Irreducibility and singularities of some nested Hilbert schemes

On the Hilbert function of Artinian local complete intersections of codimension three

“…The table shows a curious "dividing wall" of unknown but existing components. It would be important to understand whether this is an artifact of our current state of knowledge or a genuinely existing "transitory phase" between the irreducible moduli for small n and the highly reducible ones for larger n. [87,88] exist [72,18] Hilbert scheme none none [48] exist [66,69] known [26] punctual Hilbert scheme none none [16,44] exist known Quot scheme none none [101] exist known [83] punctual Quot scheme none none [5,43] exist [105, 7.10.5] known flag Hilbert scheme none exist [118] exist known Table 1. Nontrivial components of moduli spaces.…”

Irreducibility of Some Nested Hilbert Schemes

“…In [Add16, §3.A] the irreducibility of S [n,n+1,n+2] is proved. In [RT22], Ryan and Taylor study the irreducibility, singularities and Picard groups of nested Hilbert schemes S [n,n+1,n+2] . In [BE16], Bulois and Evain studied irreducible components of nested Hilbert schemes supported at a single point using the connection between nested Hilbert schemes and commuting varieties of parabolic subalgebras.…”

“…In [RS21, Proposition 3.7] the authors prove the existence of tuples n 1 < • • • < n k , for each k 5, such that the nested Hilbert scheme (A 2 ) [n 1 ,...,n k ] is reducible. We refer the reader to [RT22], [RS21] and the references therein for more results related to irreducibility of nested Hilbert schemes.…”

“…The proofs of the above results proceed by combining some of the ideas in [EL99], [BE16] and [RT22].…”