Identifying quadric bundle structures on complex projective varieties

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.      

The determinant bundle on the moduli space of stable triples over a curve

We construct a holomorphic Hermitian line bundle over the moduli space of stable triples of the form (E₁, E₂,?), where E₁ and E₂ are holomorphic vector bundles over a fixed compact Riemann surfaceX, and?: E₂ → E₁ is a holomorphic vector bundle homomorphism. The curvature of the Chern connection of this holomorphic Hermitian line bundle is computed. The curvature is shown to coincide with a constant scalar multiple of the natural Kähler form on the moduli space. The construction is based on a result of Quillen on the determinant line bundle over the space of Dolbeault operators on a fixed C^∞ Hermitian vector bundle over a compact Riemann surface.     

On the twisted endomorphism algebra of a vector bundle on a curve

Let X be a smooth projective curve, L∈Pic (X) and E a vector bundle on X. Here we study the bigraded algebra , mainly when E is a general stable bundle on X. We give informations on the degrees of the minimal generators and study a generalization of the notion of projective normality. Entrata in Redazione il 19 gennaio 2000.     

Certain relative quadric hypersurfaces in 3-dimensional projective space bundles over a projective curve

We classify the certain type of relative quadric hypersurfaces of 3-dimensional projective space bundles over a projective line or an elliptic curve whose fiber is the direct product of 2 projective lines.     

The Atiyah bundle and connections on a principal bundle

Let M be a C ^∞ manifold and G a Lie a group. Let E _G be a C ^∞ principal G-bundle over M. There is a fiber bundle C(E _G ) over M whose smooth sections correspond to the connections on E _G . The pull back of E _G to C(E _G ) has a tautological connection. We investigate the curvature of this tautological connection.     

Numerically effective line bundles associated to a stable bundle over a curve

Let G be a complex semisimple group and χ a character of a parabolic subgroup P⊂G such that the associated line bundle on G/P is ample. For a general stable G-bundle E_G over a compact Riemann surface of genus at least two, the line bundle over E_G/P defined by χ has the property that the restriction of  to any closed subvariety of E_G/P of smaller dimension is ample, although is not ample.     

Self-linked curves and normal bundle

We give necessary conditions on the degree and the genus of a smooth, integral curve C⊂P³$C\subset {\mathbb{P}}^3$ to be self-linked (i.e. locus of simple contact of two surfaces). We also give similar results for set theoretically complete intersection curves with a structure of multiplicity three (i.e. locus of 2-contact of two surfaces).     

Cycles of codimension 3 on a projective quadric

Let X be a nonsingular quadratic hypersurface in a projective space over an arbitrary field (of characteristic not two) and let CH^pX be a Chow group of codimension p, that is, a group of classes of codimension p cycles on X with respect to rational equivalency. It is proved that torsion in CH³X is either trivial or is a second order group. Torsion in CH^pX, when p 3, was studied earlier in RZhMat 1990, 9 A334 and 10 A389.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova Akademiya Nauk SSSR, Vol. 191, pp. 114–123, 1991.     

Algebraic cobordisms of a Pfister quadric

In this article we compute the ring of algebraic cobordismsof a Pfister quadric. This is a rare example of a non-cellularvariety where such a computation is known. We consider the algebraiccobordisms * of Levine and Morel, as well as the MGL^{2*, *} ofVoevodsky. The methods of computation in these two cases arequite different. However, the results do agree (which supportsthe expectation that the two theories actually coincide). Weshow that the restriction homomorphism in our case is injectivefor any field extension E/F.     

Resolutions of generic points lying on a smooth quadric

For a 0-dimensional schemeX on a smooth quadricQ we define a special type of resolution of its ideal sheaf as a locally freeO _Q. These resolutions allow to find, for schemes which are generic inQ, the minimal free resolution ofX as a subscheme of ℙ³. For almost all such schemes the graded Betti numbers in ℙ³ depend only on the Hilbert function ofX in ℙ³. Work done with financial support of M.U.R.S.T., while the authors were members of C.N.R.     

Real hypersurfaces in the complex quadric with parallel normal Jacobi operator

First we introduce the notion of parallel normal Jacobi operator for real hypersurfaces in the complex quadric . Next we give a complete classification of real hypersurfaces in the complex quadric with parallel normal Jacobi operator.     

Globally generated vector bundles on a smooth quadric surface

We give the classification of globally generated vector bundles of rank 2 on a smooth quadric surface with c1(2,2)in terms of the indices of the bundles,and extend the result to arbitrary higher rank case.We also investigate their indecomposability and give the sufficient and necessary condition on numeric data of vector bundles for indecomposability.