arXiv subjects

Felix Minddal

Publications and source records attributed to Felix Minddal.

2 recordsLinked to original sources

arXiv2026

Virtual K-theoretic invariants of the nested Hilbert scheme on $\mathbb{C}^2$

We construct a nested version of the non-commutative Hilbert scheme and embed the nested Hilbert scheme of points on $\mathbb{C}^n$ as the commutativity locus. In the $\mathbb{C}^2$-case, we exhibit this locus as the zero locus of two different sections of bundles and use this description to equip the nested Hilbert scheme of points with a perfect obstruction theory equivalent to that of Gholampour, Sheshmani and Yau. We study the torus equivariant pushforward of the virtual structure sheaf under the map of nested Hilbert schemes forgetting the largest subscheme of the nesting. Using a map of the bundles on the non-commutative Hilbert scheme, we prove that this pushforward is a twist of the virtual structure sheaf on the lower level. Using localization, we show that the twist is by a constant class with values corresponding to the equivariant Euler characteristic of a tautological class of the Hilbert scheme of points. From this, we derive a closed formula for the multivariate generating series of the equivariant virtual Euler characteristic of the nested Hilbert scheme of points.

math.AG↗

arXiv2025

Virtual invariants from the non-associative Hilbert scheme

We introduce a non-associative model for the Hilbert scheme of points in arbitrary dimension. We define a smooth ambient space, which we call the non-associative Hilbert scheme, containing the classical nested Hilbert scheme $\mathrm{NHilb}^{\underline{d}}(\mathbb{A}^n)$ as the associativity, cut out by an explicit section of an associativity bundle. This construction yields canonical perfect obstruction theories and virtual fundamental classes on $\mathrm{NHilb}^{\underline{d}}(\mathbb{A}^n)$ for all $(n,\underline d)$. Using virtual localization, we obtain closed formulas for these virtual classes as sums over admissible nested partitions. Over the punctual locus, we rewrite these as a single multivariable iterated residue formula governing all virtual integrals. Our construction works for all $n$, produces positive-dimensional virtual classes when $n$ is large compared to the number of points, and we expect that they extend the non-commutative matrix model and virtual class construction on Calabi-Yau threefolds.

math.AG↗