Extensors and the Hilbert scheme (Q2809280)

One triumph of the functorial approach in algebraic geometry is \textit{Grothendieck's} construction of the Hilbert scheme \(\mathbf{Hilb}_{p(t)}^n\), which parametrizes closed subschemes \(X \subset \mathbb P^n_k\) having Hilbert polynomial \(p(t)\) [\textit{A. Grothendieck}, in: Sem. Bourbaki 13(1960/61), No.221, 28 p. (1961; Zbl 0236.14003)]. \textit{G. Gotzmann} found the minimal value of \(r\) for which \({\mathcal O}_X\) is \(r\)-regular in terms of \(p(t)\) [Math. Z. 158, 61--70 (1978; Zbl 0352.13009)], which allows one to embed the Hilbert scheme into the Grassmann variety \({\mathbb G}^{N(r)}_{p(r)}\) (here \(N(r)=\binom{n+r}{n}\)) by associating each closed subscheme \(X\) to the vector space quotient \(H^0(\mathbb P^n, {\mathcal O} (r)) \to H^0(X, {\mathcal O}_X (r))\). \textit{A. Iarrobino} and \textit{V. Kanev} produced explicit equations of degree \(N(r+1)-p(r+1)-1\) [Power sums, Gorenstein algebras, and determinantal loci. With an appendix `The Gotzmann theorems and the Hilbert scheme' by Anthony Iarrobino and Steven L. Kleiman. Berlin: Springer (1999; Zbl 0942.14026)]. More recently \textit{M. Haiman and B. Sturmfels} [J. Algebr. Geom. 13, 725--769 (2004; Zbl 0172.14007)] produced equations of degree \(n+1\) conjectured by Bayer in his thesis.NEWLINENEWLINEIn the paper under review, the authors exploit the action of \(\text{PGL}(n+1)\) to find significantly lower degree equations when \(\text{char} \; k=0\). They use the group action and monomials corresponding to a certain Borel-fixed ideal to define the \textit{Borel open cover} for the Grassmann functor and a corresponding Borel cover for the Hilbert functor. Using the theory of \(J\)-marked sets developed by \textit{P. Lella} and \textit{M. Roggero} [``On the functoriality of marked families'', Preprint, \url{arXiv:1307.7657}], they reprove the existence of the Hilbert scheme and derive equations for the universal family in \(\mathbb P^n \times {\mathbb G}^{N(r)}_{p(r)}\) of bi-degree \((r,1)\) by exploiting properties of extensors. Consequently they find explicit equations for the Hilbert scheme in the ring of Plücker coordinates of degree at most \(\dim X + 2\). They illustrate their method by describing their equations for Hilbert schemes of two points in \(\mathbb P^n\) with \(n=2,3,4\) (each is generated by quadrics in their respective Grassmann variety), including an interesting table comparing to the equations of Iarrobino, Kleiman, Haiman and Sturmfels [loc. cit.].