POINCARÉ SERIES OF COLLECTIONS OF PLANE VALUATIONS

In earlier papers there were given formulae for the Poincaré series of multi-index filtrations on the ring [Formula: see text] of germs of functions of two variables defined by collections of valuations corresponding to (reducible) plane curve singularities and by collections of divisorial ones. It was shown that the Poincaré series of a collection of divisorial valuations determines the topology of the collection of divisors. Here we give a formula for the Poincaré series of a general collection of valuations on the ring [Formula: see text] centered at the origin and prove a generalization of the statement that the Poincaré series determines the topology of the collection.

THE ALEXANDER POLYNOMIAL OF A PLANE CURVE SINGULARITY AND INTEGRALS WITH RESPECT TO THE EULER CHARACTERISTIC

It was shown that the Alexander polynomial (of several variables) of a (reducible) plane curve singularity coincides with the (generalized) Poincaré polynomial of the multi-indexed filtration defined by the curve on the ring [Formula: see text] of germs of functions of two variables. The initial proof of the result was rather complicated (it used analytical, topological and combinatorial arguments). Here we give a new proof based on the notion of the integral with respect to the Euler characteristic over the projectivization of the space [Formula: see text] — the notion similar to (and inspired by) the notion of the motivic integration.

A plane sextic and its five cusps

SynopsisA certain plane sextic of genus 5 was encountered by Humbert and publicised by him [3] in 1894. Its striking geometrical properties clamour for elucidation; this was eventually supplied in 1951. For the canonical curve of genus 5 is the base curve C of a net N of quadrics in projective space [4], and C models a Humbert curve when all the quadrics of N have a common self-polar simplex [1]. The projection of C from one of its chords onto a plane is a 5-nodal sextic, the nodes all becoming cusps when the chord of C becomes a tangent. The properties to be elucidated become clear visually in the projection.The sextic H described here is a specialisation of the cusped curve; it emerges as linearly dependent on a pair of reducible plane sextics concocted ad hoc.