On Rees algebras and invariants for singularities over perfect fields (Q2841855)

A \textit{Rees algebra} over a ring \(B\) is a finitely generated graded \(B\)-subalgebra of the polynomial ring \(B[W]\). This concept globalizes to give the notion of Rees algebra (or \textit{Rees sheaf of algebras}) over a base scheme \(V\), say \(\mathcal G \subset \mathcal O _V[W]\). \textit{O. Villamayor} [Ann. Sci. Éc. Norm. Supér. (4) 22, No. 1, 1--32 (1989; Zbl 0675.14003); Ann. Sci. Éc. Norm. Supér. (4) 25, No. 6, 629--677 (1992; Zbl 0782.14009)] introduced the concept several years ago, with the intention to apply it to the theory of resolution of singularities of algebraic varieties. For this purpose, the most interesting situation is that where \(B\) is a regular, finitely generated algebra over a perfect field \(k\), or \(V\) is a smooth algebraic variety over such \(k\). Henceforth, we suppose these assumptions valid.NEWLINENEWLINEIn Part I, the authors review the basic theory and present some new results. Among other things, given a Rees algebra \(\mathcal G \subset {\mathcal O}_V[W]\), with \(V\) a smooth variety over a perfect field \(k\), they recall the notions of order of \(\mathcal G\) at \(x \in V\), zero set and singular locus of \(\mathcal G\), and integral closure \(\overline {\mathcal G}\) of \(\mathcal G\) in \({\mathcal O}_V[W]\). They define \textit{differential Rees algebra}, i.e., one closed under the action of differential operators, and recall the construction of the minimal differential Rees containing a given \(\mathcal G\), denoted by \({\mathbb D}(\mathcal G)\). They also define the concept of \textit{weakly equivalent} (w.e.) Rees algebras: essentially, Rees algebras (over \(V\)) \(\mathcal G\) and \(\mathcal K\) are w.e. if Sing(\(\mathcal G\)) = Sing(\(\mathcal K\)) and this equality is preserved when we take, successively, suitable ``transforms'', in a sense explained in the text. They relate Rees algebras to \textit{pairs}, that is ordered couples \((I,b)\) where \(I\) is a coherent sheaf of ideals of \(\mathcal O _V\) and \(b \geq 0\) an integer. One may associate to a pair \((I,b)\) a Rees algebra \({\mathcal O}_V[IW^b]\), and any Rees algebra is closely related to one of this type.NEWLINENEWLINETheir main new result (Theorem 3.11, called the \textit{canonicity principle}) says that Rees algebras (both over \(V\)) \(\mathcal G\) and \(\mathcal K\) are w.e. if and only if \(\overline{{\mathbb D}(\mathcal G)}=\overline{{\mathbb D}(\mathcal K)}\). Theorem 3.11 is a an immediate consequence of a more general result (Theorem 3.10), expressed in terms of certain inclusions, whose proof is rather involved.NEWLINENEWLINEIn Part II they discuss some applications of Theorem 3.11. It has been known for a long time (thanks to Hironaka's efforts) that resolution of singularities of an algebraic variety \(X\) (say, embedded in a smooth \(V\)) follows if we can resolve, in a suitable sense, certain pair \((I,b)\) associated to \(X\). But this is a local process: \(I\) is not defined on the whole \(V\), but on an étale neighborhood of a point \(x \in V\). Moreover, the pair \((I,b)\) is not unique. To verify that this process globalizes has been traditionally a hard gluing problem, requiring complicated methods. The authors reinterpret the theory in terms of Rees algebras, and use their main result to give a simple solution to the mentioned gluing problem.NEWLINENEWLINEThey also review a method to resolve Rees algebras (or, equivalently, pairs), valid in characteristic zero, which yields partial results in positive characteristic. Again one has to face a ``gluing problem'', to verify that certain procedures are well-defined. The authors once more use the canonicity principle to deal with this question.NEWLINENEWLINEThey conclude the paper by discussing an example showing that an inductive process to lower the multiplicity of a hypersurface, valid over fields of characteristic zero, may fail in positive characteristic.