A reference for modern treatment of Hartshorne's theorem on the connectedness of the Hilbert scheme My collaborator and myself were using the ideas of Hartshorne's proof of the connectedness of the Hilbert scheme (http://archive.numdam.org/article/PMIHES_1966__29__5_0.pdf) to prove some results in real algebaic geometry. We were told that the terminology in the original paper is outdated. Can someone point me to a modern treatment of the theorem along the same lines as the original?
Thanks ahead
 A: I can't claim to be an expert on these matters, but I had also read Hartshorne's paper recently, and found some more recent references (at least when the Hilbert scheme parametrizes subschemes of $\mathbf{P}^n$). I hope they can be of some use!
One reason that Hartshorne's terminology could be considered outdated is that after his work, the degeneration techniques he uses have been reinterpreted using Gröbner basis theory, in particular, the theory of generic initial ideals and Borel-fixed ideals. This method of proving connectedness of the Hilbert scheme first appears in

Alyson A. Reeves. "The radius of the Hilbert scheme." J. Algebraic Geom. 4 (1995), no. 4, 639–657. MR: 1339842.

under the assumption that the characteristic of the ground field is zero; in fact, her results are stronger than Hartshorne's. §3.1 in these notes by Maclagan give a brief account of the proof. Reeves's results were generalized to characteristic $p > 0$ in Keith Pardue's thesis, which I believe was published as

Keith Pardue. "Deformation classes of graded modules and maximal Betti numbers." Illinois J. Math. 40 (1996), no. 4, 564–585. URL: http://projecteuclid.org/euclid.ijm/1255985937. MR: 1415019.

Some of this material appears in Eisenbud's Commutative algebra, §15.9. Lastly, a generalization to ideals in an alternating algebra appears in

Irena Peeva and Mike Stillman. "Connectedness of Hilbert schemes." J. Algebraic Geom. 14 (2005), no. 2, 193–211. DOI: 10.1090/S1056-3911-04-00386-8. MR: 2123227.

They also recover some of Pardue's results, under the assumption that the characteristic of the ground field is zero.
