Linearly normal curves with degenerate general hyperplane. Characterization of the projective spaces in this paper, a characterization of the projective space will be given. Linearly normal curves with degenerate general hyperplane section. Its complement, u up a q, coincides with the quotient pp mq m c. Introduction kobayashi and ochiai 1 have given characterizations of the complex projective spaces. A point x 2 cpn corresponds to a line lx hyperplane bundle. First, if we are working over affine space, then the hyperplane is cut out by a global function, so the divisor is principal. Assume that m is not a complex projective space pnc or a complex quadric qnc. Projective planes proof let us take another look at the desargues con. In particular, h is a tautological quaternionic line bundle when the base space is a quaternionic projective space hpn.
A characterization of complex projective spaces by sections. A characterization of complex projective spaces by. Jul 18, 2014 complements of hyperplane subbundles in projective spaces bundles over \\mathbb p1\ adrien dubouloz 1 mathematische annalen volume 361, pages 259 273 2015 cite this article. Coanda, infinitely stably extendable vector bundles on projective spaces. Now consider the case r n, and let pn, pn be two projective spaces. We then study the free family of hyperplane sections of the smooth projective surface x with kodaira dimension. Here we study the holomorphic embeddings of pe into products of projective spaces and the holomorphic line bundles on pe. The hyperplane bundle h on a real projective kspace is defined as follows. In projective space, a hyperplane does not divide the space into two parts. Let us give the first nontrivial example of a vector bundle on pn. Examples lines are hyperplanes of p2 and they form a projective space of dimension 2. Line bundles on projective spaces preliminary draft 3 our description given here. Kobayashiochiai theorem 1 has been applied to obtain many important characterizations of the projective spaces, such.
Linearly normal curves with degenerate general hyperplane section edoardo ballico, nadia chiarli, silvio greco abstract. In the case c 1, we are studying hyperplane bundles i. We establish that the isomorphy type as an abstract algebraic variety of the complement of an ample hyperplane subbundle h of a p r. A direct proof that toric rank 2 bundles on projective space split 3 for some f w 2kwand 0. Vector bundles on projective space university of michigan.
Let us calculate the cohomology of projective space. This provides a good way to visualize threedimensional real linear arrangements. 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. Let v be a complex localizing banach space with countable unconditional basis and e a rank r holomorphic vector bundle on pv. The manifold of all nondegenerate nullpairs \\mathfrakn\ carries a natural kahlerian structure of hyperbolic type and of constant nonzero holomorphic sectional curvature. Main example of regular functions in projective space 19 7. 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. 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. We denote by g kv the grassmannian of kdimensional subspaces of v and by pv g 1v the projective space of v. In this note, we give two applications of 5, theorem 3. 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. Then 95 a line bundle l is positive if and only if lkdescends to an ample line bundle on x.
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. Pdf complements of hyperplane subbundles in projective. 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. The difference between a vector space and the associated af.
The tangent bundle and projective bundle let us give the first. Evan chen spring 2015 1 february 6, 2015 example 1. An affine hyperplane together with the associated points at infinity forms a projective hyperplane. Is every algebraic vector bundle over the ordered con guration space of n points in an a ne line over a eld trivial. The advantage of this extension is the symmetry of homogeneous coordinates. In fact, any vector bundle over affine space is trivial, though this is a hard theorem, the quillensuslin theorem. 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. In the case of projective space, where the tautological bundle is a line bundle, the associated invertible sheaf of sections is. We study linearly normal projective curves with degenerate general hyperplane section, in terms of the amount of degeneracy of it, giving a characterization andor. Hp a u is an arrangement of codimension 1 projective subspaces in cpd. Pglv glvzvwhere glv is the general linear group of v and z. M with respect to a line bundle l, provided that complex subspace v 1. However, for the purposes of this paper, our terminology should be. We first study the free family k of hyperplane sections of the smooth hypersurface x.
The projective space comes equipped with two line bundles, called the universal line bundle and the hyperplane bundle, denoted by opv. In particular, \\mathfrakn\ is a symplectic manifold. Every time you see a map to projective space, you should immediately simultaneously keep in mind the invertible sheaf and sections. A morphism to projective space is given by a line bundle and a choice of. Cotangent bundle over projective space and the manifold of. A point x 2 cpn corresponds to a line lx \\mathbb p1\ adrien dubouloz 1 mathematische annalen volume 361, pages 259 273.
There has been a formidable body of work dedicated to. Both methods have their importance, but thesecond is more natural. 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. Both methods have their importance, but thesecond is. In particular it depends neither on the ambient bundle pe. An introduction to hyperplane arrangements richard p. Projection from a point in pnonto a hyperplane 17 6. Vector bundles on projective space takumi murayama december 1, 20 1 preliminaries on vector bundles let xbe a quasiprojective variety over k. 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. 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. Motivated by this result, in this note, we produce some nonlinear examples, where blow up of projective.
In geometry, a hyperplane of an ndimensional space v is a subspace of dimension n. 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. 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. 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. Complements of hyperplane subbundles in projective spaces bundles over p1 article pdf available in mathematische annalen february 2014 with reads how we measure reads. The total space of h is the set of all pairs l, f consisting of a line l through the origin. Let x be a smooth dm stack of dimension nwhich can be presented as a global quotient and whose coarse moduli space is projective. Riemann surfaces jwr wednesday december 12, 2001, 8. A nondegenerate nullpair of the real projective space \pn\ consists of a point and of a hyperplane nonincident to this point. Complements of hyperplane sub bundles in projective space bundles over p 1 2011. Now this local description can be extended to global homogeneous equations for z.
Two vector bundles over the grassmannian g kv are the. Complements of hyperplane subbundles in projective. An affine space a n together with its ideal hyperplane forms a projective space p n, the projective extension of a n. An arrangement in the complex projective space pnc is a finite collection. Mhas the following holonomy invariant decomposition. Pdf on the families of hyperplane sections of some smooth. Whats abusive about this is that you cant really add a subspace. One special case of a projective hyperplane is the infinite or ideal hyperplane, which is defined with the set of all points at infinity. We next consider holomorphic line bundles over complex projective space. Second chern class of projective space, blown up in a. Complements of hyperplane subbundles in projective space bundles over p 1, preprint arxiv. X be the zero locus of a section of a positive line bundle, then topologically z q 1h for some hyperplane section h.
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. In geometry, the hyperplane separation theorem is a theorem about disjoint convex sets in ndimensional euclidean space. A family of vector spaces over xis a morphism of varieties e. The torelli problem for logarithmic bundles of hypersurface. 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. 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. In general, l k denotes the line bundle corresponding to k2z. In particular it depends neither on the ambient bundle nor on a particular ample hyperplane sub bundle with given rfold selfintersection. So a morphism of spec c h 2i is given by a choice of deformation of the line bundle s.
Explicitly, the projective linear group is the quotient group. 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. Complements of hyperplane subbundles in projective space. Stiefelwhitney classes of a projective space bundle. Pdf on the families of hyperplane sections of some. A canonical treatment of line bundles over general projective spaces. The affine structure of projective space minus a projective hyperplane. We prove that x is determined by the free family k if dimx. Keywords complex space, projective space, line bundle, complete intersected 1. The orbit space is the complex projective space of dimension d, while the orbit map, cd d1zt 0u n cp, z. Complements of hyperplane subbundles in projective spaces. Let m be an irreducible hermitian symmetric space of compact type. Let m be a compact complex space with a line bundle is said to be completely intersected l.
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