Let be a smooth, connected, dimension , quasi-projective variety embedded in . Consider integers , with , and the Hilbert scheme of aligned, finite, degree subschemes of , with multiplicities at points (possibly coinciding). The expected dimension of is . We study the locus of points where is not smooth of expected dimension, and we prove that the lines carrying this locus do not fill up .
"On the smooth locus of aligned Hilbert schemes, the -secant lemma and the general projection theorem." Duke Math. J. 162 (3) 553 - 578, 15 February 2013. https://doi.org/10.1215/00127094-2019817