Some components of the moduli space of Koszul Artin-Schelter regular algebras of dimension four
Vishal Bhatoy, Colin Ingalls, Félix LaRoche, Ravali Nookala
TL;DR
The work addresses identifying components of the moduli stack $oldsymbol{A}_4$ for Koszul Artin–Schelter regular algebras of dimension four by comparing tangent deformations to degree-zero Hochschild cohomology via the Kodaira–Spencer map. It develops a framework linking flat base deformations to first-order algebra deformations through $ ext{HH}^2_0(A_p)$, and demonstrates, in particular, that the skew polynomial family yields an isomorphism for its KS map, forming a smooth component. Using GAP-based computations, the authors compute $ ext{HH}^2_0(A_p)$ and KS maps for numerous $(14641)$-type families, establishing criteria under which KS surjectivity implies that a family maps densely onto an irreducible component, or that a bijection yields a generically finite map to the moduli stack. The results identify several families whose parameter spaces map densely onto components of $oldsymbol{A}_4$, and they discuss unirationality of these components. Overall, the paper connects Hochschild cohomology with the local geometry of moduli spaces for four-dimensional Koszul AS-regular algebras and provides concrete identifications of components via deformation-theoretic criteria.
Abstract
We compute the Hochschild cohomology and the Kodaira spencer map for known families of Koszul Artin-Schelter regular algebras of dimension four. We show that when the Kodaira Spencer map at a point is a surjection, the image of the family is a component of the moduli stack of such algebras, and when the Kodaira Spencer map is a bijection, the map to the moduli stack is generically finite. We use this to identify some components of the moduli stack.