Logarithmic vector fields and the Severi strata in the discriminant

TL;DR

This paper studies the geometric structure of Severi strata within the discriminant of plane curve singularities, revealing co-isotropic properties, equations for strata, and Cohen-Macaulay conditions, with implications for specific singularity types.

Contribution

It demonstrates that all Severi strata are co-isotropic with respect to a symplectic form and provides explicit equations for these strata, establishing Cohen-Macaulay properties for certain singularity types.

Findings

Severi strata are co-isotropic with respect to the symplectic form.

Explicit equations for the Severi strata are derived from powers of the symplectic form.

Certain Severi strata are shown to be Cohen-Macaulay, including for E6, E8, and all A_{2k} cases.

Abstract

The discriminant, $D$, in the base of a miniversal deformation of an irreducible plane curve singularity, is partitioned according to the genus of the (singular) fibre, or, equivalently, by the sum of the delta invariants of the singular points of the fibre. The members of the partition are known as the {\it Severi strata}. The smallest is the $\delta$-constant stratum, $D(\delta)$, where the genus of the fibre is $0$. It is well known, by work of Givental' and Varchenko, to be Lagrangian with respect to the symplectic form $\Omega$ obtained by pulling back the intersection form on the cohomology of the fibre via the period mapping. We show that the remaining Severi strata are also co-isotropic with respect to $\Omega$, and moreover that the coefficients of the expression of $\Omega^{\wedge k}$ with respect to a basis of $\Omega^{2k}(\log D)$ are equations for $D(\delta-k+1)$, for $k=1,\ldots,\delta$. These equations allow us to show that for $E_6$ and $E_8$, $D(\delta)$ is Cohen-Macaulay (this was already shown by Givental' for $A_{2k}$), and that, as far as we can calculate, for $A_{2k}$ all of the Severi strata are Cohen-Macaulay. Our construction also produces a canonical rank 2 maximal Cohen Macaulay module on the discriminant.