Higher Nash blowup on normal toric varieties (Q406353)

Let \(X\) be a \(d\)-dimensional algebraic variety over an algebraically closed field \(k\). To a closed point \(x \in X\) we may associate the scheme \([x^{(n)}]={\text{Spec}}({\mathcal O}_{X,x}/{{\mathcal M}_{X,x}}^{n+1})\), viewed as a closed subscheme of \(X\) concentrated at \(x\). This corresponds to a point of \(\mathrm{Hilb}_N(X)\), where \(N=\binom{d+n}{d}\), and the mapping sending a regular closed point \(x\) of \(X\) to \([x^{(n)}]\) determines a morphism \(\delta _n:X_{sm} \to \mathrm{Hilb}_N(X)\). The closure of the graph of \(\delta_n\) in \(X \times \mathrm{Hilb}_N(X)\) is called the \textit{\(n\)-th Nash blowup} of \(X\), denoted by \(\mathrm{Nash}_n(X)\). There is an induced projection \(\pi_n:\mathrm{Nash}_n(X) \to X\), which is projective, birational and an isomorphism over \(X_{sm}\). For \(n=1\) this is the usual Nash blowup, which may be defined more elementarily using tangent spaces and Grassmannians.NEWLINENEWLINEIn the paper under review, Duarte obtains several interesting results about \(\mathrm{Nash}_n(X)\) in case \(X\) is a suitable toric variety and the characteristic of \(k\) is zero. More precisely, let \(X\) be the irreducible, normal affine toric variety associated to a \(d\)-dimensional strictly convex rational polyhedron \(\sigma \subset {\mathbb R}^d\). The affine algebra of \(X\) is isomorphic to \(R=k[x^{a_1}, \dots, x_s^{a_s}]\), where we write \(a_i=(a_{i_1}, \dots, a_{i_s}) \in {\mathbb N}^s\) and \(x^{a_i}=x_1^{a_{i_1}} \dots x_s^{a_{i_s}}\), for suitable integers \(s, a_{i_1}, \dots, a_{i_s}\).NEWLINENEWLINEThe author shows that to an ideal \(I \subset k[x^{a_1}, \dots, x^{a_s}]\) one may associate a certain fan in \({\mathbb R}^d\), called the Gröbner fan of \(I\) (in its construction one uses Gröbner bases techniques).NEWLINENEWLINEDuarte proves that if \(X\) is a toric variety as above (corresponding to the cone \(\sigma\)) then for any \(n\) the \(n\)-th Nash blowup \(B_n\) of \(X\), as well as its normalization \(\bar{B_n}\), are again toric varieties. Moreover, \(\bar{B_n}\) is the normal toric variety corresponding to the Gröbner fan of the ideal \(( x^{a_1}-1, \dots, x^{a_s}-1) k[x^{a_1}, \dots, x^{a_s}]\).NEWLINENEWLINEHe also shows that if \(X\) is as above (i.e., associated to the cone \(\sigma\))then the natural projection \(B_n=\mathrm{Nash} _n (X) \to X\) is an isomorphism if and only if \(X\) is smooth. The analogous result for \(X\) a curve and \(n=1\) (the usual Nash blowup) has been known since the mid 1970's. He proves his theorem by first noticing that it easily follows from the similar statement where \(B_n\) is substituted by its normalization \(\bar{B_n}\). Then he uses his characterization of \(\bar{B_n}\) in terms of a Gröbner basis, and verifies that if \(X\) is singular the fan of this basis must be a strict subdivision of the cone \(\sigma\). This is done by using results on reduced Gröbner bases in this context of monomial subalgebras of polynomial rings.NEWLINENEWLINEThe paper is very well written.