Timeline for Does a generic curve inside the space of curves with a node at a specific point have only finitely many nodes?

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Jul 29, 2012 at 14:05	vote	accept	Ritwik
Jul 29, 2012 at 13:48	comment	added	Jason Starr		@Ritwik: The method works so long as there are "enough" curves with that singularity so that the transform curves on the blowing up give a base point free linear system. This should work for $A_k$ singularities at a single singular point $[1,0,0]$ if $k<d$ using the four curves $x^{d-k}y^k$, $x^{d-k}z^k$, $y^d$ and $z^d$. However, the method would fail, for instance, if you considered degree $3$ curves with $A_1$ singularities at two distinct points. In fact this is very closely related to the Harbourne-Hirschowitz and Segre conjectures. I recommend you learn those.
Jul 29, 2012 at 13:43	history	edited	Jason Starr	CC BY-SA 3.0	Added explanation
Jul 29, 2012 at 13:43	comment	added	Ritwik		Would this argument also work if I changed the question slightly: instead of looking at the space of curves with a strict node at [1,0,0], I look at the space of curves with a strict A_k node at [1,0,0]. I want to know if a generic element of A (or preferably closure of A) has only one singular point. Assume d is sufficiently large.
Jul 29, 2012 at 13:24	history	answered	Jason Starr	CC BY-SA 3.0