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ALGEBRAIC FOLIATIONS DEFINED BY QUASI LINES

24 pages
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


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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
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