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Let $X$ be a smooth complex projective threefold, and $\beta\in H_2(X,\mathbb Z)$ a curve class. In the Kontsevich's moduli space of stable maps, $\overline M_g(X;\beta)$, a general point $[f:C\to X]$ is represented by a map which is an immersion. In some sense this is the closest we can get, in this moduli space, to what a "curve in $X$" should look like.

I was wondering what the situation is in Donaldson-Thomas theory. Here the moduli space of curves $M=M(\textrm{ch})$ is a moduli space of torsion-free sheaves on $X$, with Chern character $\textrm{ch}=(1,0,-\beta,n)$, where $n$ keeps track of both the genus and the number of isolated and embedded points. Curves here are all embedded, but they have some isolated and embedded points in general (their number is $r=p_a-1+n$).

Question. What are the good properties that a general point of $M$ does have?

(Just some random "good" qualities: connected, reduced, Cohen-Macaulay, pure...).

Thank you in advance!

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    $\begingroup$ It need not be true that a general point in $\overline{M}_g(X,\beta)$ is represented by an immersion. For example if $g=1$, $X$ is quintic threefold, and $\beta$ is the class of the line (or twice the class of the line). In the former case, all maps lie in the boundary and collapse a component, in the later case, we also have maps with smooth domain which are not immersions (they are branched covers of the lines in $X$). There are no maps which are immersions. I don't think that there is much you can say about a general stable map (or subscheme in the case of DT theory). $\endgroup$
    – Jim Bryan
    May 12, 2014 at 20:07
  • $\begingroup$ @JimBryan: thanks for your comment. I do not remember where I read that statement - perhaps I didn't. Could it be possible that it becomes true for $g\geq 2$ or $g>2$? Or is it totally false? $\endgroup$
    – Brenin
    May 16, 2014 at 21:35
  • $\begingroup$ No it is just false. To elaborate on my previous example, all degree 2 curves on a generic quintic threefold are either smooth conic curves (genus 0) or doubled lines. Hence a stable map of genus 1 or greater of degree must either be a double cover of a line, or be degree on onto a conic and hence have a collapsing component of higher genus. Double covers of a line are always ramified, so either way you don't get immersions. $\endgroup$
    – Jim Bryan
    May 17, 2014 at 5:18
  • $\begingroup$ ...For any curve counting theory on a CY threefold, you can't escape this sort of behaviour --- multiple covers and/or collapsing components in GW theory, non-reduced structures and/or embedded points in DT theory. $\endgroup$
    – Jim Bryan
    May 17, 2014 at 5:18

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