24 Pages
English
Gain access to the library to view online
Learn more

ALGEBRAIC FOLIATIONS DEFINED BY QUASI LINES

Gain access to the library to view online
Learn more
24 Pages
English

Description

ALGEBRAIC FOLIATIONS DEFINED BY QUASI-LINES LAURENT BONAVERO AND ANDREAS HÖRING Abstract. Let X be a projective manifold containing a quasi-line l. An important di?erence between quasi-lines and lines in the projective space is that in general there is more than one quasi-line passing through two given general points. In this paper we use this feature to construct an algebraic foliation associated to a family of quasi-lines. We prove that if the singular locus of this foliation is not too large, it induces a rational fibration on X that maps the general leaf of the foliation onto a quasi-line in a rational variety. 1. Introduction 1.A. Motivation. Let X be a complex quasiprojective manifold of dimension n. A quasi-line l in X is a smooth rational curve f : P1 ?? X such that f?TX is the same as for a line in Pn, i.e. is isomorphic to OP1(2)?OP1(1)?n?1. Quasi-lines have some of the deformation properties of lines, but there are important differences: for example if x and y are general points in X there exist only finitely many deformations of l passing through the two points, but in general we do not have uniqueness1. It is now well established that given a variety X with a quasi-line l, the deformations and degenerations of l contain interesting information on the global geometry of X .

  • unique fx-leaf

  • line l? ?

  • smooth centers

  • general leaf

  • complex field

  • foliation

  • unique saturated algebraic

  • projective manifold

  • algebraic foliations

  • line through


Subjects

Informations

Published by
Reads 7
Language English

Exrait

ALGEBRAIC
FOLIATIONS DEFINED BY QUASI-LINES
LAURENT BONAVERO AND ANDREAS HÖRING
Abstract.LetXbe a projective manifold containing a quasi-linel. An important difference between quasi-lines and lines in the projective space is that in general there is more than one quasi-line passing through two given general points. In this paper we use this feature to construct an algebraic foliation associated to a family of quasi-lines. We prove that if the singular locus of this foliation is not too large, it induces a rational fibration onXthat maps the general leaf of the foliation onto a quasi-line in a rational variety.
1.Introduction
1.A.Motivation.LetXbe a complex quasiprojective manifold of dimensionn. A quasi-linelinXis asmoothrational curvef:P1֒Xsuch thatfTXis the same as for a line inPn,i.e.is isomorphic to OP1(2)⊕ OP1(1)n1Quasi-lines have some of the deformation properties of lines, but there are important differences: for example ifxandyare general points inXthere exist only finitely many deformations oflthrough the two points, but in general we do notpassing have uniqueness1 is now well established that given a variety. ItXwith a quasi-line l, the deformations and degenerations oflcontain interesting information on the global geometry ofX. Here is an example of such a result, due to Ionescu and Voica.
1.1. Theorem.[IV03, Thm.1.12]LetXbe a projective manifold containing a quasi-linel there exists a divisor. AssumeDsuch thatDl= 1and h0(X,OX(D)) =s+ 12 there exists a small deformation. Thenlofl, a fi-˜ nite composition of smooth blow-upsσ:XXwith smooth centers disjoint from ˜ land a surjective fibrationϕ:XPswith rationally connected general fibre such thatϕmaps isomorphicallyσ1(l)to a line inPs.2
A disadvantage of this statement is thata priorithere seems to be no relation between the geometry of the quasi-lineland the existence of the divisorD. The goal of this paper is to fill this gap by a construction inspired by the theory of
Date: July 27, 2009. 2000Mathematics Subject Classification.37F75, 32S65, 14D06, 14J30, 14J40, 14N10. Key words and phrases.rational curves, quasi-line, rationally connected manifold, holomorphic foliation, algebraic leaves. 1We denote bye(X l)the number of quasi-lines through two general points, see Definition 1.12 for a formal definition. 2In order to simplify the statements, we’ll simply say that there exists a rational fibration ϕ: (X l)99K(Psline). 1
complex projective manifoldsX these have been studiedswept out by linear spaces: for more than twenty years (see [Ein85, ABW92, Sat97, NO07]) and an observation common to all these papers is that if the codimension of the linear space is small, then eitherXis special (a projective space, hyperquadricetc.) or it admits a fibration such that the fibres are linear spaces. A powerful tool in their theory is the family of lines contained in the linear spaces. The guiding philosophy of this paper is that the rich geometry of a family of quasi-lines can be used to construct a natural family of subvarieties that induces a (rational) fibration onX.
1.B.Setup and main results.LetXbe a projective manifold of dimensionn containing a quasi-lineltool used in this paper is an intrinsic foliation. The main Fxassociated to the quasi-lines passing through a general pointxofX. In case the foliation has rankn1, its leaves are natural candidates to play the role of the divisorD foliation Thein Theorem 1.1.FxTXis defined by the following heuristic principle:
“forygeneral inX, the (closure of the)Fx-leaf throughyis the smallest subvarietyVXcontainingyand such that for everyz inV, every quasi-line throughxandzis entirely contained inV”.
In a more technical language (see Section 2) we prove the following theorem.
1.2. Theorem.LetXbe a projective manifold containing a quasi-lineland let Hx⊂ C(X)be the scheme parametrising deformations and degenerations oflpass-ing through a general pointxX. Then there exists a unique saturated algebraic foliationFxTXsuch that for every general pointyX, the uniqueFx-leaf (cf. Defn. 1.13) throughyis the minimalHx-stable projective subvariety throughy.
Iflis a line or more generally ife(X, l) = 1, the foliationFxhas rank one: the leaf through a general pointyis the unique quasi-line passing throughxandy. This leads immediately to the following question.
1.3. Question.LetXbe a projective manifold containing a quasi-linel. Letx be a general point inX, and denote byFx we Canthe corresponding foliation. construct a rational fibration ϕ: (X, l)99K(Y, l:=ϕ(l))
onto a projective varietyYsuch that lis a quasi-line withe(Y, l) = 1, and the generalFx-leaves are preimages of deformations ofl?
Suppose for a moment that such a fibration exists: fix two general pointsxandy inX, and denote byFxandFy hypothesis the Bythe corresponding foliations. uniqueFx-leaf throughyis the preimage of a quasi-line throughϕ(x)andϕ(y). Analogously the uniqueFy-leaf throughxis the preimage of a quasi-line through ϕ(y)andϕ(x). Since both quasi-lines are deformations oflpassing through two given general points the conditione(Y, l) = 1implies that they are identical. Hence the two leaves are identical. More formally we have a natural necessary condition for the existence of the fibration. 2