Florian Berchtold, Juergen Hausen
Demushkin's Theorem says that any two toric structures on an affine variety X are conjugate in the automorphism group of X. We provide the following extension: Let an (n-1)-dimensional torus T act effectively on an n-dimensional affine toric variety X. Then T is conjugate in the automorphism group of X to a subtorus of the big torus of X.
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Let X be a complete toric variety with homogeneous coordinate ring S. In this article, we compute upper and lower bounds for the codimension in the critical degree of ideals of S generated by dim(X)+1 homogeneous polynomials that don't vanish simultaneously on X.
We introduce a notion of embedding codimension of an arbitrary local ring, establish some general properties, and study in detail the case of arc spaces of schemes of finite type over a field. Viewing the embedding codimension as a measure of singularities, our main result can be interpreted as saying that the singularities of the arc space are maximal at the arcs that are fully embedded in the singular locus of the underlying scheme, and progressively improve as we move away from said locus. As an application, we complement a theorem of Drinfeld, Grinberg, and Kazhdan on formal neighborhoods …
We prove that there are only finitely many families of codimension two nonsingular subvarieties of quadrics $\Q{n}$ which are not of general type, for $n=5$ and $n\geq 7$. We prove a similar statement also for the case of higher codimension. The case $n=6$ has been recently settled by Fania-Ottaviani. Keywords: Codimension two, Grassmannians, Lifting, Low codimension, Not of General Type, Polynomial Bound, Quadrics
We show that a Gorenstein subcanonical codimension 3 subscheme Z in X = P^N, N > 3, can be realized as the locus along which two Lagrangian subbundles of a twisted orthogonal bundle meet degenerately, and conversely. We extend this result to singular Z and all quasiprojective ambient schemes X under the necessary hypothesis that $Z$ is strongly subcanonical in a sense defined below. A central point is that a pair of Lagrangian subbundles can be transformed locally into an alternating map. In the local case our structure theorem reduces to that of Buchsbaum-Eisenbud and says that Z is Pfaffian.…
We obtain finiteness theorems for algebraic cycles of small codimension on quadric fibrations X over curves over perfect fields k. For example, if k is finitely generated over Q and the fibration has odd relative dimension at least 11, then CH^{i}(X) is finitely generated for i<=4.
In this note, we will give a partial answer for arithmetic analogues of Grothendieck's standard conjectures due to H. Gillet and C. Soule. (Remark : I changed the title of this note.)
The Drinfeld-Grinberg-Kazhdan theorem and embedding codimension of the arc space
论文扩展了Drinfeld, Grinberg和Kazhdan定理至任意余代数的弧，并证明了非奇异位势的不可约子集的嵌入余维数在通用情况下恒定。同时将最大除数集的有限形式模型与底层变种的奇点不变量联系起来。
We classify codimension two analytic submanifolds X of projective space having the property that any line through a general point p having contact to order two with X at p automatically has contact to order three. We give applications to the study of the Debarre--de Jong conjecture, and of n-dimensional varieties whose Fano variety of lines has dimension 2n-4.
On Abelian extensions in mixed characteristic and ramification in codimension one
针对Paul Roberts关于正则局部环整闭包性质的定理在模情形下不成立的问题，通过修改余维一p上分歧的相关表述，将该定理推广至混合特征下非分歧正则局部环的模情形（伽罗瓦群p挠元被p零化）。
A Gorenstein A-algebra R of codimension 2 is a perfect finite A-algebra such that R=Ext^2(R,A) holds as R-modules, A being a Cohen-Macaulay local ring with dim(A)-dim_A(R)=2. I prove a structure theorem for these algebras improving on an old theorem of M. Grassi. Special attention is paid to the question how the ring structure of R is encoded in its Hilbert resolution. It is shown that R is automatically a ring once one imposes a weak depth condition on a determinantal ideal derived from a presentation matrix of R over A. The interplay of Gorenstein algebras and Koszul modules as introduced by…
We present a polynomial partitioning theorem for finite sets of points in the real locus of an irreducible complex algebraic variety of codimension at most two. This result generalizes the polynomial partitioning theorem on the Euclidean space of Guth and Katz, and its extension to hypersurfaces by Zahl and by Kaplan, Matoušek, Sharir and Safernová. We also present a bound for the number of incidences between points and hypersurfaces in the four-dimensional Euclidean space. It is an application of our partitioning theorem together with the refined bounds for the number of connected components …
We present a variant of the classical Darboux-Jouanolou Theorem. Our main result provides a characterization of foliations which are pull-backs of foliations on surfaces by rational maps. As an application, we provide a structure theorem for foliations on 3-folds admitting an infinite number of extremal rays.
The purpose of this paper is to prove the following theorem of uniform Artin-Rees properties: Let $A$ be an excellent (in fact J-2) ring and let $N\subset M$ be two finitely generated $A$-modules such that ${\rm dim}(M/N)\leq 1$. Then there exists an integer $s\geq 1$ such that, for all integers $n\geq s$ and for all ideals $I$ of $A$, $I^{n}M\cap N=I^{n-s}(I^{s}M\cap N)$.
Terracini loci and a codimension one Alexander-Hirschowitz theorem
本文在Alexander-Hirschowitz定理基础上，完整刻画了二维射影空间中Terracini轨迹存在余维数为1的不可约分支的充要条件，推进了该领域的分类研究。
We prove Bertini type theorems for the inverse image, under a proper morphism, of any Schubert variety in an homogeneous space. Using generalisations of Deligne's trick, we deduce connectedness results for the inverse image of the diagonal in $X^2$ where $X$ is any isotropic grassmannian. We also deduce simple connectedness properties for subvarieties of $X$. Finally we prove transplanting theorems {à} la Barth-Larsen for the Picard group of any isotropic grassmannian of lines and for the Neron-Severi group of some adjoint and coadjoint homogeneous spaces.
In this paper we show that any smoothable complex projective variety, smooth in codimension two, with klt singularities and numerically trivial canonical class admits a finite cover, étale in codimension one, that decomposes as a product of an abelian variety, and singular analogues of irreducible Calabi-Yau and irreducible symplectic varieties.
In this manuscript we sharpen the lower bound on the codimension of the irreducible components of the Noether-Lefschetz locus of surfaces in projective toric threefolds given in [BG17]. We also provide a simpler proof of Theorem 4.11 in [BG17], which allows one to avoid some technical assumptions.
We establish a structure theorem for degree three codimension one foliations on projective spaces of dimension $n\ge 3$, extending a result by Loray, Pereira, and Touzet for degree three foliations on $\mathbb P^3$. We show that the space of codimension one foliations of degree three on $\mathbb{P}^n$, $n\ge 3$, has exactly $18$ distinct irreducible components parameterizing foliations without rational first integrals, and at least $6$ distinct irreducible components parameterizing foliations with rational first integrals.
We prove that the cone of mobile divisors and the cone of curves birationally movable in codimension 1 are dual in the $(K+B)$-negative part for a klt pair $(X/Z,B)$. We also prove the structure theorem and the contraction theorem for the expanded cone of curves birationally movable in codimension 1. The duality of the cones gives a partial answer to the problem posed by Sam Payne.
Based on the celebrated result on zeros of holomorphic 1-forms on complex varieties of general type by Popa and Schnell, we study holomorphic 1-forms on $n$-dimensional varieties of Kodaira dimension $n-1$. We show that a complex minimal smooth projective variety $X$ of Kodaira dimension $κ(X)=\dim X-1$ admits a holomorphic 1-form without zero if and only if there is a smooth morphism from $X$ to an elliptic curve. Furthermore, for a general smooth projective variety (not necessarily minimal) $X$ of Kodaira codimension one, we give a structure theorem for $X$ given that $X$ admits a holomorphi…
An affine variety $X$ of dimension $\ge 2$ is called {\em flexible} if its special automorphism group SAut$(X)$ acts transitively on the smooth locus $X_{reg}$ \cite{AKZ}. Recall that the special automorphism group SAut$(X)$ is the subgroup of the automorphism group Aut$(X)$ generated by all one-parameter unipotent subgroups \cite{AKZ}. Given a normal, flexible, affine variety $X$ and a closed subvariety $Y$ in $X$ of codimension at least 2, we show that the pointwise stabilizer subgroup of $Y$ in the group SAut$(X)$ acts infinitely transitively on the complement $X\backslash Y$, that is, $m$-…
Some remarks on smooth projective varieties of small degree and codimension
本文针对小次数小余维光滑射影簇研究，给出余维2的高维Fano或Calabi-Yau簇为完全交的Ballico-Chiantini定理的快速证明，并将光滑射影簇为完全交的次数界从约0.63·n^(1/2)提升至约0.79·n^(2/3)。
Suppose that g > 2, that n > 0 and that m > 0. In this paper we show that if E is an irreducible smooth variety which dominates a divisor D in M_{g,n}[m], the moduli space of n-pointed, smooth curves of genus g with a level m structure, then the closure of the image of the monodromy representation pi_1(E,e)--> Sp_g(Zhat) has finite index in Sp_g(Zhat). A similar result is proved for codimension 1 families of the universal principally polarized abelian variety of dimension g > 2. Both results are deduced from a general "non-abelian strictness theorem". The first result is used in arXiv:1001.500…
In its most basic form, Dubreil's Theorem states that for an ideal $I$ defining a codimension $2$, arithmetically Cohen--Macaulay subscheme of projective $n$-space, the number of generators of $I$ is bounded above by the minimal degree of a minimal generator plus $1$. By introducing a new ideal $J$ which is the complete intersection of $n-1$ general linear forms, we are able to extend Dubreil's Theorem to an ideal $I$ defining a locally Cohen--Macaulay subscheme $V$ of any codimension. Our new bound involves the lengths of the Koszul homologies of the cohomology modules of $V$, with respect to…
A General Noether-Lefschetz Theorem and applications
本文将经典 Noether-Lefschetz 定理推广到任意光滑射影三维流形，证明一般奇异成员不在 Noether-Lefschetz 轨迹中，并应用于证明余维二 Chow 群的自然映射是单射，从而给出 Nori 猜想的一个较弱版本。
We prove a singular Darboux type theorem for homogeneous polynomial closed $2$-forms of degree one on $\mathbb{C}^n$. As application, we classify non-integrable codimension one distributions, of degree one, and arbitrary classes on projective spaces.
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