Hugo Galeano, Marcos Jardim, Alan Muniz
We establish a full classification of degree $2$ codimension one distributions on $\mathbb{P}^3$ according to invariants of their tangent sheaves.
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In this paper we classify codimension 1 Mukai foliations on complex projective manifolds
We give a classification of embedded smooth projective varieties swept out by rational homogeneous varieties whose Picard number and codimension are one.
We generalize the definition of alpha invariant to arbitrary codimension. We also give a lower bound of these alpha invariants for K-semistable Q-Fano varieties and show that we can characterize projective spaces among all K-semistable Fano manifolds in terms of higher codimensional alpha invariants. Our results demonstrate the relation between alpha invariants of any codimension and volumes of Fano manifolds in the characterization of projective spaces.
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).
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。
Fake weighted projective spaces
我们定义伪加权射影空间作为加权射影空间的推广。我们引入余维1上的基本群和余维1上的泛覆盖的概念。证明了对任意伪加权射影空间，其余维为1的泛覆盖是加权射影空间。
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 foliations in positive characteristic
研究正特征域上光滑射影簇的余维一叶状结构几何，旨在为射影流形上余维一全纯叶状结构的结构研究提供应用。
Let X\subsetneq\mathbb{P}_{\mathbb{C}}^{N} be an n-dimensional nondegenerate smooth projective variety containing an m-dimensional subvariety Y. Assume that either m>\frac{n}{2} and X is a complete intersection or that m\geq\frac{N}{2}, we show deg(X)|deg(Y) and codim_{span(Y)}Y\geq codim_{\mathbb{P}^{N}}X, where span(Y) is the linear span of Y. These bounds are sharp. As an application, we classify smooth projective n-dimensional quadratic varieties swept out by m\geq[\frac{n}{2}]+1 dimensional quadrics passing through one point.
Higher codimension nef and effective cycles on the Hilbert scheme of 3 points in projective 3-space
本文计算了三维射影空间中三点希尔伯特概形的高余维nef锥与有效锥，推广了已有结果，并引入新Chow环基。
Ran proved that smooth codimension 2 varieties in ${\bf P}^{m+2}$ are $j$-normal if $(j+1)(3j-1)\le m-1$, in this paper we extend this result to small codimension projective varieties. Let $X$ be a r codimension subvariety of $\pro$, we prove that if the set $Σ_{(j+1)}$ of $(j+1)$-secants to $X$ through a generic external point is not empty, $2(r+1)j\leq m-r$ and $(j+1)((r+1)j-1)\leq m-1$ then $X$ is $j$-normal. If $X$ is given by the zero locus of a section of a rank $r$ vector bundle $E$ on $\pro$, we prove that $\textrm{deg} Σ_{j+1}=\frac{1}{(j+1)!}\prod_{i=0}^{j}c_{r}(E(-i))$. Moreover we …
Codimension 3 Arithmetically Gorenstein Subschemes of projective $N$-space
研究了$\mathbb{P}^N$的每个算术Cohen-Macaulay子概型是否为glicci的最低维开情形，即$\mathbb{P}^3$的每个零概型是否为glicci.我们证明了在$\PP^3$中一般位置的一组$n \geq 56$点不允许严格下降的Gorenstein联络或biliaison。为了证明这个定理，我们在$\mathbb{P}^3$中建立了关于算术Gorenstein零格式的一些重要结果。
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)。
The Generalized Hodge conjecture for 1-cycles and codimension two algebraic cycles
本文证明了“光滑射影簇$X$上的（广义）Hodge猜想对余维为2的圈成立“是一个双有理不变命题，即如果该命题对$X$成立，则对所有与$X$双有理等价的光滑簇$X'$也成立.我们还证明了1-圈的类似结果。作为直接推论，Hodge猜想对维数小于或等于5的光滑有理射影流形成立，广义Hodge猜想对维数小于或等于4的光滑有理射影流形成立.
We give examples of infinitely extendable (not as cones) arithmetically Cohen-Macaulay and arithmetically Gorenstein subvarieties of projective spaces and which are not complete intersections. The proof uses the computation of the dimension of the Hilbert scheme of codimension $2$ subschemes of projective spaces due to G. Ellingsrud and of arithmetically Gorenstein codimension $3$ subschemes due to J. O. Kleppe and R.-M. Miró-Roig.
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.
In this article we give a necessary and sufficient condition to characterize projective submanifolds in ${\mathbb P}^N$ with codimensions 2 and 3. The conditions involve the Chern classes of the manifold and a very ample line bundle on the manifold. This generalizes our earlier characterization for hypersurfaces. The higher codimensional cases are proposed as a general question.
We prove that a representation of the fundamental group of a quasi-projective manifold into the group of formal diffeomorphisms of one variable either is virtually abelian or, after taking the quotient by its center, factors through an orbicurve.
Critical loci for projective reconstruction from three views in four dimensional projective space are defined by an ideal generated by maximal minors of suitable $4 \times 3$ matrices, $N,$ of linear forms. Such loci are classified in this paper, in the case in which $N$ drops rank in codimension one, giving rise to reducible varieties. This leads to a complete classification of matrices of size $(n+1) \times n$ for $n \le 3,$ which drop rank in codimension one. Instability of reconstruction near non-linear components of critical loci is explored experimentally.
This article studies codimension one foliations on projective man-ifolds having a compact leaf (free of singularities). It explores the interplay between Ueda theory (order of flatness of the normal bundle) and the holo-nomy representation (dynamics of the foliation in the transverse direction). We address in particular the following problems: existence of foliation having as a leaf a given hypersurface with topologically torsion normal bundle, global structure of foliations having a compact leaf whose holonomy is abelian (resp. solvable), and factorization results.
Generic Projection Methods in Castelnuovo Regularity of Projective Varieties
Castelnuovo-Mumford正则性是射影代数簇的一个重要不变量。Eisenbud和Goto提出的一个著名猜想给出了正则性的余维和度的界，即，给定簇X的Castelnuovo-Mumford正则性小于或等于deg（X）-余维数（X）+1。一般的投影方法被证明是有效的研究规则性的光滑投影簇的维数最多为4（cf。[BM}，[K2]，[L]，[Pi]和[R1]），因为向量丛的上同调有很好的消灭定理（例如Kodaira-Kawamata-Viehweg消灭定理）和关于从X到同维超曲面的一般投影的纤维的详细信息。这里我们用类似于[K 2]的方法证明了对任意光滑五重函数的$\reg{X}\le(deg(X)-codimension(X)+1)+10$和对任意光滑六重函数的$\reg{X}\le(deg(X)-codimension(X)+1)+20$.此外，使用类似的方法，我们给出了P^N中任意（不一定是局部Cohen-Macaulay）射影曲面X的正则性的界。也就是说，我们证明了$\reg{X}\le(d-e+1)d-(2e+1)$，其中d=deg（X）和e=余维（X）。这是不依赖于光滑度的曲面的第一个边界。
We construct Calabi-Yau 3-folds as orbifolds embedded in weighted projective space in codimension 4. For each Hilbert series that is realised, there are at least two different components of Calabi-Yau 3-folds.
A general method to express in terms of Gauss sums the number of rational points of subschemes of projective schemes over finite fields is applied to the image of the triple embedding $\mathbb{P}^1\hookrightarrow\mathbb{P}^3$. As a consequence, we obtain a non-trivial description of the value of a Kloosterman-sum-like exponential sum.
Birational geometry in codimension 2 of symplectic resolutions
本文证明了如下猜想：（i）辛簇$W$的两个投影辛分解在余维为2的情况下，通过$W$上的Mukai初等变换联系起来.经典单复李代数中的幂零轨道闭包;（ii）.商辛簇
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…
On the Picard group of low-codimension subvarieties
我们介绍了一种方法来确定是否n维光滑的子簇的环境空间的维数最多为2n 2继承皮卡德群从环境空间（因为它发生时，环境空间是一个射影空间，根据结果的巴思和拉森）。作为应用，我们给出了一个肯定的答案（直到一些温和的自然数值条件），当周围空间是一个Grassmannian的线（从而改进了Barth，Van de Ven和Sommese的结果）或产品的两个射影空间的相同的维度。
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 …
Factoriality and Neron-Severi groups of a projective codimension two complete intersection with isolated singularities
对于一个射影簇$Z$和任何整数$p$，定义$Z$的$p$ -th Néron-Severi群$NS_p(Z)$为循环映射$A_{p}(Z)\to H_{2p}(Z; \mathbb{C})$的像。现在令$X\subset \Ps^{2m+1}$（$m\geq 1$）是维数$2m-1$的射影簇，具有孤立奇点，一个次数为$k$的光滑超曲面与一个次数为$n>max\{k, 2m+1\}$的超曲面的完全相交，令$F$是一个包含$X$的次数为$n$的一般超曲面。证明了自然映射$NS_m(X)\to NS_m(F)$是满射的，并且如果$dim NS_m(F)=1$，则$dim NS_m(X)=1$ .特别是$dim NS_m(X)=1$当且仅当$dim NS_m(F)=1$。当$X$是一个三重（即$m=2$），我们推出了一个新的表征$X$的阶乘，即$X$是阶乘当且仅当$dim NS_2(F)=1$。这使我们能够给出阶乘三重的例子，在某些情况下有许多奇点。在对上述结果的证明过程中，我们证明了$F$的中上同调与$X$的循环类之商在包含$X$的$n$次超曲面诱导的单值作用下是不可约的.作为推论，我们导出了具有孤立奇点的射影完全交的Noether-Lefschetz定理，并利用最近关于余维2的Hodge猜想的结果，在$X\subset \Ps^{5}$是三重的情形下，我们导出了包含$X$的次数为$n$…
We show that for every smooth generic projective hypersurface $X\subset\mathbb P^{n+1}$, there exists a proper subvariety $Y\subsetneq X$ such that $\operatorname{codim}_X Y\ge 2$ and for every non constant holomorphic entire map $f\colon\mathbb C\to X$ one has $f(\mathbb C)\subset Y$, provided $°X\ge 2^{n^5}$. In particular, we obtain an effective confirmation of the Kobayashi conjecture for threefolds in $\mathbb P^4$.
In the present note we study configurations of codimension 2 flats in projective spaces and classify those with the smallest rate of growth of the initial sequence. Our work extends those of Bocci, Chiantini in P^2 and Janssen in P^3 to projective spaces of arbitrary dimension.
Some Examples of Gorenstein Liaison in Codimension Three
Gorenstein联络是将射影空间中余维2簇联络的著名结果推广到更高余维的自然概念。本文研究了${\mathbb P}^3$中的点和${\mathbb P}^4$中的曲线，试图看看典型的余维~2结果能推广多远。虽然结果是令人满意的小程度，我们发现在每种情况下，我们不能决定的结果的例子。这些例子是余维~2定理所希望的推广的反例的候选者。
In this article, we give the structure of codimension one foliations with canonical singularities and numerically trivial canonical class on varieties with klt singularities. Building on recent works of Spicer, Cascini - Spicer and Spicer - Svaldi, we then describe the birational geometry of rank two foliations with canonical singularities and canonical class of numerical dimension zero on complex projective threefolds.
We classify singular fibres of a projective Lagrangian fibration over codimension one points. As an application, we obtain a canonical bundle formula for a projective Lagrangian fibration over a smooth manifold.
We give the construction of weighted Lagranngiann Grassmannians$wLGr(3,6)$ and weighted partial $A_3$ flag variety $wFl_{1,3}$ coming from the symplectic Lie group $Sp(6,\mathbb C)$ and the general linear group $GL(4,\mathbb C)$ respectively. We give general formulas for their Hilbert series in terms of Lie theoretic data. We use them as key varieties (Format) to construct some families of polarized 3-folds in codimension 7 and 9. At the end, we list all the distinct weighted flag varieties in codimension $4\le c\le 10$.
Improved, sublinear projective Schwartz-Zippel and (sub)quadratic dimension growth bounds in arbitrary codimension
本文改进了投影Schwartz-Zippel界，通过亚线性方法在低维情况下获得亚二次维度增长界，并将二次界扩展到任意余维数。
Let $X$ a projective manifold equipped with a codimension $1$ (maybe singular) distribution whose conormal sheaf is assumed to be pseudoeffective. By a theorem of Jean-Pierre Demailly, this distribution is actually integrable and thus defines a codimension $1$ holomorphic foliation $\F$. We aim at describing the structure of such a foliation, especially in the non abundant case: It turns out that $\F$ is the pull-back of one of the "canonical foliations" on a Hilbert modular variety. This result remains valid for "logarithmic foliated pairs".
Using twisted and formal-local methods, we prove that every separated algebraic space which is the (open/flat) pushout of affine schemes has enough Azumaya algebras. As a corollary we show that, under mild hypothesis, every cohomological Brauer class is representable by an Azumaya algebra away from a closed subset of codimension $\geq 3$, generalizing an early result of Grothendieck. Next, we show that $\mathrm{Br}(X)=\mathrm{Br}'(X)$ when $X$ is an algebraic space obtained from a quasi-projective scheme by contracting a curve. This is the first method of descending an Azumaya algebra along a …
We study the behavior of the bilinkage process in codimension $3$. In particular, we construct a smooth canonically embedded and linearly normal surface of general type of degree $18$ in $\mathbb{P}^5$, this is probably the highest degree such surface may have. Next, we apply our construction to find a geometric description of Tonoli Calabi--Yau threefolds in $\mathbb{P}^6$.
Take a smooth, connected and non-degenerate projective curve $X\subset \mathbb {P}^r$, $r\ge 2b+2\ge 6$, defined over an algebraically closed field with characteristic $0$ and let $σ_b(X)$ be the $b$-secant variety of $X$. We prove that the $X$-rank of $q$ is at least $b+1$ for a non-empty codimension $1$ locally closed subset of $σ_b(X)$.
In this paper we study projective flat deformations of projective spaces. We prove that the singular fibers of projective flat deformations of projective spaces appear either in codimension 1 or over singular points of the base. We also describe projective flat deformations of projective spaces with smooth total space, and discuss flatness criteria.
We prove that a Fano complete intersection of codimension $k$ and index 1 in the complex projective space ${\mathbb P}^{M+k}$ for $k\geqslant 20$ and $M\geqslant 8k\log k$ with at most multi-quadratic singularities is birationally superrigid. The codimension of the complement to the set of birationally superrigid complete intersections in the natural parameter space is shown to be at least $\frac12 (M-5k)(M-6k)$. The proof is based on the techniques of hypertangent divisors combined with the recently discovered $4n^2$-inequality for complete intersection singularities.
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.
Unlikely intersections of codimension one foliations
该研究聚焦复射影流形上的奇异全纯叶状结构族，探究其整体相交所定义的余维数意外偏低的叶状结构，推进了复几何与代数几何领域的叶状结构相交理论研究。
In this work, we study dominant rational maps preserving singular holomorphic codimension one foliations on projective manifolds and that exhibit non-trivial transverse dynamics.
Minimal free resolutions of projective subschemes of small degree
讨论了不可约投射子概型X的最小自由分解。如果X也是约化的，我们关注它的度等于2加上余维的情况。若余维数为2，则描述了所有可能的分次贝蒂数的集合.总的来说，只有部分结果。然后讨论了线性子空间上的Cohen-Macaulay结构。对于这种二阶曲线，Notari，Spreafico和作者已经获得了分类。总的来说，分类问题似乎相当困难。Vatne已经证明，这种分类的结构，其程度是最多为3，相当于哈茨霍恩猜想光滑簇的余维2。
Let $X = \mathbb{P}(E_1) \times_C \mathbb{P}(E_2)$ where $C$ is a smooth curve and $E_1$, $E_2$ are vector bundles over $C$.In this paper we compute the pseudo effective cones of higher codimension cycles on $X$.
Anti-flips of the blow-ups of the projective spaces at torus invariant points
本文通过反翻转显式构造光滑环面法诺簇，使其同构于射影空间在余维一环面不变点处的爆破。
Obstructions for codimension one multiple fibers of Lagrangian and Calabi--Yau fibrations
该研究证明了两类纤维化不存在余维一多重纤维的结论，为拉格朗日纤维化结构、相关猜想及卡拉比-丘纤维化研究提供了关键进展，仅存在一个奇维K-平凡簇的例外情形。
Let $X$ be a submanifold of dimension $n$ of the complex projective space $\mathbb P^N$ ($n<N$), and let $E$ be a vector bundle of rank two on $X$ . If $n\geq\frac{N+3}{2}\geq 4$ we prove a geometric criterion for the existence of an extension of $E$ to a vector bundle on the first order infinitesimal neighborhood of $X$ in $\mathbb P^N$ in terms of the splitting of the normal bundle sequence of $Y\subset X\subset\mathbb P^N$, where $Y$ is the zero locus of a general section of a high twist of $E$. In the last section we show that the universal quotient vector bundle on the Grassmann variety $…
Suppose that $X_A\subset \mathbb{P}^{n-1}$ is a toric variety of codimension two defined by an $(n-2)\times n$ integer matrix $A$, and let $B$ be a Gale dual of $A$. In this paper we compute the Euclidean distance degree and polar degrees of $X_A$ (along with other associated invariants) combinatorially working from the matrix $B$. Our approach allows for the consideration of examples that would be impractical using algebraic or geometric methods. It also yields considerably simpler computational formulas for these invariants, allowing much larger examples to be computed much more quickly than…
The purpose of this thesis is to obtain the degree of the exceptional component of the space of holomorphic foliations of degree two and codimension one in P3. I construct a parameter space as an explicit fiber bundle over the variety of complete flags. Using tools from equivariant intersection theory, especially Bott's formula, the degree is expressed as an integral over our parameter space.
Let $f \colon X \to Y$ be the blow-up of a smooth projective variety $Y$ along its codimension two smooth closed subvariety. In this paper, we show that the moduli space of stable sheaves on $X$ and $Y$ are connected by a sequence of flip-like diagrams. The result is a higher dimensional generalization of the result of Nakajima and Yoshioka, which is the case of $\dim Y=2$. As an application of our general result, we study the birational geometry of the Hilbert scheme of two points.
On the birational geometry of the parameter space for codimension 2 complete intersections
本文研究了2阶完全交集参数空间\bar{H}的理化几何，证明了其MMP收缩的存在性，并在特定情况下完成了MMP，进而证明了点Hilbert方案中存在完整曲线。
Well formedness vs weak well formedness
本文比较了加权射影空间中良构簇的两种定义，证明二者确实不同，并证明它们在维数至少为3的拟光滑加权完全交上等价。
On Smooth Divisors of a Projective Hypersurface
本文证明了P^n，n>4（特征为零的代数闭域上的射影空间）超曲面的光滑除子次数的一个有效界.我们的结果如下从一个强有力的（因为程度的除数是不参与）推广的“特殊性定理”的Gruson-Peskine，我们证明持有任何顺利，subcanonical，余维2，投影verieties维至少3。
In this paper, we study the cones of higher codimension (pseudo)effective cycles on point blow-ups of projective space. We determine bounds on the number of points for which these cones are generated by the classes of linear cycles, and for which these cones are finitely generated. Surprisingly, we discover that for (very) general points, the higher codimension cones behave better than the cones of divisors. For example, for the blow-up $X_r^n$ of $\mathbb P^n$, $n>4$, at $r$ very general points, the cone of divisors is not finitely generated as soon as $r> n+3$, whereas the cone of curves is …
The Gauss map of a projective foliation
本文研究了射影空间中全纯一维层的高斯映射，重点探讨其在$n=3$时的双射性质。
We provide explicit graded constructions of orbifold del Pezzo surfaces with rigid orbifold points of type $\left\{k_i\times\frac{1}{r_i}(1,a_i): 3\le r_i \le 10,k_i \in \ZZ_{\ge 0}\right\}$; as well-formed and quasismooth varieties embedded in some weighted projective space. In particular, we present a collection of 147 such surfaces such that their image under their anti-canonical embeddings can be described by using one of the following sets of equations: a single equation, two linearly independent equations, five maximal Pfaffians of $5\times 5$ skew symmetric matrix, and nine $2\times 2$ …
We consider the set $K(n,c,\rtt)$ of codimension one holomorphic foliations on $¶^n,\,\, n\geq3$, with Chern class $c$, and with a compact, connected Kupka set of radial transversal type. We will prove that foliations in this set, have a rational first integral and define an irreducible component of the space of foliations.
We consider weak Fano manifolds with small contractions obtained by blowing up successively curves and subvarieties of codimension 2 in products of projective spaces. We give a classification result for a special case. In the process of proof, we describe explicitly the structure of nef cones and compute the self intersection numbers of anti-canonical divisors for such weak Fano manifolds.
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