Resolving Toric Varieties with Nash Blowups

Abstract

of the paper. In §2 we introduce the toric dictionary and, following Gonzalez-Sprinberg [6], translate the question into convex geometry. In §3 we summarize the effect of the iterated Nash blow-up, in the toric case, using the notion of a resolution tree. In §4 we spell out what happens in the 2-dimensional toric case in terms of continued fractions. In §5 we digress briefly on the classification of quasi-smooth affine toric varieties, those corresponding to simplicial cones in the toric dictionary. Then in §6 we give an account of our computer investigations.Notation and conventions. We slightly abuse terminology in two ways. First, as we are concerned with the normalized Nash blow-up throughout, we refer to it simply as the Nash blow-up. Second, as we are concerned with rational polyhedra and rational polyhedral cones throughout, we refer to them simply as polyhedra and cones. We denote the natural numbers, including 0, by Z + , and we likewise denote the nonnegative rational numbers, including 0, by Q + . We denote the span of v 1 , . . . , v k with coefficients in S by S v 1 , . . . , v k . Thus, for example, the first quadrant in Q 2 is denoted Q + e 1 , e 2 .Acknowledgements. The very helpful advice provided by Kevin Purbhoo is gratefully acknowledged. We also thank Jeffrey Lagarias for an inspiring conversation, Dave Bayer for recommending the use of 4ti2, and Sam Payne and Greg Smith for informing us of their parallel work on the subject.

Toric geometry and the Semple–Nash modification

“…Let us consider a vector ν such that ν ∈ int(θ)∩ int(σ). Then we get that (1) and Proposition 3.6 the closure of any orbit contained T Γ contains orb(σ, Γ) thus T Γ ⊂ T Λ .…”

“…The recent paper [1] suggests that it would be interesting to develop an approach from a computational viewpoint to the iteration of Semple-Nash modification.…”

“…We consider the set J : We now consider the blowing up of the monomial ideal J . A cone τ (1) ∈ Σ(d) determines a vertex m (1) of N σ (J ) by (18) and also the finitely generated semigroup Γ (2) τ (1)…”

Local Euler obstructions of toric varieties

“…One can quite explicitly describe both the Nash blowup of a toric variety, and its normalization as a toric variety, see [3], [9]. It would be interesting to prove Matsui and Takeuchi's formula for the local Euler obstruction directly from the algebraic definition, for instance if one could describe the Nash sheaf as a module over the Cox ring of X A .…”

“…It isn't easy to see a clear pattern. This might be analogous to the computations of the Nash blow-up of toric varieties in [3], which in principle could be be used to compute the local Euler obstruction. The authors write "Almost every straightforward conjecture one might make about the patterns in the Nash resolution seems to be false.…”