de Jong alterations

de Jong alterations
Name	de Jong alterations
Field	Algebraic Geometry
Introduced	1996
Introduced by	A. J. de Jong
Related	Resolution of singularities, Modifications, Proper morphisms, Étale morphisms

de Jong alterations de Jong alterations are a method in Algebraic Geometry introduced by Aise Johan de Jong in 1996 that provides a replacement for resolution techniques by producing proper, generically finite morphisms from smooth varieties. The result influenced work by researchers at institutions such as the Institute for Advanced Study, Princeton University, Harvard University, University of Cambridge, and University of California, Berkeley and connects to conjectures and theorems studied by mathematicians like Jean-Pierre Serre, Alexander Grothendieck, Pierre Deligne, David Mumford, and Michael Artin.

Introduction

De Jong alterations assert that for a variety over a field, there exists a proper, surjective, generically finite morphism from a regular variety that is an alteration of the original. The theorem is situated alongside foundational results like the Hironaka resolution of singularities for characteristic zero, the work of Oscar Zariski on birational geometry, and later advances by Temkin, Gabber, and Illusie. Applications span areas treated by scholars at the Max Planck Institute for Mathematics, IHÉS, and Rutgers University and inform results in the study of étale cohomology, l-adic cohomology, and the Langlands program.

Statement and construction

The basic statement: given an integral, separated scheme of finite type X over a field k, there exists a proper, surjective morphism Y → X with Y regular and the morphism generically finite. De Jong's original construction uses successive alterations, resolution-like techniques, and the existence of dominant morphisms to projective space, invoking tools developed by Grothendieck in the Éléments de géométrie algébrique and building on work by Nagata on compactification and Raynaud on flattening. The construction makes essential use of finite extensions of function fields studied by Claude Chevalley and uses moduli arguments related to work by Mumford and Deligne–Mumford stacks.

Properties and applications

Alterations guarantee that many cohomological and structural properties of X can be studied after pulling back to Y, linking to comparisons used in proofs by Deligne of the Weil conjectures and in the theory of ℓ-adic sheaves developed by Pierre Deligne and Nicholas Katz. They yield semistable reduction statements akin to results by Grothendieck and Serre and are a cornerstone in approaches to purity theorems used by Faltings and Kisin. Alterations preserve properties under generically finite base change and facilitate the construction of alterations compatible with group actions studied by Serre and Langlands-related work by Bernstein and Jacquet in arithmetic geometry contexts.

Comparisons with other alteration theorems

Compared with Hironaka resolution of singularities (valid in characteristic zero), de Jong alterations work in positive characteristic and avoid producing a birational morphism; instead they yield generically finite covers. The technique complements the improvements by Temkin and refinements by Gabber on alterations with controlled ramification and relates to the semistable reduction theorem of Deligne–Mumford and the desingularization strategies of Villamayor, Bierstone, and Milman. In contrast to the canonical resolution approaches pursued by Wlodarczyk and Kollar, de Jong's method is noncanonical but highly flexible, matching needs in arithmetic applications such as those considered by Fontaine, Faltings, and Scholze.

Examples and key cases

Key examples include alterations of curves leading to classical stable reduction results from Deligne–Mumford; alterations of surfaces where constructions echo work by Zariski and Lipman; and higher-dimensional cases relevant to investigations by Abhyankar on ramification. Explicit constructions appear in papers by de Jong and in subsequent expositions by Raynaud–Gruson flattening techniques and in seminars at IHÉS and MSRI. Important special cases include alterations compatible with group actions studied in contexts involving Galois representations and applications to compactifications related to work by Satake and Baily–Borel.

Technical tools and proofs

The proof uses techniques from Éléments de géométrie algébrique by Grothendieck, compactification results of Nagata, flattening stratifications by Raynaud and Gruson, and étale-local arguments familiar from the work of Artin. It employs constructions of dominant morphisms to projective space, alteration by finite covers studied by Chevalley and Zariski, and moduli-theoretic reductions influenced by Mumford and Deligne–Mumford. Later refinements and alternative proofs draw on rigid-analytic methods used by Bosch–Lütkebohmert–Raynaud and on perfectoid techniques explored by Scholze.

Category:Algebraic geometry