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  • $\begingroup$ By the way, a result of Hartschorne says that the Hilbert scheme of subschemes of projective space is connected (see : numdam.org/article/PMIHES_1966__29__5_0.pdf). So, if I am not mistaken, the total Betti number should not vary too much along the Hibert scheme... $\endgroup$ Commented May 26, 2017 at 11:39
  • $\begingroup$ @Libli: The issue is that the Hilbert scheme may become highly disconnected when one removes points corresponding to singular varieties - I admittedly don't have an example of the top of my head, but I would be extremely surprised if one did not exist. $\endgroup$ Commented May 26, 2017 at 12:20
  • $\begingroup$ might be the case $\endgroup$ Commented May 26, 2017 at 12:26
  • $\begingroup$ There is a bound on the sum of the Betti numbers due to Thom and Milnor. $\endgroup$ Commented May 26, 2017 at 13:32
  • $\begingroup$ @JasonStarr Does the Milnor-Thom approach provide a bound for any subvariety? I was under the impression that their theorem only covered the cases of hypersurfaces. $\endgroup$ Commented May 26, 2017 at 15:25