Masaki Kashiwara, Kari Vilonen
In this note we sketch a proof of a fundamental conjecture, the codimension-three conjecture, for microdifferential holonomic systems with regular singularities. It states that any regular holonomic E-module extends beyond a codimension-three analytic subset.
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The codimension-three conjecture states that any regular holonomic module extends uniquely beyond an analytic subset with codimension equal to or larger than three. We give a proof of this conjecture.
We prove an analogue for holonomic DQ-modules of the codimension-three conjecture for microdifferential modules recently proved by Kashiwara and Vilonen. Our result states that any holonomic DQ-module having a lattice extends uniquely beyond an analytic subset of codimension equal to or larger than three in a Lagrangian subvariety containing the support of the DQ-module.
We establish the multiplicity conjecture of Herzog, Huneke, and Srinivasan about the multiplicity of graded Cohen-Macaulay algebras over a field, for codimension two algebras and for Gorenstein algebras of codimension three. In fact, we prove stronger bounds than the conjectured ones allowing us to characterize the extremal cases. This may be seen as a converse to the multiplicity formula of Huneke and Miller that inspired the conjectural bounds.
In this paper, we prove that the statement: ``The (Generalized) Hodge Conjecture holds for codimension-two cycles on a smooth projective variety $X$" is a birationally invariant statement, that is, if the statement is true for $X$, it is also true for all smooth varieties $X'$ which are birationally equivalent to $X$. We also prove the analogous result for 1-cycles. As direct corollaries, the Hodge Conjecture holds for smooth rational projective manifolds with dimension less than or equal to five, and, the Generalized Hodge Conjecture holds for smooth rational projective manifolds with dimensi…
We prove the Tate conjecture for divisor classes and the Mumford-Tate conjecture for the cohomology in degree 2 for varieties with $h^{2,0}=1$ over a finitely generated field of characteristic 0, under a mild assumption on their moduli. As an application of this general result, we prove the Tate and Mumford-Tate conjectures for some classes of algebraic surfaces with $p_g=1$.
It is conjectured that the dual variety of every smooth nonlinear subvariety of dimension $> \frac{2N}{3}$ in projective $N$-space is a hypersurface, an expectation known as the duality defect conjecture. This would follow from the truth of Hartshorne's complete intersection conjecture but nevertheless remains open for the case of subvarieties of codimension $> 2$. A combinatorial approach to proving the conjecture in the codimension $2$ case was developed by Holme, and following this approach Oaland devised an algorithm for proving the conjecture in the codimension $3$ case for particular $N$…
In his work on birational classification of foliations on projective surfaces, Brunella showed that every regular foliation on a rational surface is algebraically integrable with rational leaves. This led Touzet to conjecture that every regular foliation on a rationally connected manifold is algebraically integrable with rationally connected leaves. Druel proved this conjecture for the case of weak Fano manifolds. In this paper, we extend this result showing that Touzet's conjecture is true for codimension one foliations on threefolds with nef anti-canonical bundle.
In this note, we show how the classical Hodge index theorem implies the Hodge index conjecture of Beilinson for height pairing of homologically trivial codimension two cycles over function field of characteristic 0. Such an index conjecture has been used in our paper on Gross-Schoen cycles to deduce the Bogomolov conjecture and a lower bound for Hodge class (or Faltings height) from some conjectures about metrized graphs which have just been recently proved by Zubeyir Cinkir.
Effective nonvanishing for weighted complete intersections of codimension two
本研究证明Kawamata有效非零猜想对余维数为2的拟光滑加权完全交成立，验证了满足条件的丰沛Cartier除子对应的线性系非空，推进了该猜想在特定代数簇类中的验证。
Generic vanishing conjecture for divisors
受Pareschi和Popa的通用消失猜想启发，本文证明了余维数为1情形的对应类比。
For $d \ge 4$, the Noether-Lefschetz locus $\mathrm{NL}_d$ parametrizes smooth, degree $d$ surfaces in $\mathbb{P}^3$ with Picard number at least $2$. A conjecture of Harris states that there are only finitely many irreducible components of the Noether-Lefschetz locus of non-maximal codimension. Voisin showed that the conjecture is false for sufficiently large $d$, but is true for $d \le 5$. She also showed that for $d=6, 7$, there are finitely many \emph{reduced}, irreducible components of $\mathrm{NL}_d$ of non-maximal codimension. In this article, we prove that for any $d \ge 6$, there are …
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 prove the conjecture that two projective symplectic resolutions for a symplectic variety $W$ are related by Mukai's elementary transformations over $W$ in codimension 2 in the following cases: (i). nilpotent orbit closures in a classical simple complex Lie algebra; (ii). some quotient symplectic varieties.
Suppose that 2d-2 tangent lines to the rational normal curve z\mapsto (1 : z : ... : z^d) in d-dimensional complex projective space are given. It was known that the number of codimension 2 subspaces intersecting all these lines is always finite; for a generic configuration it is equal to the d^{th} Catalan number. We prove that for real tangent lines, all these codimension 2 subspaces are also real, thus confirming a special case of a general conjecture of B. and M. Shapiro. This is equivalent to the following result: If all critical points of a rational function lie on a circle in the Riemann…
In this paper, the generalized Bloch Conjecture on zero cycles for the quotient of certain complete intersections with trivial canonical bundle is proved to hold. As an application of Bloch-Srinivas method on the decomposition of the diagonal, we compute the rational coefficient Lawson homology for 1-cycles and codimension two cycles for these quotient varieties. The (Generalized) Hodge Conjecture is proved to hold for codimension two cycles (and hence also for 2-cycles) on these quotient varieties.
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.)
We prove a conjecture of Gavril Farkas claiming that for all integers r \geq 2 and g \geq \binom{r+2}{2} there exists a component of the locus \mathcal{S}^r_g of spin curves with a theta characteristic L such that h^0(L) \geq r+1 and h^0(L)\equiv r+1 (mod 2) which has codimension \binom{r+1}{2} inside the moduli space \mathcal{S}_g of spin curves of genus g.
A new tautological equation of $\Mbar_{3,1}$ in codimension 3 is derived and proved, using the invariance condition explained in earlier works.
For a fixed $d \ge 5$, the Noether-Lefschetz locus parametrizes smooth degree $d$ surfaces in $\mathbb{P}^3$ with Picard number greater than $1$. This is a countable union of proper algebraic varieties. It is known (due to works of Voisin, Green and others) that the largest irreducible component is of codimension (in the space of all smooth surface in $\mathbb{P}^3$ of degree $d$) equal to $d-3$. The main object of study in this article is: For fixed $r$ greater than $2$ and less than $d$, the locus parametrizing degree $d$ surfaces in $\mathbb{P}^3$ with Picard number at least equal to $r$. I…
On minimal Gorenstein Hilbert function
本文研究 full Perazzo 代数的极小 Hilbert 函数猜想，在低余维长度四、五及特定子类中加以证明，并新证相关最小项渐近结果的一部分。
We establish a full classification of degree $2$ codimension one distributions on $\mathbb{P}^3$ according to invariants of their tangent sheaves.
Representability of codimension three cycles
本文通过表面零循环模理性等价来定义四维空间中三维循环的可表示性，探讨其核心方法与贡献。
Gorenstein liaison seems to be the natural notion to generalize to higher codimension the well-known results about liaison of varieties of codimension~2 in projective space. In this paper we study points in ${\mathbb P}^3$ and curves in ${\mathbb P}^4$ in an attempt to see how far typical codimension~2 results will extend. While the results are satisfactory for small degree, we find in each case examples where we cannot decide the outcome. These examples are candidates for counterexamples to the hoped-for extensions of codimension~2 theorems.
In this paper, we study the higher codimensional cycle structure of the Hilbert scheme of three points in the projective plane. In particular, we compute all Chern (and Segre) classes of all tautological bundles on it and compute the nef (effective) cones of cycles in codimensions 2 and 3 (dimensions 2 and 3).
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.
Codimension one distributions of degree 3 on the three-dimensional projective space
本文对三维射影空间上余维1的3次分布进行分类，给出切丛Chern类与奇点方案分量，并利用切丛稳定性证明模空间的存在性。
Castelnuovo-Mumford regularity of unprojections and the Eisenbud-Goto regularity conjecture
针对Eisenbud-Goto正则性猜想，本文提出一种基于非投影过程的反例构造方法，可构造任意固定维数≥3、余维数≥2的反例，同时揭示了奇点对Castelnuovo-Mumford正则性的影响。
Arithmetic in the Boij Söderberg Cone
本文通过将猜想转化为Boij-Söderberg锥体中的算术问题，研究了格雷码模的Betti数下界，并在五重和六重情况下证明了新的结果，揭示了Betti表、交换代数和丢番图方程之间的联系。
Divisorial contractions to codimension three orbits
本文研究连通代数群作用下中心为余维三单连通轨道的等变极值收缩，证明此类收缩均为加权爆破，推广了 Kawakita 的经典结果。
We study codimension one holomorphic distributions on the projective three-space, analyzing the properties of their singular schemes and tangent sheaves. In particular, we provide a classification of codimension one distributions of degree at most 2 with locally free tangent sheaves, and show that codimension one distributions of arbitrary degree with only isolated singularities have stable tangent sheaves. Furthermore, we describe the moduli space of distributions in terms of Grothendieck's Quot-scheme for the tangent bundle. In certain cases, we show that the moduli space of codimension one …
Codimension two Artinian algebras $A$ have the strong and weak Lefschetz properties provided the characteristic is zero or greater than the socle degree. It is open to what extent such results might extend to codimension three AG algebras - the most promising results so far have concerned the weak Lefschetz property for such algebras. We here show that every standard-graded codimension three Artinian Gorenstein algebra $A$ having low maximum value of the Hilbert function - at most six - has the strong Lefschetz property, provided that the characteristic is zero. When the characteristic is grea…
In this mostly expository note, we explain a proof of Tate's two conjectures [Tat65] for algebraic cycles of arbitrary codimension on certain products of elliptic curves and abelian surfaces over number fields.
The resolution property holds away from codimension three
本文在温和假设下验证了Gross猜想，证明约化、分离且优良的概形在余维数至少三的闭子集之外具有可解性质，方法涉及形式局部下降、平坦仿射邻域及构造局部自由层。
On the injective self-maps of algebraic varieties
本文证明了宫西猜想在非奇异簇、曲面及局部完全交集且余维数2处正则的簇上成立，并讨论了诱导非奇异轨迹自同构的实例及孤立奇点情形。
Generic vanishing subschemes of codimension three
本文分类了维数至少为6的不可分解主极化阿贝尔簇中余维数为三的几何非退化GV子概形，证明其为雅可比簇中平移的$\pm W_{g-3}(C)$，并利用theta对偶性计算了相关上同调及刻画超椭圆情形。
Let $X$ be a smooth projective variety over a finite field $\F$. We discuss the unramified cohomology group $H^3_\nr(X,\Q/\Z(2))$. Several conjectures put together imply that this group is finite. For certain classes of threefolds, $H^3_\nr(X,\Q/\Z(2))$ actually vanishes. It is an open question whether this holds true for arbitrary threefolds. For a threefold $X$ equipped with a fibration onto a curve $C$, the generic fibre of which is a smooth projective surface $V$ over the global field $\F(C)$, the vanishing of $H^3_\nr(X,\Q/\Z(2))$ together with the Tate conjecture for divisors on $X$ impl…
On the Finkelberg-Ginzburg Mirabolic Monodromy Conjecture
本文计算旗点 Harish-Chandra D-模的单值性，证明 Finkelberg–Ginzburg 猜想在秩 1 的特殊参数处即失效，并利用 Opdam 移位算子和缠结算子处理高秩共振。
Let $X$ be a non-singular projective threefold over an algebraically closed field of any characteristic, and let $A^2(X)$ be the group of algebraically trivial codimension 2 algebraic cycles on $X$ modulo rational equivalence with coefficients in $\mathbb Q$. Assume $X$ is birationally equivalent to another threefold $X'$ admitting a fibration over an integral curve $C$ whose generic fiber $X'_{\bar η}$, where $\bar η=Spec(\bar {k(C)})$, satisfies the following three conditions: (i) the motive $M(X'_{\bar η})$ is finite-dimensional, (ii) $H^1_{et}(X_{\bar η},\mathbb Q_l)=0$ and (iii) $H^2_{et}…
In the present paper, we provide general bounds for the torsion in the codimension 2 Chow groups of the products of projective homogeneous surfaces. In particular, we determine the torsion for the product of four Pfister quadric surfaces and the maximal torsion for the product of three Severi-Brauer surfaces. We also find an upper bound for the torsion of the product of three quadric surfaces with the same discriminant.
Proof of Miyanishi's conjecture on endomorphisms of varieties
本文证明了关于变种内端射的Miyanishi猜想，通过证明birational端射在特定条件下为自同构，并推广了Ax的定理。
In this note we are going to prove that if we have a fibration of smooth projective varieties $X\to S$ over a surface $S$ such that $X$ is of dimension four and that the geometric generic fiber has finite dimensional motive and the first étale cohomology of the geometric generic fiber with respect to $\mathbb {Q}_l$ coefficients is zero and the second étale cohomology is spanned by divisors, then $A^3(X)$ (codimension three algebraically trivial cycles modulo rational equivalence) is dominated by finitely many copies of $A_0(S)$. Meaning that there exists finitely many correspondences $Γ_i$ on…
Let $M$ be a submanifold of ${\Bbb P}^N$ of dimension $n>2$. Suppose that $(M,{\Cal O}_M(1))\cong{\Bbb P}({\Cal E}),{\Cal O}(1))$ for some vector bundle ${\Cal E}$ on a surface $S$. Then $N\ge 2n-1$ by Barth-Lefschetz Theorem. We are interested in the case $N=2n-1$. In 1994 Ionescu and Toma gave a classification of the cases where $S$ is not of general type. Here we propose a conjecture concerning this remaining case, which is verified for $n\le 1100$ by a computer programm.
We study the problem of whether an arbitrary codimension three graded artinian Gorenstein algebra has the Weak Lefschetz Property. We reduce this problem to checking whether it holds for all compressed Gorenstein algebras of odd socle degree. In the first open case, namely Hilbert function (1,3,6,6,3,1), we give a complete answer in every characteristic by translating the problem to one of studying geometric aspects of certain morphisms from $\mathbb P^2$ to $\mathbb P^3$, and Hesse configurations in $\mathbb P^2$.
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