A Simplified Proof of Desingularization and Applications

A Proof of Desingularization over fields of characteristic zero

We present a proof of embedded desingularization for closed subschemes which does not make use of Hilbert-Samuel function and avoids Hironaka's notion of normal flatness. This proof, already sketched in [A course on constructive desingularization and equivariance. In {\em Resolution of singularities (Obergurgl, 1997)}, vol. 181 {\em Progr. Math.}, Birkhäuser, 2000.] page 224, is done by showing that desingularization of a closed subscheme $X$, in a smooth sheme W, is achieved by taking an algorithmic principalization for the ideal $I(X)$, associated to the embedded scheme $X$.

An introduction to constructive desingularization

These are introductional notes to resolution of singularities and Log principalization of ideals over fields of characteristic zero. We refer to `A simplified proof of desingularization and applications', A. Bravo, S. Encinas and O. Villamayor, (to appear in {\em Revista Matemática Iberoamericana}), for more details. The notes follow the algorithmic proof in `A new proof of desingularization over fields of characteristic zero', S. Encinas and O. Villamayor U., {\em Revista Matemática Iberoamericana}, 19 (2) (2003), which provides an important conceptual simplification over the original proofs …

Absolute desingularization in characteristic zero

We present a survey on the recent progress with desingularization of quasi-excellent schemes of characteristic zero. Due to expository goals of the paper we concentrate on giving examples and outlining main ideas, and we refer to other sources for detailed proofs and technical details. This survey will be published in the proceedings of the conference "Motivic integration and its interactions with model theory and non-archimedean geometry" (ICMS, May 2008).

Tame distillation and desingularization by $p$-alterations

We strengthen Gabber's $l'$-alteration theorem by avoiding all primes invertible on a scheme. In particular, we prove that any scheme $X$ of finite type over a quasi-excellent threefold can be desingularized by a $\mathrm{char}(X)$-alteration, i.e. an alteration whose order is only divisible by primes non-invertible on $X$. The main new ingredient in the proof is a tame distillation theorem asserting that, after enlarging, any alteration of $X$ can be split into a composition of a tame Galois alteration and a $\mathrm{char}(X)$-alteration. The proof of the distillation theorem is based on the …

Motivic Steenrod 代数与无需消解的 Thom 障碍

Motivic Steenrod algebra and Thom obstructions without desingularization

本文给出无需消解的证明，确认特征零域上双稳定模ℓ motivic上同调运算等同于motivic Steenrod代数，并应用此证明复代数环上所有Thom消解障碍消失。

A strong desingularization theorem

Let $X$ be a closed subscheme embedded in a scheme $W$ smooth over a field ${\bf k}$ of characteristic zero, and let ${\mathcal I}(X)$ be the sheaf of ideals defining $X$. Assume that the set of regular points of $X$ is dense in $X$. We prove that there exists a proper, birational morphism, $π: W_r\longrightarrow W$, obtained as a composition of monoidal transformations, so that if $X_r\subset W_r$ denotes the strict transform of $X\subset W$ then: 1) The morphism $π:W_r\longrightarrow W$ is an embedded desingularization of $X$ (as in Hironaka's Theorem); 2) The {\em total transform} of ${\mat…