Principal gradient schemes have regular reduced closed subschemes

Equations of tropical varieties

We introduce a scheme-theoretic enrichment of the principal objects of tropical geometry. Using a category of semiring schemes, we construct tropical hypersurfaces as schemes over idempotent semirings such as $\mathbb{T} = (\mathbb{R}\cup \{-\infty\}, \mathrm{max}, +)$ by realizing them as solution sets to explicit systems of tropical equations that are uniquely determined by idempotent module theory. We then define a tropicalization functor that sends closed subschemes of a toric variety over a ring R with non-archimedean valuation to closed subschemes of the corresponding tropical toric vari…

Parabolic reductions of principal bundles

In this paper we describe the structure of the space of parabolic reductions, and their compactifications, of principal $G$-bundles over a smooth projective curve over an algebraically closed field of arbitrary characteristic. We first prove estimates for the dimensions of moduli spaces of stable maps to the twisted flag varieties $E/P$ and Hilbert schemes of closed subschemes of $E/P$ with same Hilbert polynomials as that of a $P$-reductions of $E$. This generalizes the earlier results of Mihnea Popa and Mike Roth to connected reductive groups and the results of Y. I. Holla and M. S. Narasimh…

Schematic HN stratification for families of principal bundles and lambda modules

For a family of principal bundles with a reductive structure group on a family of curves in characteristic zero, it is known that the Harder Narasimhan type of its restriction to each fiber varies semicontinuously over the parameter scheme of the family. This defines a stratification of the parameter scheme by locally closed subsets, known as the Harder-Narasimhan stratification. In this note, we show how to endow each Harder-Narasimhan stratum with the structure of a locally closed subscheme of the parameter scheme, which enjoys the universal property that under any base change the pullback f…

Determinantal schemes and Buchsbaum-Rim sheaves

Let $ϕ$ be a generically surjective morphism between direct sums of line bundles on $\proj{n}$ and assume that the degeneracy locus, $X$, of $ϕ$ has the expected codimension. We call $B_ϕ = \ker ϕ$ a (first) Buchsbaum-Rim sheaf and we call $X$ a standard determinantal scheme. Viewing $ϕ$ as a matrix (after choosing bases), we say that $X$ is good if one can delete a generalized row from $ϕ$ and have the maximal minors of the resulting submatrix define a scheme of the expected codimension. In this paper we give several characterizations of good determinantal schemes. In particular, it is shown …

On the Grothendieck-Serre conjecture on principal bundles in mixed characteristic

Let R be a regular local ring. Let G be a reductive R-group scheme. A conjecture of Grothendieck and Serre predicts that a principal G-bundle over R is trivial if it is trivial over the quotient field of R. The conjecture is known when R contains a field. We prove the conjecture for a large class of regular local rings not containing fields in the case when G is split.

Proof of Grothendieck--Serre conjecture on principal bundles over regular local rings containing a finite field

Let R be a regular local ring, containing a finite field. Let G be a reductive group scheme over R. We prove that a principal G-bundle over R is trivial, if it is trivial over the fraction field of R. If the regular local ring R contains an infinite field this result is proved in [FP]. Thus the conjecture is true for regular local rings containing a field.

多中心变形空间的复形

A polyptych of multi-centered deformation spaces

研究将Verdier变形空间等进行扩展，引入与浸入链相关的变形空间概念。基于多中心扩张形式得到镶板同构，能产生不同长度变形空间间的典范同构，还可用于给出变形空间分层的几何描述。

A proof of the Grothendieck-Serre conjecture on principal bundles over regular local rings containing infinite fields

Let R be a regular local ring, containing an infinite field. Let G be a reductive group scheme over R. We prove that a principal G-bundle over R is trivial, if it is trivial over the fraction field of R.

非主理想 BCM 阈值

BCM-thresholds of non-principal ideals

本文引入并研究非主理想 BCM 阈值，展示其与弱 F-正则环经典 F-阈值一致，并获得参数理想 F-阈值的相关结果及混合特征版本的乘数结果。

On the discreteness and rationality of F-jumping coefficients

This paper studies the jumping coefficients of principal ideals of regular local rings. Recently M. Blickle, M. Mustata and K. Smith showed that, when $R$ is of essentially finite type over a field and $F$-finite, bounded intervals contain finitely many jumping coefficients and that those are rational. In a later paper they extended these results to principal ideals of $F$-finite complete regular local rings. The aim of this paper is to extend these results on the discreteness and rationality of jumping coefficients to principal ideals of arbitrary (i.e. not necessarily $F$-finite) excellent r…

On Grothendieck-Serre conjecture concerning principal G-bundles over regular semi-local domains containing a finite field: I

In three preprints [Pan2],[Pan3] and the present one we prove Grothendieck-Serre's conjecture concerning principal G-bundles over regular semi-local domains R containing a finite field (here G is a reductive group scheme). The present preprint contains main geometric presentation theorems which are necessary for that. The preprint [Pan2] contains reduction of the Grothendieck-Serre's conjecture to the case of a simple simply-connected group scheme. The preprint [Pan3] contains a proof of Grothendieck-Serre's conjecture for regular semi-local domains R containing a finite field. One of the main…

On Grothendieck-Serre conjecture concerning principal G-bundles over regular semi-local domains containing a finite field: II

In three preprints [Pan1], [Pan3] and the present one we prove Grothendieck-Serre's conjecture concerning principal G-bundles over regular semi-local domains R containing a finite field (here $G$ is a reductive group scheme). The preprint [Pan1] contains main geometric presentation theorems which are necessary for that. The present preprint contains reduction of the Grothendieck--Serre's conjecture to the case of semi-simple simply-connected group schemes (see Theorem 1.0.1). The preprint [Pan3] contains a proof of that conjecture for regular semi-local domains R containing a finite field. The…

Proof of Grothendieck--Serre conjecture on principal G-bundles over regular local rings containing a finite field

Let R be a regular local ring, containing a finite field. Let G be a reductive group scheme over R. We prove that a principal G-bundle over R is trivial, if it is trivial over the fraction field of R. In other words, if K is the fraction field of R, then the map of pointed sets H^1_{et}(R,G) \to H^1_{et}(K,G), induced by the inclusion of R into K, has a trivial kernel. Certain arguments used in the present preprint do not work if the ring R contains a characteristic zero field. In that case and, more generally, in the case when the regular local ring R contains an infinite field this result is…

Two purity theorems and the Grothendieck--Serre's conjecture concerning principal G-bundles

In a series of papers [Pan0], [Pan1], [Pan2], [Pan3] we give a detailed and better structured proof of the Grothendieck--Serre's conjecture for semi-local regular rings containing a finite field. The outline of the proof is the same as in [P1],[P2],[P3]. If the semi-local regular ring contains an infinite field, then the conjecture is proved in [FP]. Thus the conjecture is true for regular local rings containing a field. A proof of Grothendieck--Serre conjecture on principal bundles over a semi-local regular ring containing an arbitrary field is given in [Pan3]. That proof is heavily based on …

虚拟Cartier除子与胀开

Virtual Cartier divisors and blow-ups

本文基于“虚拟有效Cartier除子”概念证明正则浸入子概形胀开的泛性质，并在导出代数几何中构造拟光滑闭浸入的胀开，推进了胀开相关理论研究。

有限概形的 Castelnuovo-Mumford 正则性

Castelnuovo-Mumford regularity of finite schemes

本文研究了有限子概形的 Castelnuovo-Mumford 正则性，证明了其正则性接近已知上界当且仅当存在唯一的有理正规曲线使得该曲线与概形交集的正则性等于原概形的正则性。

Hypersurfaces in projective schemes and a moving lemma

Let X/S be a quasi-projective morphism over an affine base. We develop in this article a technique for proving the existence of closed subschemes H/S of X/S with various favorable properties. We offer several applications of this technique, including the existence of finite quasi-sections in certain projective morphisms, and the existence of hypersurfaces in X/S containing a given closed subscheme C, and intersecting properly a closed set F. Assume now that the base S is the spectrum of a ring R such that for any finite morphism Z -> S, Pic(Z) is a torsion group. This condition is satisfied if…

Matrix factorizations and semi-orthogonal decompositions for blowing-ups

We study categories of matrix factorizations. These categories are defined for any regular function on a suitable regular scheme. Our paper has two parts. In the first part we develop the foundations; for example we discuss derived direct and inverse image functors and dg enhancements. In the second part we prove that the category of matrix factorizations on the blowing-up of a suitable regular scheme X along a regular closed subscheme Y has a semi-orthogonal decomposition into admissible subcategories in terms of matrix factorizations on Y and X. This is the analog of a well-known theorem for…

Nice triples and Grothendieck--Serre's conjecture concerning principal G-bundles over reductive group schemes

In a series of papers [Pan0], [Pan1], [Pan2], [Pan3] we give a detailed and better structured proof of the Grothendieck--Serre's conjecture for semi-local regular rings containing a finite field. The outline of the proof is the same as in [P1],[P2],[P3]. If the semi-local regular ring contains an infinite field, then the conjecture is proved in [FP]. Thus the conjecture is true for regular local rings containing a field. The present paper is the one [Pan1] in that new series. Theorem 1.1 is one of the main result of the paper. It is also one of the key steps in the proof of the Grothendieck--S…

Schémas de Hilbert invariants et théorie classique des invariants

Let W be an affine variety equipped with an action of a reductive group G. The invariant Hilbert scheme is a moduli space which classifies the G-stable closed subschemes of W such that the affine algebra is the direct sum of simple G-modules with previously fixed finite multiplicities. In this thesis, we first study the invariant Hilbert scheme, denoted H. It parametrizes the GL(V)-stable closed subschemes Z of W=n1 V \oplus n2 V^* such that k[Z] is isomorphic to the regular representation of GL(V) as GL(V)-module. If dim(V)<3, we show that H is a smooth variety, so that the Hilbert-Chow morph…

On the Grothendieck-Serre Conjecture about principal bundles and its generalizations

Let $U$ be a regular connected affine semi-local scheme over a field $k$. Let $G$ be a reductive group scheme over $U$. Assuming that $G$ has an appropriate parabolic subgroup scheme, we prove the following statement. Given an affine $k$-scheme $W$, a principal $G$-bundle over $W\times_kU$ is trivial if it is trivial over the generic fiber of the projection $W\times_kU\to U$. We also simplify the proof of the Grothendieck-Serre conjecture: let $U$ be a regular connected affine semi-local scheme over a field $k$. Let $G$ be a reductive group scheme over $U$. A principal $G$-bundle over $U$ is t…

Cotangent bundle and micro-supports in mixed characteristic case

For a regular scheme and a prime number $p$, we define the FW-cotangent bundle as a vector bundle on the closed subscheme defined by $p=0$, under a certain finiteness condition. For a constructible complex on the etale site of the scheme, we introduce the condition to be micro-supported on a closed conical subset in the FW-cotangent bundle. At the end of the article, we compute the singular supports in some cases.

Bernstein-Sato polynomials for general ideals vs. principal ideals

We show that given an ideal I generated by regular functions f_1,...,f_r on the smooth complex variety X, the Bernstein-Sato polynomial of I is equal to the reduced Bernstein-Sato polynomial of the function g=\sum_{i=1}^rf_iy_i on the product of X with an r-dimensional affine space. By combining this with results from [BMS], we relate invariants and properties of I to those of g. We also use the result on Bernstein-Sato polynomials to show that the Strong Monodromy Conjecture for Igusa zeta functions of principal ideals implies a similar statement for arbitrary ideals.

Crystalline boundedness principle

We prove that an $F$-crystal $(M,\vph)$ over an algebraically closed field $k$ of characteristic $p>0$ is determined by $(M,\vph)$ mod $p^n$, where $n\ge 1$ depends only on the rank of $M$ and on the greatest Hodge slope of $(M,\vph)$. We also extend this result to triples $(M,\vph,G)$, where $G$ is a flat, closed subgroup scheme of ${\bf GL}_M$ whose generic fibre is connected and has a Lie algebra normalized by $\vph$. We get two purity results. If ${\got C}$ is an $F$-crystal over a reduced ${\bf F}_p$-scheme $S$, then each stratum of the Newton polygon stratification of $S$ defined by ${\g…