Carlo Madonna
We show that Horrocks' criterion for the splitting of rank two vector bundles in P^3 can be extended, with some assumptions on the Chern classes, on non singular hypersurfaces in P^4. Extension of other splitting criterion are studied.
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Let X be a smooth projective hypersurface. In this note we show that any six generated arithmetically Cohen-Macaulay vector bundle over X splits if dim X >= 6.
In this paper, we give a complete classification of initialized and ACM line bundles on a smooth quartic hypersurface on P^3$.
In this paper, we investigate when there exists a wild hypersurface bundle over a smooth proper toric variety in positive characteristic. In particular, we determine the possibilities for toric varieties with Picard number at most three or toric Fano varieties of dimension at most four. Moreover, we can construct wild hypersurface bundles over them.
We prove that any rank two arithmetically Cohen-Macaulay vector bundle on a general hypersurface of degree at least three in $\mathbb{P}^5$ must be split.
We prove that any rank two arithmetically Cohen-Macaulay vector bundle on a general hypersurface of degree at least six in projective four space must be split.
Let $X$ be a smooth projective hypersurface. In this note we show that any rank 3 arithmetically Cohen-Macaulay vector bundle over $X$ splits when dim $X \geq 7$. We also find a splitting criterion for rank 4 arithmetically Cohen-Macaulay vector bundles on $X$ when dim $X \geq 9$.
Rank-two weak Fano bundles on a four-dimensional quadric hypersurface $Q^4$
本文对四维二次超曲面Q^4上的秩2弱法诺丛分类，经线丛扭转后这类丛分为三类，含文献[APW94]构造的特定稳定丛。
Using monads, we construct a large class of stable bundles of rank 2 and 3 on 3-fold hypersurfaces, and study the set of all possible Chern classes of stable vector bundles.
Let $X$ be a smooth projective hypersurface of dimension $\geq 5$ and let $E$ be an arithmetically Cohen-Macaulay bundle on $X$ of any rank. We prove that $E$ splits as a direct sum of line bundles if and only if $H^i_*(X, \wedge^2 E) = 0$ for $i = 1,2,3,4$. As a corollary this result proves a conjecture of Buchweitz, Greuel and Schreyer for the case of rank 3 arithmetically Cohen-Macaulay bundles.
Let $X$ be a smooth quintic hypersurface in $\mathbb{P}^3$, let $C$ be a smooth hyperplane section of $X$, and let $H=\mathcal{O}_X(C)$. In this paper, we give a necessary and sufficient condition for the line bundle given by a non-zero effective divisor on $X$ to be initialized and aCM with respect to $H$.
In this paper we show that on a general hypersurface of degree $r=3,4,5,6$ in ${\bf P}^5$ a rank 2 vector bundle $E$ splits if and only if $h^1 E(n)=h^2 E(n)=0$ for all $n \in \bf Z$.
Verlinde bundles are vector bundles $V_k$ arising as the direct image $π_*(\mathcal L^{\otimes k})$ of polarizations of a proper family of schemes $π\colon \mathfrak X \to S$. We study the splitting behavior of Verlinde bundles in the case where $π$ is the universal family $\mathfrak X \to |\mathcal O(d)|$ of hypersurfaces of degree $d$ in $\mathbb P^n$ and calculate the cohomology class of the locus of jumping lines of the Verlinde bundles $V_{d+1}$ in the cases $n=2,3$.
Let $X$ be a smooth quartic hypersurface in $\mathbb{P}^3$. By the Brill-Noether theory of curves on K3 surfaces, if a rank 2 aCM bundle on $X$ is globally generated, then it is the Lazarsfeld-Mukai bundle $E_{C,Z}$ associated with a smooth curve $C$ on $X$ and a base point free pencil $Z$ on $C$. In this paper, we will focus on the classification of such bundles on $X$ to investigate aCM bundles of rank 2 on $X$. Concretely, we will give a necessary condition for a rank 2 vector bundle of type $E_{C,Z}$ to be indecomposable initialized and aCM, in the case where the class of $C$ in Pic($X$) i…
Restricted tangent bundle of rational curves on projective hypersurfaces
本文完全分类了一般射影超曲面包含具有平衡受限切丛的有理曲线的所有次数参数，并给出了有理正规曲线情形下的显式构造方法。
Let $E$ be an indecomposable rank two vector bundle on the projective space $\PP^n, n \ge 3$, over an algebraically closed field of characteristic zero. It is well known that $E$ is arithmetically Buchsbaum if and only if $n=3$ and $E$ is a null-correlation bundle. In the present paper we establish an analogous result for rank two indecomposable arithmetically Buchsbaum vector bundles on the smooth quadric hypersurface $Q_n\subset\PP^{n+1}$, $n\ge 3$. We give in fact a full classification and prove that $n$ must be at most 5. As to $k$-Buchsbaum rank two vector bundles on $Q_3$, $k\ge2$, we pr…
Singularity of cubic hypersurfaces and hyperplane sections of projectivized tangent bundle of projective space
研究立方超曲面的奇点性质，并通过线性代数方法描述射影空间射影化切丛的超平面截面，扩展了Mazouni-Nagaraj的高维结果。
Correspondence between Projective bundles over $\mathbb{P}^{2}$ and rational Hypersurfaces in $\mathbb{P}^{4}$
该代数几何研究证明$\u2119^{3}$上零相关丛在超平面的限制对应的射影丛同构于$\u2119^{4}$中非奇异二次曲面沿直线的爆破，且对任意$d\geq2$存在对应d次超曲面，建立了两类几何对象的对应关系。
Deformations of the tangent bundle of a projective hypersurface
本文研究了射影超曲面切丛的无穷小形变，证明了其形变空间的维数公式及无阻碍性，并通过构造显式双有理模型证明包含切丛的稳定丛模空间不可约分支是有理簇。
We investigate the complex analytic structure of the complement of a non-singular hypersurface with unitary flat normal bundle when the corresponding line bundle admits a Hermitian metric with semipositive curvature.
We investigate the arrangement of hypersurfaces on a nonsingular varieties whose associated logarithmic vector bundle is arithmetically Cohen-Macaulay (for short, aCM), and prove that the projective space is the only smooth complete intersection with Picard rank one that admits an aCM logarithmic vector bundle. We also obtain a number of results on aCM logarithmic vector bundles over several specific varieties. As an opposite situation we investigate the Torelli-type problem that the logarithmic cohomology determines the arrangement.
Let D = {D_{1},...,D_{l}} be an arrangement of smooth hypersurfaces with normal crossings on the complex projective space P^n and let Ω^{1}_{P^n}(log D) be the logarithmic bundle attached to it. Following [1], we show that Ω^{1}_{P^n}(log D) admits a resolution of lenght 1 which explicitly depends on the degrees and on the equations of D_{1},...,D_{l}. Then we prove a Torelli type theorem when all the D_{i}'s have the same degree d and l >= {{n+d} \choose {d}}+3: indeed, we recover the components of D as unstable smooth hypersurfaces of Ω^{1}_{P^n}(log D). Finally we analyze the cases of one q…
We use a generalization of Horrocks monads for arithmetic Cohen-Macaulay (ACM) varieties to establish a cohomological characterization of linear and Steiner bundles over projective spaces and quadric hypersurfaces. We also study resolutions of bundles on ACM varieties by line bundles, and characterize linear homological dimension in the case of quadric hypersurfaces.
Let $\cMx$ be the moduli space of stable vector bundles of rank $n\geq 3$ and determinant $ξ$ over a connected Riemann surface $X$, with $n$ and $d(ξ)$ coprime. Let $D$ be a Calabi-Yau hypersurface of $\cMx$. Denote by $U_D$ the restriction of the universal bundle to $X\times D$. It is shown that the restriction $(U_D)_x$ to $x\times D$ is stable, for any $x\in X$. Furthermore, for a general curve the connected component of the moduli space of semistable sheaves over $D$, containing $(U_D)_x$, is isomorphic to $X$. It is also shown that $U_D$ is stable for any polarisation, and the connected c…
Let $\Q_n \subset \mathbb P^{n+1}$ be a smooth quadric hypersurface. Here we prove that the tensor product of an $m$-Qregular sheaf on $\Q_n$ and an $l$-Qregular vector bundle on $\Q_n$ is $(m+l)$-Qregular.
Let $X$ be a smooth hypersurface of degree $d$ in $\CP^3$. By totally algebraic calculations, we prove that on the third Demailly-Semple jet bundle $X_3$ of $X$, the bundle $\ocal_{X_3}(1)$ is big for $d\geq 11$, and that on the fourth Demailly-Semple jet bundle $X_4$ of $X$, the bundle $\ocal_{X_4}(1)$ is big for $d\geq 10$, improving a recent result of Diverio.
Let $X \subset \mathbb{P}^n$ be a smooth hypersurface. Given a sequence of integers $\vec{a} = (a_1, \ldots, a_{n-2})$ with $a_1 \leq \cdots \leq a_{n-2}$, let $F_{\vec{a}}(X)$ be the parameter space of lines $L$ on $X$ such that $N_{L/X} \cong \mathcal{O}(a_1) \oplus \cdots \oplus \mathcal{O}(a_{n-2})$. The loci $F_{\vec{a}}(X)$ form a stratification of the Fano scheme of lines on $X$. We show that for general hypersurfaces, the $F_{\vec{a}}(X)$ have the expected dimension and, in this case, compute the class of $\overline{F_{\vec{a}}(X)}$ in the Chow ring of the Grassmannian of lines in $\ma…
Normal Bundles of Rational Normal Curves on Hypersurfaces
本文研究超曲面上有理正规曲线的法丛分裂类型，给出所有可能分裂类型的显式超曲面例子，并计算相应超曲面空间的维数，对二次曲面情形给出最大秩下界。
Let SU_X(3) be the moduli space of semi-stable vector bundles of rank 3 and trivial determinant on a curve X of genus 2. It maps onto P^8 and the map is a double cover branched over a sextic hypersurface called the Coble sextic. In the dual P^8 there is a unique cubic hypersurface, the Coble cubic, singular exactly along the abelian surface of degree 1 line bundles on X. We give a new proof that these two hypersurfaces are dual. As an immediate corollary, we derive a Torelli-type result.
We investigate the logarithmic bundles associated to arrangements of hypersurfaces with a fixed degree in a smooth projective variety. We then specialize to the case when the variety is a quadric hypersurface and a multiprojective space to prove a Torelli type theorem in some cases.
Recently it has been proved that any arithmetically Cohen-Macaulay (ACM) bundle of rank two on a general, smooth hypersurface of degree at least three and dimension at least four is a sum of line bundles. When the dimension of the hypersurface is three, a similar result is true provided the degree of the hypersurface is at least six. We extend these results to complete intersection subvarieties by proving that any ACM bundle of rank two on a general, smooth complete intersection subvariety of sufficiently high multi-degree and dimension at least four splits. We also obtain partial results in t…
We prove a weak version of a bigness criterion for equivariant vector bundles on toric varieties.
We show that the Nakai--Moishezon ampleness criterion holds for real line bundles on complete schemes. As applications, we treat the relative Nakai--Moishezon ampleness criterion for real line bundles and the Nakai--Moishezon ampleness criterion for real line bundles on complete algebraic spaces. The main ingredient of this paper is Birkar's characterization of augmented base loci of real divisors on projective schemes.
We give a cohomological criterion for a parabolic vector bundle on a curve to be semistable. It says that a parabolic vector bundle $E$ with rational parabolic weights is semistable if and only if there is another parabolic vector bundle $F$ with rational parabolic weights such that the cohomologies of the vector bundle underlying the para- bolic tensor product $E \otimes F$ vanish. This criterion generalizes the known semistability criterion of Faltings for vector bundles on curves and significantly improves the result in [Bis07].
A criterion of containment for tropical hypersurfaces
本文利用 Newton 多面体的位似与平移，给出热带 n 变量多项式对应热带超曲面之间包含关系的充要判据。
Holonomy and bundle stability for curves on Fano hypersurfaces
本文证明了在Fano超曲面及射影空间中，存在具有稳定或半稳定法丛的曲线，推广了先前仅限于特定环境空间的结果。
In this note, we give a cohomological characterization of all rank 2 split vector bundles on Hirzebruch surfaces.
We observe that the proof of the Bogomolov stable restriction theorem can be adapted to give an ampleness criterion for globally generated rank 2 vector bundles on certain surfaces. This applies to the Lazarsfeld-Mukai bundles, to congruences of lines in P^3, and possibly to the construction of surfaces with ample cotangent bundle (help welcome!).
The moduli space M_0 of semi-stable rank 2 vector bundles with fixed trivial determinant over a non-hyperelliptic curve C of genus 3 is isomorphic to a quartic hypersurface in P^7 (Coble's quartic). We show that M_0 is self-dual and that its polar map associates to a stable bundle E \in M_0 a bundle F which is characterized by dim H^0(C, E \otimes F) = 4. The projective space PH^0(C, E \otimes F) is equipped with a net of quadrics Πand it is shown that the map which associates to E \in M_0 the isomorphism class of the plane quartic Hessian curve of Πis a dominant map to the moduli space of gen…
Hirzebruch-Milnor classes of hypersurfaces with nontrivial normal bundles and applications to higher du Bois and rational singularities
本文将 Hirzebruch-Milnor 类推广到法丛非平凡且无整体定义函数的超曲面，并证明在射影奇点轨迹假设下，谱类可刻画最小指数，未规范化类可检测高阶 du Bois 与有理奇点。
Let $ \mathcal{D} = \{D_{1}, \ldots, D_{\ell}\} $ be an arrangement of smooth hypersurfaces with normal crossings on the complex projective space $ \mathbb{P}^{n} $ and let $ Ω^{1}_{\mathbb{P}^{n}}(log \mathcal{D}) $ be the logarithmic bundle attached to it. Our aim is to study the injectivity of the correspondence $ \mathcal{D} \longrightarrow Ω^{1}_{\mathbb{P}^{n}}(log \mathcal{D}) $. In order to do that, we first show that $ Ω^{1}_{\mathbb{P}^{n}}(log \mathcal{D}) $ admits a resolution of length $ 1 $ depending on the degrees and on the equations of $ D_{1}, \ldots, D_{\ell} $. Then, we pro…
Let $C$ be a curve of genus two. We denote by $SU_C(3)$ the moduli space of semi-stable vector bundles of rank 3 and trivial determinant over $C$, and by $J^d$ the variety of line bundles of degree $d$ on $C$. In particular, $J^1$ has a canonical theta divisor $Θ$. The space $SU_C(3)$ is a double cover of $P^8=|3Θ|$ branched along a sextic hypersurface, the Coble sextic. In the dual $\check{P}^8=|3Θ|^*$, where $J^1$ is embedded, there is a unique cubic hypersurface singular along $J^1$, the Coble cubic. We prove that these two hypersurfaces are dual, inducing a non-abelian Torelli result. More…
We study smooth quadric surfaces in the Pfaffian hypersurface in $\mathbb{P}^{14}$ parameterising $6 \times 6$ skew-symmetric matrices of rank at most 4, not intersecting the Grassmannian $\mathbb{G}(1,5)$. Such surfaces correspond to quadratic systems of skew-symmetric matrices of size 6 and constant rank 4, and give rise to a globally generated vector bundle $E$ on the quadric. We analyse these bundles and their geometry, relating them to linear congruences of lines in $\mathbb{P}^5$.
Finite automorphism groups of hypersurfaces via Bott vanishing
本文通过Bott消失定理给出上同调判据，证明驯特征下光滑Deligne-Mumford栈中超曲面的自同构群概形离散，并应用于环面轨道簇，获得拟光滑加权射影超曲面及射影空间乘积中超曲面的有限性结果，同时指出方法失效的例外窗口。
In the present note we use rank-2-bundles over ${\bb P}^3$ to construct octic hypersurfaces with many nodes. We give an example with 128 nodes.
Let $X$ be a compact connected Riemann surface of genus $g$, with $g\geq 2$, and ${\cal M}_ξ$ a smooth moduli space of fixed determinant semistable vector bundles of rank $n$, with $n\geq 2$, over $X$. Take a smooth anticanonical divisor $D$ on ${\cal M}_ξ$. So $D$ is a Calabi-Yau variety. We compute the number of moduli of $D$, namely $\dim H^1(D, T_D)$, to be $3g-4 + \dim H^0({\cal M}_ξ, K^{-1}_{{\cal M}_ξ})$. Denote by $\cal N$ the moduli space of all such pairs $(X',D')$, namely $D'$ is a smooth anticanonical divisor on a smooth moduli space of semistable vector bundles over the Riemann su…
We consider principal bundles over homogeneous spaces G/P, where P is a parabolic subgroup of a semisimple and simply connected complex linear algebraic group G. We prove that a holomorphic principal H--bundle, where H is a complex reductive group, is homogeneous if the adjoint vector bundle ad(E) is homogeneous. We also show that E is homogeneous if its associated vector bundle for any finite dimensional faithful H--module is homogeneous.
Let $f_s: X_s \to {\bf {P}}^2$ be the blowing-up of $s$ distinct points and $E$ a vector bundle on $X_s$. Here we give a cohomological criterio which is equivalent to $E \cong f_s^\ast (A)$ with $A$ a direct sum of line bundles. We also some cohomological characterizations of very particular rank 2 vector bundles on ${\bf {P}}^2$.
A criterion for associated bundles and the weak Beauville--Bogomolov decomposition
本文给出纤维丛成为伴随纤维丛的判别准则，证明满足特定条件的满射纤维丛均为伴随纤维丛，进而推出丰反典范除子代数簇的弱博韦尔-博戈莫洛夫分解。
In this paper we give the classification of rank 3 vector bundles without "inner" cohomology on a quadric hypersurface \Q_n (n>3) by studying the associated monads.
It is known that a vector bundle E on a smooth projective curve Y defined over an algebraically closed field is semistable if and only if there is a vector bundle F on Y such that the cohomologies of E\otimes F vanish. We extend this criterion for semistability to vector bundles on curves defined over perfect fields. Let X be a geometrically irreducible smooth projective curve defined over a perfect field k, and let E be a vector bundle on X. We prove that E is semistable if and only if there is a vector bundle F on $X$ such that the cohomologies of E\otimes F vanish. We also give an explicit …
Generalized determinantal representation of hypersurfaces
本文提出了一种超曲面的广义行列式表示方法，并展示了其在向量丛和曲线退化余 locus 中的应用。
Let $X$ be a connected smooth complex projective variety of dimension $n \geq 1$. Let $D$ be a simple normal crossing divisor on $X$. Let $G$ be a connected complex Lie group, and $E_G$ a holomorphic principal $G$-bundle on $X$. In this article, we give criterion for existence of a logarithmic connection on $E_G$ singular along $D$.
Irregular bundles on Hopf surfaces
本文研究经典 Hopf 曲面上秩 2 向量丛的模空间，刻画由不规则丛形成的超曲面。
Hodge rank of ACM bundles and Franchetta's conjecture
本文证明一般高维超曲面上低正则性ACM丛中非平凡Hodge类蕴含平凡直和项，并由此推出次数不低于3、维数至少4的光滑超曲面族不存在通用Ulrich丛，给出Franchetta型结论。
In this paper we show that on a general sextic hypersurface $X\subset \bf P^4$, a rank 2 vector bundle $E$ splits if and only if $h^1(E(n))=0$ for any $n \in \bf Z$. We get thus a characterization of complete intersection curves in $X$
Given a vector bundle $E$ on an irreducible projective variety $X$ we give a necessary and sufficient criterion for $E$ to be a direct image of a line bundle under an étale morphism. The criterion in question is the existence of a Cartan subalgebra bundle of the endomorphism bundle $\text{End}(E)$. As a corollary, a criterion is obtained for $E$ to be the direct image of the structure sheaf under an étale morphism. The direct image of a parabolic line bundle under any ramified covering map has a natural parabolic structure. Given a parabolic vector bundle, we give a similar criterion for it to…
Let X be a smooth hypersurface in projective space. We discuss in this paper when X can be defined by an equation det M = 0 (resp. pf M = 0), where M is a matrix (resp. a skew-symmetric matrix) with homogeneous entries. Standard homological algebra methods show that this is equivalent to produce a line bundle (resp. a rank 2 vector bundle) E of a certain type on X . We discuss a number of applications for hypersurfaces of small dimension. An Appendix by F.-O. Schreyer proves (using Macaulay 2) that a general form of degree d in P^3 (resp. P^4) can be written as the pfaffian of a skew-symmetric…
Here we define the concept of Qregularity for coherent sheaves on quadrics. In this setting we prove analogs of some classical properties. We compare the Qregularity of coherent sheaves on $\Q_n\subset \mathbb P^{n+1}$ with the Castelnuovo-Mumford regularity of their extension by zero in $\mathbb P^{n+1}$. We also classify the coherent sheaves with Qregularity $-\infty$. We use our notion of Qregularity in order to prove an extension of Evans-Griffiths criterion to vector bundles on Quadrics. In particular we get a new and simple proof of the Knörrer's characterization of ACM bundles.
A Cohomological criterion for the splitting of vector bundles on $\mathbb{P}^{n_1}\times\cdots\times\mathbb{P}^{n_s}$
本文提出了一种判断向量丛在乘积射影空间上是否分裂的上同调准则，并推广了在对称乘积射影空间上的消失上同调准则。
We use certain special prehomogeneous representations of algebraic groups in order to construct aCM vector bundles, possibly Ulrich, on certain families of hypersurfaces. Among other results, we show that a general cubic hypersurface of dimension seven admitsan indecomposable Ulrich bundle of rank nine, and that a general cubic fourfold admits an unsplit aCM bundle of rank six.
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