Will give a Algebraic Geometry and Commutative Algebra Seminar entitled:
Resolving configuration hypersurfaces and their Nash blow up
Abstract: A configuration (the choice of a subspace W in a vector space V with distinguished basis E) over a field gives rise to a hyperplane arrangement, a matroid, and a configuration hypersurface. Bloch, Esnault and Kreimer observed that that projective configuration hypersurfaces XW permit a natural rational surjection from a certain incidence variety. They came across it in a search for a resolution of singularities and surmised they had one. We give simple examples showing that this is not so, and exhibit a purely combinatorial property governs the smoothness of this incidence variety. Taking a closer look at the incidence variety, it also permits a second morphism, given by a "Hadamard square". We discuss that the image of this map is the Nash blow up of XW, and that this map is the normalization map. While one can show directly that the incidence variety is normal, using Serre's criterion, we verify nice properties about its affine cone: it is in all positive characteristics F-rational. We may sketch the proof of this in the talk; it uses a duality result and a strategy that was employed before for showing that XW and its cone are F-regular. Time permitting, we then give fewer or more details of a construction of an embedded resolution of singularities of the incidence variety (and hence also of XW); it is based on the fact that replacing XW with the incidence variety made it smooth in all points that lie on the torus. The main tool is theory of Tevelev that asserts a regularization process for closures (in toric varieties) of closed smooth subsets of tori. Joint with D Bath, G Denham, M Schulze.
Date: 11-24-2025
Time: 11:00 am
Location: 258 Hurley Hall
Originally published at math.nd.edu.