in a manifold -with -boundary M, the Milnor fiber, obtained from a compact Riemann surface of genus g by removing r discs. Here g = ∂ + 1 – r, where ∂ = (1/2)(µ -1+r) is the algebraic geometer’s version of µ. General fiber = the special fiber plus the Milnor fiber (topologically) Thus a topological model of the nearby smooth curve is obtained by making a connected sum of this bordered manifold M with the normalization of the singularity, i.e. with the bordered manifold obtained by removing a small neighborhood of the singularity on the singular curve.
Conjectural structure of the stable reduction
Since the stable reduction of such a degeneration arises algebraically by blowing up, the normalization of the singular curve should in general be a component of the stable limit. Since also the arithmetic genus is constant in a family, it seems the stable limit should be in general obtained topologically from the normalization by wedging on the result of collapsing the r boundary curves of the Milnor fiber, i.e. algebraically by adding in a non singular curve of genus g, touching the normalization once at each branch.
- If a smooth curve degenerates to a singularity with r = 4 branches and µ = 3, i.e. ∂ = 3, then g = 0, the Milnor fiber is a sphere with 4 discs removed, and we should adjoin to the normalization a smooth rational curve meeting each of the 4 branches once.
- For degenerations with singular general fiber, the stable reduction should be a specialization of this one. If three nodes coalesce at the previous isolated singularity with µ = 3 = ∂, and r = 4, the “Milnor fiber” M of the singular family is a sequential union of 4 discs, each meeting the previous one at one point and the next one at a different point. Thus the first and last discs meet the others in only one point, but the two middle discs meet each other and one other disc.
Thus the stable reduction is obtained by collapsing the 4 boundary circles of M and attaching the result to the normalization at the 4 resulting points. Then the first and last two components are not stable, so they are then each collapsed to a point.
Hence the stable reduction is created by joining to the normalization at 4 points, a reducible rational curve consisting of two smooth rational components with one point in common. This is a specialization of the previous case, formed by collapsing a loop surrounding two of the 4 marked points on the previous smooth rational component.
- If a family of curves with three nodes degenerates so that the three nodes not only come together, but we also collapse three disjoint “transverse homology 1-cycles”, one passing through each node, we get a singularity with r = 1, µ = total # of “vanishing cycles” (starting from a smooth curve) = 6, and g = 3. The Milnor fiber in the generic singular curve is the result of removing one disc from an irreducible curve of arithmetic genus 3 with three nodes. Then the obvious candidates for a stable limit are obtained by joining the normalization to a specialization of an irreducible curve of arithmetic genus 3 with three nodes. The specialization should be either stable or become stable after marking the point where it meets the normalization. There seem to be several such possibilities, but the generic case should be where the joined component is irreducible of arithmetic genus 3.
- For a family of smooth plane quartics approaching a double conic, the Milnor fiber as defined above is the entire nearby smooth fiber. Then it seems one should get the stable limit topologically by removing the singular locus, i.e. the whole conic, and replacing it by a smooth curve of genus 3.
- If a smooth space curve acquires a triple point with independent branches it appears the Milnor fiber is a sphere with three discs removed, so the stable limit should be the normalization plus a smooth curve of genus zero meeting all three branches.
Are these “obvious” candidates for the stable limit correct? If so, would seem that to determine the topological type of the stable limit, once it is assumed to exist, that often no blowing up and down or base change is needed. And since Milnor’s book appeared in 1968, this presumably would have been known for a long time.
Although I had long ago read the original sources:
and Mumford: http://www.jmilne.org/math/Documents/woodshole3.pdf,
I was discouraged by the apparent complexity of computing examples, and did not notice the connection with Milnor fibers until recently, while watching an expert (Joe Harris) explain some simple cases. He was using the method of blowing up, but his answers sparked this question. How much truth is there in this simple topological intuition? Is this something one learns on day 1 nowadays, or is it too optimistic? One reservation is that Milnor’s theory was apparently proved by him only in the case of hypersurfaces, curves on smooth surfaces.
It seems that to answer this in the affirmative, one only needs to know that the part of the stable reduction residual to the normalization, i.e. the "tail", is connected and meets the normalization once at each branch of the singularity. I.e. then it seems to follow that after smoothing all the nodes of the tail except the ones where it meets the normalization, one has an irreducible smooth tail of precisely the genus given by the Milnor number calculation.
I want to express my gratitude for a recent conversation with Enrico Arbarello, who observed that this method should also apply to degenerations of pointed curves. Moreover he remarked the usefulness of examining morphisms between various manifestations of such curves, such as the normalization morphism. As a consequence, it seems normalization should commute with stable degeneration, and hence the Milnor method should apply as stated to pointed curves.
E.g. consider a 2 pointed curve where the 2 points are coming together. A neighborhood of the limit is a disc containing the two points. Hence the limit should have as a limiting tail, the sphere obtained by shrinking to a point the circular boundary of the disk. Thus the tail is a sphere with 2 marked points.