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Suppose $X$ is a smooth projective surface over $\mathbb{C}$ with irregularity $0$ $(q_1(X)=0)$. I want to understand the curves on the Hilbert scheme of $n$-points on $X$.

By the work of Fogarty, we know that $X^{[n]}$ is a smooth projective variety of dimension $2n$. Given a curve $C$ on $X$, there are two ways we can get a curve on $X^{[n]}$:

  1. The curve given by fixing $n-1$ general points on $X$ and letting an $n$-th point move along $C$.
  2. If $C$ admits a $g^1_n$, then the fibers of the map $C\to \mathbb{P}^1$ give a rational curve on $X^{[n]}$.

Lastly, we have those curve classes contracted by the Hilbert-Chow morphism. Are there other constructions of curves on $X^{[n]}$ not coming from $X$?

Motive: In my recent paper, I have encountered a divisor on $X^{[n]}$. Now I am trying to determine whether that is nef or not. To show it is NOT nef, I am trying to intersect it with different curve classes on $X^{[n]}$, but so far the intersection has been non-negative.

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    $\begingroup$ I don't see why the "in other words" is related. There are certainly other constructions: If $C$ admits a degree $d\leq n$ map to a curve $D$ then letting $d$ points move along $C$ and fixing $n-d$ points of $X$ does the trick, but this doesn't shed much light on your final question. $\endgroup$
    – Will Sawin
    Commented Aug 12 at 19:34
  • $\begingroup$ Dear @Will, I'm just trying to get more info about the cone of curves on the Hilbert scheme. Let me edit the question if the last question confuses people. Thanks for the idea though. $\endgroup$
    – Rio
    Commented Aug 12 at 20:51
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    $\begingroup$ Are you trying to determine the cone of effective curve classes (or, duality, the cone of nef divisors)? If so, I recommend looking at the articles of Coskun, Huizenga, et al. $\endgroup$
    – Jason Starr
    Commented Aug 13 at 4:33
  • $\begingroup$ Dear @JasonStarr, I have edited the question. Thanks for your suggestion. $\endgroup$
    – Rio
    Commented Aug 13 at 12:41

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