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@Andreas, why is that? I've heard this before and I can easily prove the holomorphic case, but I can't think off the top of my head why the continuous result is true.
Two questions: 1) is your $K_{\mathbb P^2}$ Calabi-Yau? If you mean by the total space of the canonical bundle the scheme $Spec_{\mathbb P^2} Sym(\omgea_{\mathbb P^2})$ then I believe this has canonical bundle $\pi^*\mathcal O_{\mathbb P^2}(-6)$. So did you mean the total space of the line bundle which has sheaf of sections $\omega_{\mathbb P^2}$, which I agree would be Calabi-Yau. 2)Also, aren't the GV invariants conjectured to be nonnegative (for example see p. 19 of arxiv.org/abs/hep-th/9111025), so is that simply not believed anymore, or are they talking about different things?
I don't know of a general statement for $S^{[2]}$, but I know that in the simple case of, say, a surface with one node, the singular locus of $S^{[2]}$ is known. I don't remember the details precisely, but I believe it looks locally like $S\times C$, where $C$ is the smooth plane conic $S$ induces (alternatively the exceptional divisor of the blow-up of $S$ at the node). Less geometrically, the singular locus as one might expect consists of length 2 subschemes which are support at least partially at the node. It would probably not be too hard to generalize this to any nodal surface.
Not to be pedantic, but did you switch A) and B)? Also for your argument in B), $\dim H^0$ of the fiber being the number of connected components doesn't necessarily apply if the fibers have embedded components, which I believe normality rules out, so would reduced geometric fibers be enough there? Also I don't see how semicontinuity finishes it. It implies there is an open set of $S$ upon which the minimal number of components is achieved, but why does this imply it is all of $S$?
