  We study punctual quot-schemes of torsion-free sheaves $E_Y$ on smooth
projective curves, surfaces and Calabi--Yau fourfolds via their virtual
geometry. Our goal is to give a complete description of the virtual fundamental
classes and their tautological integrals. In the fourfold case, we first
construct these classes under additional conditions. We use novel methods
relying on the wall-crossing of Joyce. Our results include -the dependence of
the cobordism classes on the torsion-free sheaf $E_Y$ where $Y$ is a surface,
-relations to the previous results in the literature, which addressed the case
of a trivial $E_Y$, -a new 12-fold correspondence relating Segre and Verlinde
invariants for curves, surfaces and Calabi-Yau fourfolds based on the one
observed by Arbesfeld-Johnson-Lim-Oprea-Pandharipande in dimensions one and
two, -a closed formula for the Nekrasov genus, which gives a compact analogue
of Nekrasov's conjecture. As our techniques are orthogonal to the original
literature, we make our work independent by proving a new combinatorial
identity in arXiv:2111.09868

Algebraic Geometry;Combinatorics