Unexpected but recurrent phenomena for Quot and Hilbert schemes of points

We investigate some aspects of the geometry of two classical generalisations of the Hilbert schemes of points. Precisely, we show that parity conjecture for QuotᵣᵈA³ already fails for d=8 and r=2 and that lots of the elementary components of the nested Hilbert schemes of points on smooth quasi-projective varieties of dimension at least 4 are generically non-reduced. Finally, we give an infinite family of elementary components of the classical Hilbert schemes of points.