Hyperplane bundle projective space pdf

A morphism to projective space is given by a line bundle and a choice of. In particular, h is a tautological quaternionic line bundle when the base space is a quaternionic projective space hpn. In general, l k denotes the line bundle corresponding to k2z. Aug 31, 2011 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 r1 over the projective line.

Complements of hyperplane subbundles in projective spaces bundles over p1 article pdf available in mathematische annalen february 2014 with reads how we measure reads. We denote by g kv the grassmannian of kdimensional subspaces of v and by pv g 1v the projective space of v. Main example of regular functions in projective space 19 7. 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 r1 over the projective line depends only on the the rfold selfintersection of h. Complements of hyperplane sub bundles in projective spaces bundles over p1 article pdf available in mathematische annalen february 2014 with reads how we measure reads. Pdf on the families of hyperplane sections of some. Pdf complements of hyperplane subbundles in projective. There has been a formidable body of work dedicated to.

Let v be a complex localizing banach space with countable unconditional basis and e a rank r holomorphic vector bundle on pv. Hp a u is an arrangement of codimension 1 projective subspaces in cpd. In fact, any vector bundle over affine space is trivial, though this is a hard theorem, the quillensuslin theorem. Let us calculate the cohomology of projective space. Let us give the first nontrivial example of a vector bundle on pn. The torelli problem for logarithmic bundles of hypersurface. In particular, \\mathfrakn\ is a symplectic manifold. In geometry, a hyperplane of an ndimensional space v is a subspace of dimension n.

A direct proof that toric rank 2 bundles on projective space split 3 for some f w 2kwand 0. A nondegenerate nullpair of the real projective space \pn\ consists of a point and of a hyperplane nonincident to this point. We then study the free family of hyperplane sections of the smooth projective surface x with kodaira dimension. Examples lines are hyperplanes of p2 and they form a projective space of dimension 2. In projective space, a hyperplane does not divide the space into two parts. Stiefelwhitney classes of a projective space bundle. Evan chen spring 2015 1 february 6, 2015 example 1. Line bundles on projective space daniel litt we wish to show that any line bundle over pn k is isomorphic to om for some m. Projection from a point in pnonto a hyperplane 17 6. For instance, if a consists of the three coordinate hyperplanes x1 0, x2 0, and x3 0, then a projective drawing is given by 2 1 3 the line labelled i is the projectivization of the hyperplane xi 0. Keywords complex space, projective space, line bundle, complete intersected 1.

Coanda, infinitely stably extendable vector bundles on projective spaces. In mathematics, especially in the group theoretic area of algebra, the projective linear group also known as the projective general linear group or pgl is the induced action of the general linear group of a vector space v on the associated projective space pv. A point x 2 cpn corresponds to a line lx \\mathbb p1\ adrien dubouloz 1 mathematische annalen volume 361, pages 259 273. A characterization of complex projective spaces by sections. We study linearly normal projective curves with degenerate general hyperplane section, in terms of the amount of degeneracy of it, giving a characterization andor. In one version of the theorem, if both these sets are closed and at least one of them is compact, then there is a hyperplane in between them and even two parallel hyperplanes in between them separated by a gap. In the case c 1, we are studying hyperplane bundles i. Pglv glvzvwhere glv is the general linear group of v and z. We prove that x is determined by the free family k if dimx. The space v may be a euclidean space or more generally an affine space, or a vector space or a projective space, and the notion of hyperplane varies correspondingly since the definition of subspace differs in these settings. Complements of hyperplane subbundles in projective. We study linearly normal projective curves with degenerate general hyperplane section, in terms of the amount of degeneracy of it, giving a characterization andor a description of such curves. X be the zero locus of a section of a positive line bundle, then topologically z q 1h for some hyperplane section h.

The difference between a vector space and the associated af. Assume that m is not a complex projective space pnc or a complex quadric qnc. A point x 2 cpn corresponds to a line lx hyperplane bundle. The tangent bundle and projective bundle let us give the first. Vector bundles on projective space takumi murayama december 1, 20 1 preliminaries on vector bundles let xbe a quasiprojective variety over k. However, for the purposes of this paper, our terminology should be. Both methods have their importance, but thesecond is more natural. The projective space comes equipped with two line bundles, called the universal line bundle and the hyperplane bundle, denoted by opv. In particular it depends neither on the ambient bundle nor on a particular ample hyperplane sub bundle with given rfold selfintersection.

This provides a good way to visualize threedimensional real linear arrangements. In geometry, the hyperplane separation theorem is a theorem about disjoint convex sets in ndimensional euclidean space. Complements of hyperplane subbundles in projective space bundles over p 1, preprint arxiv. Kobayashiochiai theorem 1 has been applied to obtain many important characterizations of the projective spaces, such. An affine hyperplane together with the associated points at infinity forms a projective hyperplane. Dual bundle 6,6 dual cell decomposition 5,3 dual curve 26, 4 dual kummer surface 76,3 dual projective space 15, dual schubert cycle 20,0 effective divisor, 238 effective 0cycle 66,7 elementary divisors 30,6 elementary invariant polynomials 402, elliptic curve 222, 225, 238, 286, 564, 575, 586. We next consider holomorphic line bundles over complex projective space. Complements of hyperplane sub bundles in projective space bundles over p 1 2011. The orbit space is the complex projective space of dimension d, while the orbit map, cd d1zt 0u n cp, z. Cotangent bundle over projective space and the manifold of. Mhas the following holonomy invariant decomposition.

A characterization of complex projective spaces by. We establish that the isomorphy type as an abstract algebraic variety of the complement of an ample hyperplane sub bundle h of a p r. We establish that the isomorphy type as an abstract algebraic variety of the complement of an ample hyperplane subbundle h of a projective space bundle of rank r1 over the projective line. An arrangement in the complex projective space pnc is a finite collection. The anticanonical line bundle o1 is its dual, which should go to 1 2z. There is a tautological bundle over cpn, denoted o 1 for reasons which will soon become clear. The hyperplane bundle h on a real projective kspace is defined as follows.

Characterization of the projective spaces in this paper, a characterization of the projective space will be given. We first study the free family k of hyperplane sections of the smooth hypersurface x. Two vector bundles over the grassmannian g kv are the. Is every algebraic vector bundle over the ordered con guration space of n points in an a ne line over a eld trivial. 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 r1 over the projective line. Both methods have their importance, but thesecond is. Riemann surfaces jwr wednesday december 12, 2001, 8. Second chern class of projective space, blown up in a. M with respect to a line bundle l, provided that complex subspace v 1. Let x be a smooth dm stack of dimension nwhich can be presented as a global quotient and whose coarse moduli space is projective. An introduction to hyperplane arrangements richard p. A point x 2 cpn corresponds to a line lx projective space, and hence by the yoneda philosophy, this can be taken as the denition of projective space.

So a morphism of spec c h 2i is given by a choice of deformation of the line bundle s. Algebraic vector bundles over complements of hyperplane arrangements in affine spaces over a field igor kriz in questions related to algebraic models of chiral conformal eld theory 1, the author encountered the following question. Linearly normal curves with degenerate general hyperplane. Its complement, u up a q, coincides with the quotient pp mq m c. An affine space a n together with its ideal hyperplane forms a projective space p n, the projective extension of a n. Now consider the case r n, and let pn, pn be two projective spaces. Motivated by this result, in this note, we produce some nonlinear examples, where blow up of projective. One special case of a projective hyperplane is the infinite or ideal hyperplane, which is defined with the set of all points at infinity. Linearly normal curves with degenerate general hyperplane section. Here we study the holomorphic embeddings of pe into products of projective spaces and the holomorphic line bundles on pe. First, if we are working over affine space, then the hyperplane is cut out by a global function, so the divisor is principal. X is the projection to the coarse moduli space, then q o x o x, so an invertible sheaf on x has the same global sections 110 when pulled back to x. Let m be a compact complex space with a line bundle is said to be completely intersected l. Then 95 a line bundle l is positive if and only if lkdescends to an ample line bundle on x.

Now this local description can be extended to global homogeneous equations for z. Complements of hyperplane subbundles in projective spaces. In this note, we give two applications of 5, theorem 3. Introduction kobayashi and ochiai 1 have given characterizations of the complex projective spaces. A family of vector spaces over xis a morphism of varieties e. The manifold of all nondegenerate nullpairs \\mathfrakn\ carries a natural kahlerian structure of hyperbolic type and of constant nonzero holomorphic sectional curvature. Line bundles on projective spaces preliminary draft 3 our description given here. In the case of projective space, where the tautological bundle is a line bundle, the associated invertible sheaf of sections is. We establish that the isomorphy type as an abstract algebraic variety of the complement of an ample hyperplane subbundle h of a p r. The total space of h is the set of all pairs l, f consisting of a line l through the origin. In particular it depends neither on the ambient bundle pe.