Let $\mathbb{C}[[x,y]]/f(x,y)$ be a reduced plane curve singularity. The base of a versal family can be taken to be (an open subset in) $\Lambda = \mathbb{C}[x,y]/(f,\partial_x f, \partial_y f)$; the curve over $g \in \Lambda$ being the locus cut out by $f+g$. One can consider the closed loci $\Lambda_h$ where the sum of the $\delta$-invariants ("virtual number of nodes near the origin") of the fibre is at least $h$.

The smallest of these -- where the $\delta$-invariant is the same as that of the central fibre -- is sometimes called the equigeneric stratum, and it was shown by Diaz and Harris that the reduced subvariety of $T_0\Lambda$ underlying the tangent cone to $\Lambda_\delta$ is the image of the conductor ideal $I \subset \mathbb{C}[x,y]/f$ inside $ \mathbb{C}[x,y]/(f,\partial_x f, \partial_y f)$.

Is there a description of the tangent cones of the other $\Lambda_h$ ?


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