Tangent cones to Severi strata - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-19T15:44:38Z http://mathoverflow.net/feeds/question/68685 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/68685/tangent-cones-to-severi-strata Tangent cones to Severi strata Vivek Shende 2011-06-24T01:58:43Z 2011-06-24T02:05:56Z <p>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 $h$. </p> <p>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 $\mathbb{C}[x,y]/(f,\partial_x f, \partial_y f)$. </p> <blockquote> <blockquote> <p>Is there a description of the tangent cones of the other $\Lambda_h$ ?</p> </blockquote> </blockquote>