COUNTING CONICS IN COMPLETE INTERSECTIONS

COUNTING CONICS IN COMPLETE INTERSECTIONS

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COUNTING CONICS IN COMPLETE INTERSECTIONS LAURENT BONAVERO, ANDREAS HORING Abstract. We count the number of conics through two general points in complete intersections when this number is finite and give an application in terms of quasi-lines. 1. Introduction Let X be a complex projective 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. Let X be a smooth projective variety containing a quasi-line l. Following Ionescu and Voica [IV03], we denote by e(X, l) the number of quasi-lines which are deformations of l and pass through two given general points of X. We denote by e0(X, l) the number of quasi-lines which are deformations of l and pass through a general point x of X with a given general tangent direction at x. Note that one always has e0(X, l) ≤ e(X, l), but in general the inequality may be strict [IN03, p.1066]. 1.1. Theorem. Let X ? Pn+r be a general smooth n-dimensional complete intersec- tion of multi-degree (d1, .

  • equations

  • general conic

  • plane πc

  • plane

  • zero locus

  • pass through

  • conic-connected

  • equations corresponding

  • unique plane


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Ajouté le 18 juin 2012
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COUNTING CONICS IN COMPLETE INTERSECTIONS
¨ LAURENT BONAVERO, ANDREAS HORING
Abstract.We count the number of conics through two general points in complete intersections when this number is finite and give an application in terms of quasilines.
1.Introduction LetXbe a complex projective manifold of dimensionn. A quasilinelinXis a 1n smoothrational curvef:P֒Xsuch thatf TXis the same as for a line inP,i.e. is isomorphic to n1 O1(2)⊕ O1(1). P P LetXbe a smooth projective variety containing a quasilinelIonescu and. Following Voica [IV03], we denote bye(X, l) the number of quasilines which are deformations of land pass through two given general points ofX. We denote bye0(X, l) the number of quasilines which are deformations ofland pass through a general pointxofXwith a given general tangent direction atx. Note that one always hase0(X, l)e(X, l), but in general the inequality may be strict [IN03, p.1066]. n+r 1.1. Theorem.LetXPbe ageneralsmoothndimensional complete intersec tion of multidegree(d1, . . . , dr). Assume moreover that n+ 1 d1+∙ ∙ ∙+dr= +r. 2 Then (1)the family of conics contained inXis a nonempty, smooth and irreducible component of the Chow schemeC(X), (2)a general conicCcontained inXis a quasiline ofXand r Y 1 e0(X, C) =e(X, C() = di1)!di!. 2 i=1 The numerical assumptiond1+∙ ∙ ∙+dr= (n+ 1)/2 +rassures that ifCis a conic in X, thenKXC=nThis numerical condition is of course necessary for a curve+ 1. to be a quasiline. Note that varieties appearing in our theorem are Fano varieties of dimensionnand index (n+1)/2; they are well known to be the boundary Fano varieties with Picard number one being conicconnected (see [IR07], Theorem 2.2). Using a degeneration argument, one can strengthen parts of the statement. n+r 1.2. Corollary.LetXPbe a smoothndimensional complete intersection of multidegree(d1, . . . , dr). Ifd1+∙ ∙ ∙+dr= (n+ 1)/2 +r, the varietyXcontains a conic that is a quasiline.
Date: September 9, 2009. Keywords :conics, quasilines, complete intersections.A.M.S. classification :14N10, 14M10.
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