Philippe Ellia, Davide Franco
We prove the following: (a) Let X be a smooth, codimension two subvariety of P6. If X lies on a hyperquintic or if deg(X)<74, then X is a complete intersection. (b) Let X be a smooth, subcanonical threefold in P5. If X lies on a hyperquartic, then X is a complete intersection.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
继续探索
In this paper we extend the properties of ordinary points of curves [10] to ordinary closed points of one-dimensional affine reduced schemes and then to ordinary subvarieties of codimension one.
We extend results of our previous papers, on ordinary multiple points of curves, and on the computation of their conductor, to ordinary multiple subvarieties of codimension one.
We consider smooth codimension two subcanonical subvarieties in $\mathbb{P}^n$ with $n \geq 5$, lying on a hypersurface of degree $s$ having a linear subspace of multiplicity $(s-2)$. We prove that such varieties are complete intersections. We also give a little improvement to some earlier results on the non existence of rank two vector bundles on $\mathbb{P}^4$ with small Chern classes.
We show that for a smooth hypersurface $X\subset \bbP^n$ of degree at least 2, there exist arithmetically Cohen-Macaulay (ACM) codimension two subvarieties $Y\subset X$ which are not an intersection $X\cap{S}$ for a codimension two subvariety $S\subset\bbP^n$. We also show there exist $Y\subset X$ as above for which the normal bundle sequence for the inclusion $Y\subset X\subset\bbP^n$ does not split.
Extendable codimension two subvarieties in a general hypersurface
本文在一般超曲面中构造可扩展余维二子簇，并证明当第一陈类远小于次数时，不存在全局生成不可分解 ACM 丛。
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.
We introduce a method to determine if n-dimensional smooth subvarieties of an ambient space of dimension at most 2n − 2 inherit the Picard group from the ambient space (as it happens when the ambient space is a projective space, according to results of Barth and Larsen). As an application, we give an affirmative answer (up to some mild natural numerical conditions) when the ambient space is a Grassmannian of lines (thus improving results of Barth, Van de Ven and Sommese) or a product of two projective spaces of the same dimension.
We investigate a method proposed by E. Arrondo and J. Caravantes to study the Picard group of a smooth low-codimension subvariety X in a variety Y when Y is homogeneous. We prove that this method is strongly related to the signature σ_Y of the Poincare pairing on the middle cohomology of Y. We give under some topological assumptions a bound on the rank of Picard group Pic(X) in terms of σ_Y and remove these assumptions for grassmannians to generalise the main result of E. Arrondo and J. Caravantes.
We make precise the structure of the first two reduction morphisms associated with codimension two nonsingular subvarieties of quadrics $\Q{n}$, $n\geq 5$. We give a coarse classification of the same class of subvarieties when they are assumed to be not of log-general-type. Keywords: Adjunction Theory, classification, codimension two, conic bundles, low codimension, non log-general-type, quadric, reduction, special variety
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.
Fix integers $a\geq 1$, $b$ and $c$. We prove that for certain projective varieties $V\subset{\bold P}^r$ (e.g. certain possibly singular complete intersections), there are only finitely many components of the Hilbert scheme parametrizing irreducible, smooth, projective, low codimensional subvarieties $X$ of $V$ such that $$ h^0(X,\Cal O_X(aK_X-bH_X)) \leq λd^{ε_1}+c(\sum_{1\leq h < ε_2}p_g(X^{(h)})), $$ where $d$, $K_X$ and $H_X$ denote the degree, the canonical divisor and the general hyperplane section of $X$, $p_g(X^{(h)})$ denotes the geometric genus of the general linear section of $X$ o…
Let $X$ be a codimension two nonsingular subvariety of a nonsingular quadric $\Q{n}$ of dimension $n\geq 5$. We classify such subvarieties when they are scrolls. We also classify them when the degree $d\leq 10$. Both results were known when $n=4$. Keywords: Classification; liaison; low codimension; low degree; quadric; scroll; vector bundle
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 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 this paper, we introduce a notion of adjoint ideal sheaves along closed subvarieties of higher codimension and study its local properties using characteristic $p$ methods. When $X$ is a normal Gorenstein closed subvariety of a smooth complex variety $A$, we formulate a restriction property of the adjoint ideal sheaf $\adj_X(A)$ of $A$ along $X$ involving the l.c.i. ideal sheaf $\mathcal{D}_X$ of $X$. The proof relies on a modification of generalized test ideals of Hara and Yoshida.
We give the first examples of nef line bundles on smooth projective varieties over finite fields which are not semi-ample. More concretely, we find smooth curves on smooth projective surfaces over finite fields such that the normal bundle has degree zero, but no positive multiple of the curve moves in a family of disjoint curves. This answers questions by Keel and Mumford. The proof uses an obstruction theory, in the spirit of homotopy theory, which links the infinitely many obstructions to moving higher and higher multiples of a given codimension-one subvariety. On 3-folds over a finite field…
The monodromy of families of subvarieties on abelian varieties
本文结合几何论证与表示论，证明阿贝尔簇中高余维非平凡子簇族经一般秩一局部系统扭转后具有大单值性。
We obtain criteria for detecting complete intersections in projective varieties. Motivated by a conjecture of Hartshorne concerning subvarieties of projective spaces, we investigate situations when two-codimensional smooth subvarieties of rational homogeneous varieties are complete intersections.
Let $V$ be a subvariety of codimension $\leq g$ of the moduli space $\cA_g$ of principally polarized abelian varieties of dimension $g$ or of the moduli space $\tM_g$ of curves of compact type of genus $g$. We prove that the set $E_1(V)$ of elements of $V$ which map onto an elliptic curve is analytically dense in $V$. From this we deduce that if $V \subset \cA_g$ is complete, then $V$ has codimension equal to $g$ and the set of elements of $V$ isogenous to a product of $g$ elliptic curves is countable and analytically dense in $V$. We also prove a technical property of the conormal sheaf of $V…
We prove that in characteristic p>0 the locus of stable curves of p-rank at most f is pure of codimension g-f in the moduli space of stable curves. Then we consider the Prym map and analyze it using tautological classes. We study the locus of curves with an etale double cover of p-rank 0 in some detail. In particular, in genus 2 we obtain a formula for the number of such curves. We end with several examples illustrating our formula.
For $Y \subset X$ a locally complete intersection of codimension p, Spencer Bloch [2] constructed the semi-regularity map $π: H^{1}(\mathcal{N}_{Y/X}) \to H^{p+1}(Ω_{X/k}^{p-1})$. As an analogue, we construct a map $\tildeπ: H^{1}(\mathcal{N}_{Y/X}) \to H^{p+1}(Ω_{X/\mathbb{Q}}^{p-1})$, without assuming local complete intersections. While the semi-regularity map $π$ is expected to be injective, we show $\tildeπ$ is a zero map. We use this zero map to interpret how to eliminate obstructions to deforming cycles, an idea by Mark Green and Phillip Griffiths in [9].
We construct a new 20-dimensional family of projective 6-dimensional irreducible holomorphic symplectic manifolds. The elements of this family are deformation equivalent with the Hilbert scheme of three points on a K3 surface and are constructed as natural double covers of special codimension 3 subvarieties of the Grassmanian G(3,6). These codimension 3 subvarieties are defined as Lagrangian degeneracy loci and their construction is parallel to that of EPW sextics, we call them the EPW cubes. As a consequence we prove that the moduli space of polarized IHS sixfolds of K3-type, Beauville-Bogomo…
Arrondo, Sols and De Cataldo proved that there are only finitely many families of codimension two subvarieties not of general type in the smooth quadric of dimension $n+2$ for $n\ge 2 $, $n\neq 4$. In this paper we drop the assumption $n\neq 4$ from the previous result (obviously the assumption $n\ge 2$ cannot be removed).
In this paper we prove that given a pair $(X,D)$ of a threefold $X$ and a boundary divisor $D$ with mild singularities, if $(K_X+D)$ is movable, then the orbifold second Chern class $c_2$ of $(X,D)$ is pseudo-effective. This generalizes the classical result of Miyaoka on the pseudo-effectivity of $c_2$ for minimal models. As an application we give a simple solution to Kawamata's effective non-vanishing conjecture in dimension $3$, where we prove that $H^0(X, K_X+H)\neq 0$, whenever $K_X+H$ is nef and $H$ is an ample, effective, reduced Cartier divisor. Furthermore, we study Lang-Vojta's conjec…
In this article we develop a new way of systematically constructing infinitely many families of smooth subvarieties $X$ of any given dimension $m$, $m \geq 3$, and any given codimension in $\mathbb P^N$, embedded by complete subcanonical linear series, and, in particular, in the range of Hartshorne's conjecture. We accomplish this by showing the existence of everywhere non--reduced schemes called ropes, embedded in $\mathbb P^N$, and by smoothing them. In the range $3 \leq m < N/2$, we construct smooth subvarieties, embedded by complete subcanonical linear series, that are not complete interse…
Some applications of microlocalization for local complete intersection subvarieties
本文提出适用于局部完全交子簇的微局部化构造，证明最小指数与最小 Bernstein–Sato 根及最小非零谱数一致，并推出其可构造性与等奇异族中的恒定性。
We prove Barth-type connectedness results for low-codimension smooth subvarieties with good numerical properties inside certain "easy" ambient spaces (such as homogeneous varieties, or spherical varieties). The argument employs some basics from the theory of cones of cycle classes, in particular the notion of bigness of a cycle class.
We work over an algebraically closed field of arbitrary characteristic. Ellingsrud-Peskine proved that smooth surfaces in P^4 are subject to strong limitations. Their whole argument is derived from the fact that the sectional genus of surfaces of degree d lying on a hypersurface of degree s varies in an interval of length \frac{d(s-1)^2}{2s}. The aim of the present paper is to show that for smooth codimension two subvarieties of P^n, n\geq 5, one can get a similar result with an interval whose length depends only on s. The main point is Lemma 1.1 whose proof is a direct application of the posi…
Subvarieties of complete intersections of large degree
研究大度数完全交集的子簇类型及其代数超曲率性，证明了在特定条件下非一般类型的子簇必须位于由直线覆盖的区域。
Classification of Smooth Minimal Kähler Fourfolds Without Effective Divisors and Surfaces
研究紧致凯勒四维流形\(X\)，其典范丛伪有效且无余维数一二的子簇，证明\(K_X\)是挠线丛，借助Beauville - Bogomolov分解定理得出\(X\)是复环面商或不可约全纯辛流形。
Resolutions of toric subvarieties by line bundles and applications
本文利用实环面分层，用环面 Frobenius 前推的线丛直和项构造环面子簇结构层的长度 k 分解，并据此证明虚拟合冲定理、Rouquier 维数及导出范畴生成等猜想。
Two birational subvarieties of P^n are called Cremona equivalent if there is a Cremona modification of P^n mapping one to the other. If the codimension of the varieties is at least 2 then they are always Cremona Equivalent. For divisors the question is much more subtle and a general answer is unknown. In this paper I study the case of rational quartic surfaces and prove that they are all Cremona equivalent to a plane.
We show that points in the intersection of the tropicalizations of subvarieties of a torus lift to algebraic intersection points with expected multiplicities, provided that the tropicalizations intersect in the expected dimension. We also prove a similar result for intersections inside an ambient subvariety of the torus, when the tropicalizations meet inside a facet of multiplicity 1. The proofs require not only the geometry of compactified tropicalizations of subvarieties of toric varieties, but also new results about the geometry of finite type schemes over non-noetherian valuation rings of …
Combining theorems of Voisin and Marian, Shen, Yin and Zhao, we compute the dimensions of the orbits under rational equivalence in the Mukai system of rank two and genus two. We produce several examples of algebraically coisotropic and constant cycle subvarieties.
We consider the problem to determine which blow-ups along subvarieties in products of two projective spaces are log Fano. By describing the nef cones of such blow-ups with special centers, we give a partial classification result. For each example, explicit boundary divisors are also given.
The goal of this article is twofold. On one hand, we study the subvarieties of projective varieties which possess partially ample normal bundle; we prove that they are G2 in the ambient space. This generalizes results of Hartshorne and Bădescu-Schneider. We work with the cohomological partial ampleness introduced by Totaro. On the other hand, we define the concept of a partially ample subvariety, which generalizes the notion of an ample subvariety introduced by Ottem. We prove that partially ample subvarieties enjoy the stronger G3 property. Moreover, we present an application to a connectedne…
Let $X$ be a codimension 1 subvariety of dimension $>1$ of a variety of minimal degree $Y$. If $X$ is subcanonical with Gorenstein canonical singularities admitting a crepant resolution, then $X$ is Arithmetically Gorenstein and we characterise such subvarieties $X$ of $Y$ via apolarity as those whose apolar hypersurfaces are Fermat.
We propose two conjectures on a moduli theoretic approach to constructing Lagrangian subvarieties of hyperkähler varieties arising from the Kuznetsov components of cubic fourfolds or Gushel--Mukai fourfolds. Then we verify the conjectures in several cases, recovering classical examples. As a corollary, we confirm a conjecture of O'Grady in several instances on the existence of Lagrangian covering families for hyperkähler varieties.
The aim of this paper is to prove the existence of large complete subvarieties in moduli spaces of rank two stable sheaves with arbitrary $c_1$ and sufficiently large $c_2$ on algebraic surfaces. Then we study the restriction of these sheaves to curves of high degree embedded in the surface. In the final section we gives a relation with the spin strata defined by Pidstrigach and Tyurin.
Let $X$ be a quasi-affine algebraic variety isomorphic to the complement of a closed subvariety of dimension at most $n-3$ in $\C^n$. We find some conditions under which an isomorphism of two closed subvarieties of $X$ can be extended to an automorphism of $X$.
On a counterexample to a conjecture of J. Harris for octic surfaces
研究八次Fermat曲面中直线与(3,3)型完全交曲线的和，证明除有限个参数外，对应的Noether-Lefschetz轨迹是余维31的子簇，相交于余维32的子簇，构成J. Harris猜想的可能反例。
Let $π: Z \to X$ be Galois cover of smooth projective curves with Galois group $W$ a Weyl group of a simple Lie group $G$. For a dominant weight $λ$, we consider the intermediate curve $Y_λ= Z/\Stab(λ)$. One can realise a Prym variety $P_λ\subset \Jac(Y_λ)$ and we denote $φ_λ$ the restriction of the principal polarisation of $\Jac(Y_λ)$ upon $P_λ$. For two dominant weights $λ$ and $μ$, we construct a correspondence $Δ_{λμ}$ on $Y_λ\times Y_μ$ and calculate the pull-back of $φ_μ$ by $Δ_{λμ}$ in terms of $φ_λ$.
Totally geodesic subvarieties of the moduli space of curves and linear systems
本研究在曲线模空间全测地子簇的一般曲线上构造线性系统，推导了维数≥2的全测地子簇的秩界，并完成了至多含两个零点的层中此类子簇的分类。
A Hodge Theoretic Generalization of $\mathbb{Q}$-Homology Manifolds II: Local Complete Intersections
研究复代数簇$Z$的奇点不变量$\HRH(Z)$，聚焦局部完全交子簇，将其与伯恩斯坦 - 佐藤多项式等著名不变量联系，超曲面情形可由这些不变量完全刻画$\HRH(Z)$，高余维情形更复杂。
On a conjecture on Hodge loci of linear combinations of linear subvarieties
本文针对 Movasati 关于包含相交 $k$-平面的三次和四次超曲面 Hodge 轨迹维数的猜想，在 $k \geq 5$ 时给出反例，并在 $k=3,4$ 时提供支持该猜想的新证据。
We prove that in the parameter space of $M$-dimensional Fano complete intersections of index one and codimension two the locus of varieties that are not birationally superrigid has codimension at least $\frac12 (M-9)(M-10)-1$.
We investigate the relation between codimension two smooth complete intersections in a projective space and some naturally associated graded algebras. We give some examples of log-concave polynomials and we propose two conjectures for these algebras.
We develop reductions for classifications of singularities of orbit closures in module varieties. Then we show that the orbit closures for representations of Dynkin quivers are regular in codimension two.
In this note we study two features of submanifolds (subvarieties) with ample normal bundles in a compact Kähler manifold X. First, we study how algebraic X can be, i.e. we investigate the algebraic dimension. Second, we study curves with ample normal bundles in case X is projective. We prove in various cases that the class of the curve is in the interior of the Mori cone.
We describe the topology of critical loci of coamoeba of generic affine planes in four-space.
This paper classifies toric Fano 3-folds with singular locus { 1/k(1,1,1) } for any positive integer k, building on the work of Batyrev and Watanabe-Watanabe. This is achieved by completing an equivalent problem in the language of Fano polytopes. Furthermore we identify birational relationships between entries of the classification. For a fixed value k>4, there are exactly two such Fano 3-folds linked by a blow-up in a torus-invariant line.
In this paper, we obtain two extension theorems for cohomology classes and holomorphic sections defined on analytic subvarieties, which are defined as the supports of the quotient sheaves of multiplier ideal sheaves of quasi-plurisubharmonic functions with arbitrary singularities. The first result gives a positive answer to a question posed by Cao-Demailly-Matsumura, and unifies a few well-known injectivity theorems. The second result generalizes and optimizes a general $L^2$ extension theorem obtained by Demailly.
Lagrangian subvarieties of hyperspherical varieties related to $G_2$
本文研究了与 $G_2$ 相关的两个 $S$-对偶超球面簇，并验证了其拉格朗日子簇的不可约分支数量相等的猜想。
Intersecting subvarieties of abelian schemes with group subschemes I
本文建立阿贝尔簇上Habegger有界高度定理的族版本，利用Ax--Schanuel定理和Yuan--Zhang的adelic线丛相交理论，将Silverman特殊化定理推广到高维基并得到朝向Zhang的ICM猜想的有界高度结果。
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。
Universal codimension two cycle on a very general cubic threefold
本文证明非常一般的三次三维簇无万有余维二闭链，从而得出其稳定非有理的结论。
New formulas are given for Chow forms, discriminants and resultants arising from (not necessarily normal) toric varieties of codimension 2. Exact descriptions are also given for the secondary polygon and for the Newton polygon of the discriminant.
Generic linkage is used to compute a prime ideal such that the radical of the initial ideal of the prime ideal is equal to the radical of a given codimension two monomial ideal that has a Cohen-Macaulay quotient ring.
We define higher categorical invariants (gerbes) of codimension two algebraic cycles and provide a categorical interpretation of the intersection of divisors on a smooth proper algebraic variety. This generalization of the classical relation between divisors and line bundles furnishes a new perspective on the Bloch-Quillen formula.
We define the monomial invariants of a projective variety $Z$; they are invariants coming from the generic initial ideal of $Z$. Using this notion, we generalize a result of Cook: If $Z$ is an integral variety of codimension two, satisfying the additional hypothesis $s_Z=s_Γ,$ then its monomial invariants are connected.
扫码添加微信好友，提出您的宝贵建议 👇
💡 备注请填写：网站反馈
发表机构由 AI 从论文文本抽取，如发现错误，欢迎反馈修正建议。