Severi strata in the discriminant of plane curve singularities — equations and conjectures. 

David Mond (University of Warwick, UK)

This is joint work with Paul Cadman (former PhD student at Warwick)
and Duco van Straten.

Let C be a plane curve singularity defined by an irreducible
germ f, and let S be  the base space of an R_e-versal deformation F of
f. Note that F  induces a deformation of C, and S can be partitioned
according to the sum of the delta invariants of the singular points on the
fibres of this deformation. The subsets in this partition are
known as Severi strata, and the stratum with highest value is the classical
delta constant stratum.

Many years ago Givental and Varchenko showed that the delta constant
stratum is Lagrangian with respect to a symplectic form omega obtained by
pulling back the intersection form on the vanishing cohomology using a period
mapping.  We show that all of the Severi strata are co-isotropic with respect to
omega, and we give equations for them all, as coefficients of wedge powers of
omega with respect to a basis for the module of differential forms with  logarithmic
poles along the  discriminant of the deformation.

We observe some unexpected and unexplained structures, and make some
conjectures concerning the Severi strata and the cohomology of the normalisation of
the fibres of the deformation.