Classification theory and the study of projective varieties which are covered by rational curves of minimal degrees naturally leads to the study of families of singular rational curves. Since families of arbitrarily singular curves are hard to handle, it has been shown in a previous paper that there exists a partial resolution of singularities which transforms a bundle of possibly badly singular curves into a bundle of nodal and cuspidal plane cubics.
In cases which are of interest for classification theory, the total spaces of these bundles will clearly be projective. It is, however, generally false that an arbitrary bundle of plane cubics is globally projective. For that reason the question of projectivity seems to be of interest, and the present work gives a characterization of the projective bundles.
In this short article, we determine the bigness of the tangent bundle $T_X$ of the projective bundle $X=\mathbb{P}_C(E)$ associated to a vector bundle $E$ on a smooth projective curve $C$.
We study global deformations of certain projective bundles over projective spaces. We show that any projective global deformation of a projective bundle over $\bP^1$ carries the structure of a projective bundle over some projective space. Furthermore, we construct examples in arbitrary dimension $\ge 3$ of almost homogeneous Fano projective bundles over $\bP^2$ which can be globally deformed to non-almost homogeneous manifolds.
We establish that the isomorphy type as an abstract algebraic variety of the complement of an ample hyperplane sub-bundle H of a projective space bundle of rank r-1 over the projective line depends only on the the r-fold self-intersection of H . In particular it depends neither on the ambient bundle nor on a particular ample hyperplane sub-bundle with given r-fold self-intersection. Our proof exploits the unexpected property that every such complement comes equipped with the structure of a non trivial torsor under a vector bundle on the affine line with a double origin.
The aim of this note is to investigate the relation between two types of non-singular projective varieties of Picard rank 2, namely the Projective bundles over Projective spaces and certain Blow-up of Projective spaces.
We study structures of embedded projective manifolds swept out by cubic varieties. We show if an embedded projective manifold is swept out by high-dimensional smooth cubic hypersurfaces, then it admits an extremal contraction which is a linear projective bundle or a cubic fibration. As an application, we give a characterization of smooth cubic hypersurfaces. We also classify embedded projective manifolds of dimension at most five swept out by copies of the Segre threefold P^1\timesP^2. In the course of the proof, we classify projective manifolds of dimension five swept out by planes.
An almost Fano bundle is a vector bundle on a smooth projective variety that its projectivization is an almost Fano variety. In this paper, we prove that almost Fano bundles exist only on almost Fano manifolds and study rank 2 almost Fano bundles over projective spaces.
In previous papers it was shown that the left and right O-module structure of the jet bundles on the projective line differed. In this paper we show that similar statements hold for jet bundles on projective space in any dimension. We also consider some classes associated to jet bundles in the algebraic K-theory of projective space.
We study ample vector bundles on smooth projective stacks. In particular, we prove that the tangent bundle for the weighted projective stack $\mathbb{P}(a_0,...,a_n)$ is ample. A result of Mori shows that the only smooth projective varieties with ample vector bundle are isomorphic to $\mathbb{P}^N$ for some $N$. Extending our geometric spaces from varieties to projective stacks, we are able to provide a new example. This leaves the open question of knowing if the only smooth projective stacks are the weighted projective stacks.
This short paper shows that a projective bundle over a smooth variety with a section can be uniquely recovered from the embedding of the section in its second infinitesimal neighborhood within the bundle.
We show the existence of rank 6 Ulrich bundles on a smooth cubic fourfold. First, we construct a simple sheaf E of rank 6 as an elementary modification of an ACM bundle of rank 6 on a smooth cubic fourfold. Such an E appears as an extension of two Lehn-Lehn-Sorger-van Straten sheaves. Then we prove that a general deformation of E(1) becomes Ulrich. In particular, this says that general cubic fourfolds have Ulrich complexity 6.
In this paper, we study the structure of projective space bundles whose relative anti-canonical line bundle is nef. As an application, we get a characterization of abelian varieties up to finite etale covering.
We give several mild conditions on a toric bundle on a nonsingular toric variety under which the projectivization of the toric bundle is Frobenius split.
The automorphism group of a projective bundle P(E) over a simplicial toric variety is described when the bundle E is a direct sum of line bundles. Applications to study of moduli of complete intersections on toric varieties, including Calabi Yau m-folds and Del Pezzo surfaces, are indicated.
We define Schwarzenberger bundles on any smooth projective variety X. We introduce the notions of jumping pairs of a Steiner bundle E on X and determine a bound for the dimension of its jumping locus. We completely classify Steiner bundles whose set of jumping pairs have maximal dimension, proving that they are all Schwarzenberger bundles.
Given two equivariant vector bundles over an algebraic GKM manifold with the same equivariant Chern classes, we show that the genus zero equivariant Gromov--Witten theory of their projective bundles are naturally isomorphic.
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.
We investigate the universal cover of projective threefolds whose tangent bundle is a direct sum of subbundles in case the Kodaira dimension is not 1 and 2. We also prove results on Fano manifolds with splitting tangent bundles in any dimension.
Hartshorne's conjecture about vector bundles on projective space states that any rank 2 vector bundle on n-dimensional projective space splits as soon as n is at least 7. Klyachko has shown that Hartshorne's conjecture is true when the vector bundles are torus equivariant. Moreover, recent work of Ilten and Süss generalizes Klyachko's work to the case of a smaller rank torus action on projective space. In this note we give a new, direct proof that torus rank 2 bundles split that avoids a description of the category of torus equivariant vector bundles.
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 prove an elementary but somewhat unexpected result about projective embeddings of smooth varieties X whose cotangent bundles are numerically effective. Specifically, we show that the degree of X in any projective embedding must grow (almost) exponentially in the dimension of X.
Our goal is to study the syzygies of the projective embeddings defined by ample line bundles on a fake projective plane S. The syzygies are studied in terms of the property $N_p$. For various kinds of ample line bundles, we give explicit lower bounds for their powers above which the property $N_p$ is satisfied.
We show that the moduli space of rank three toric vector bundles on smooth projective toric varieties satisfies Murphy's law, that is, every singularity type of finite type arises somewhere on this space.
We study certain moduli spaces of stable vector bundles of rank two on cubic and quartic threefolds. In many cases under consideration, it turns out that the moduli space is complete and irreducible and a general member has vanishing intermediate cohomology. In one case, all except one component of the moduli space has such vector bundles.
Generalizing work done by Miyaoka and others in the case of divisors and of curves, we compute the cones of effective cycles of arbitrary dimension on a projective bundle over a complex projective curve in terms of the numerical data in an associated Harder-Narasimhan filtration. An application to cycles on projective bundles over a smooth complex projective base of arbitrary dimension is also given.
An explicit projective embedding of the moduli space of marked cubic surfaces is given. This embedding is equivariant under the Weyl group of type E6. The image is defined by a system of linear and cubic equations. To express the embedding in a most symmetric way, the target would be 79-dimensional, however the image lies in a 9-dimensional linear subspace.
In this article we prove new results on projective normality and normal presentation of adjunction bundle associated to an ample and globally generated line bundle on higher dimensional smooth projective varieties with nef canonical bundle. As one of the consequences of the main theorem, we give bounds on very ampleness and projective normality of pluricanonical linear systems on varieties of general type in dimensions three, four and five. These improve known such results.
We consider some conditions under which a smooth projective variety X is actually the projective space. We also extend to the case of positive characteristic some results in the theory of vector bundle adjunction. We use methods and techniques of the so called Mori theory, in particular the study of rational curves on projective manifolds.
In this paper, we attempt to determine the quantum cohomology of projective bundles over the projective space P^n. In contrast to the previous examples, the relevant moduli spaces in our case frequently do not have expected dimensions. It makes the calculation more difficult. We overcome this difficulty by using excessive intersection theory.
We systematically study the splitting of vector bundles on a smooth, projective variety, whose restriction to the zero locus of a regular section of an ample vector bundle splits. First, we find ampleness and genericity conditions which ensure that the splitting of the vector bundle along the subvariety implies its global splitting. Second, we obtain a simple splitting criterion for vector bundles on the Grassmannian and on partial flag varieties.
We classify the Ulrich vector bundles of arbitrary rank on smooth projective varieties of minimal degree. In the process, we prove the stability of the sheaves of relative differentials on rational scrolls.
We give a canonical birational map between the moduli space of pfaffian vector bundles on a cubic surface and the space of complete pentahedra inscribed in the cubic surface. The universal situation is also considered, and we obtain a rationality result. As a by-product, we provide an explicit normal form for five general lines in $P_5$. Applications to the geometry of Palatini threefolds and Debarre-Voisin's Hyper-Kähler manifolds are also discussed.
We generalise the variant of the Babylonian tower theorem for vector bundles on projective spaces proved by I. Coanda and G. Trautmann (2006) to the case of principal $G$-bundles over projective spaces, where $G$ is a linear algebraic group defined over an algebraically closed field. In course of the proofs some new insight into the structure of such principal $G$-bundles is obtained.
We determine the splitting (isomorphism) type of the normal bundle of a generic genus-0 curve with 1 or 2 components in any projective space, as well as the (sometimes nontrivial) way the bundle deforms locally with a general deformation of the curve. We deduce an enumerative formula for divisorial loci of smooth rational curves whose normal bundle is of non-generic splitting type.
In this paper we characterize smooth complex projective varieties that admit a quadric bundle structure on some dense open subset in terms of the geometry of certain families of rational curves.
We extend our classification of special Cremona transformations whose base locus has dimension at most three to the case when the target space is replaced by a (locally) factorial complete intersection.
According to Horrocks (1966), a vector bundle E on the projective n-space extends stably to the projective N-space, N>n, if there exists a vector bundle on the larger space whose restriction to the smaller one is isomorphic to E plus a direct sum of line bundles. We show that E extends stably to the projective N-space for every N>n if and only if E is the cohomology of a free monad (with three terms). The proof uses the method of Coanda and Trautmann (2006). Combining this result with a theorem of Mohan Kumar, Peterson and Rao (2003), we get a new effective version of the Babylonian tower theo…
Let $U^{'s}_L(n,d)$ be the moduli space of stable vector bundles of rank $n$ with determinant $L$ where $L$ is a fixed line bundle of degree $d$ over a nodal curve $Y$. We prove that the projective Poincare bundle on $Y \times U^{'s}_L(n,d)$ and the projective Picard bundle on $U^{'s}_L(n,d)$ are stable for suitable polarisation. For a nonsingular point $x \in Y$, we show that the restriction of the projective Poincare bundle to $x \times U^{'s}_L(n,d)$ is stable for any polarisation. We prove that for arithmetic genus $g\ge 3$ and for $g=n=2, d$ odd, the Picard group of the moduli space $U'_L…
We give a complete classification of P1-bundles over a projective manifold of Picard number one which admit another smooth morphism of relative dimension one.
We prove that the kernel bundle of the evaluation morphism of global sections, namely the syzygy bundle, of a sufficiently ample line bundle on a smooth projective variety is slope stable with respect to any polarization. This settles a conjecture of Ein-Lazarsfeld-Mustopa.
In this article, we study Newton-Okounkov bodies on projective vector bundles over curves. Inspired by Wolfe's estimates used to compute the volume function on these varieties, we compute all Newton-Okounkov bodies with respect to linear flags. Moreover, we characterize semi-stable vector bundles over curves via Newton-Okounkov bodies.
We show that there exists a cubic threefold defined by an invertible polynomial that, when quotiented by the maximal diagonal symmetry group, has a derived category which does not have a full exceptional collection consisting of line bundles. This provides a counterexample to a conjecture of Lekili and Ueda.
In this note we give a complete description of all the hyperplane section of the projective bundle associated to the tangent bundle of $\mathbb{P}^2$ under its natural embedding in $\mathbb{P}^7.$ As an application one obtains a description of all possible deformations, in $\text{SL_3(\mathbb{C}}/B,$ of the co-dimension one sub-scheme which is the union of two fundamental Schubert divisors.
We investigate the jumping conics of stable vector bundles $\Ee$ of rank 2 on a smooth quadric surface $Q$ with the Chern classes $c_1=\Oo_Q(-1,-1)$ and $c_2=4$ with respect to the ample line bundle $\Oo_Q(1,1)$. We describe the set of jumping conics of $\Ee$, a cubic symmetroid in $\PP_3$, in terms of the cohomological properties of $\Ee$. As a consequence, we prove that the set of jumping conics, $S(\Ee)$, uniquely determines $\Ee$. Moreover we prove that the moduli space of such vector bundles is rational.
The Segre cubic and Castelnuovo-Richmond quartic are two projectively dual hypersurfaces in $\mathbb{P}^4$, with a long and rich history starting in the 19th century. We will explain how Kuznetsov's theory of homological projective duality lifts this projective duality to a relationship between the derived category of a small resolution of the Segre cubic and a small resolution of the Coble fourfold, the double cover of $\mathbb{P}^4$ ramified along the Castelnuovo-Richmond quartic. Homological projective duality then provides a description of the derived categories of linear sections, which w…
We use the cut and paste relation $[Y^{[2]}] = [Y][\mathbb{P}^m] + \mathbb{L}^2 [F(Y)]$ in $K_0(\text{Var}_k)$ of Galkin--Shinder for cubic hypersurfaces arising from projective geometry to characterize cubic hypersurfaces of sufficiently high dimension under certain numerical or genericity conditions. Removing the conditions involving the middle Betti number from the numerical conditions used extends the possible $Y$ to cubic hypersurfaces, complete intersections of two quadric hypersurfaces, or complete intersections of two quartic hypersurfaces. The same method also gives a family of other …
In an earlier paper we conjectured a relation between the quantum $\mathcal D$-modules of a smooth variety $X$ and the projectivisation of a direct sum of line bundles over it. In this paper we prove the conjecture when $X$ is a complete intersection in a toric variety. We also use the conjecture to show that the relations of the small quantum cohomology ring of $X$ that come from differential operators lift to the projective bundle. The basic cohomology relation of the projective bundle deforms to a relation in the small quantum cohomology.
