Covered by lines and Conic connected varieties

LE MATEMATICHE Vol. LXVI (2011) – Fasc. II, pp. 137–151 doi: 10.4418/2011.66.2.12 COVERED BY LINES AND CONIC CONNECTED VARIETIES S. MARCHESI - A. MASSARENTI - S. TAFAZOLIAN We study some properties of an embedded variety covered by lines and give a numerical criterion ensuring the existence of a singular conic through two of its general points. We show that our criterion is sharp. Conic-connected, covered by lines, QEL, LQEL, prime Fano, defective, and dual defective varieties are closely related. We study some relations between the above mentioned classes of objects using celebrated results by Ein and Zak. Contents 1 Notation and Preliminaries 139 2 Some results by Ein and Zak 141 3 Prime Fano varieties and varieties covered by lines 142 4 Some results on Conic Connectedness 145 Entrato in redazione: 30 maggio 2011 AMS 2010 Subject Classification: 14MXX, 14NXX, 14J45, 14M07. Keywords: Conic-connected varieties, Covered by lines, Dual and secant defective, Hartshorne Conjecture. 138 SIMONE MARCHESI - ALEX MASSARENTI - SAEED TAFAZOLIAN Introduction The study of rational curves on algebraic varieties is of fundamental importance in algebraic geometry. Indeed the birational geometry of a smooth projective variety is closely related to the rational curves it contains. Many tools have been introduced for this purpose, such as Mori theory (see [12]) which has been a great breakthrough in the theory of minimal models. These issues naturally lead to the study of varieties covered by rational curves and rationally connected varieties. Over an algebraically closed field of characteristic zero a variety X is said to be uniruled if for x ∈ X general point there exists a rational curve on X through x, while X is rationally connected if two general points x, y ∈ X can be connected by a rational curve on X. This subject can be explored in an abstract context, for instance by Hwang and Kebekus in [7], or in an embedded setting, by Ionescu and Russo in [8]. An intermediate point of view is to consider varieties polarized by ample divisors, for instance by Lanteri and Palleschi in [11]. We shall place ourselves in an embedded context, that is considering the variety X as a subvariety of a projective space PN and using techniques coming from classical algebraic geometry. From this point of view the so called variety of minimal rational tangents of X at a point x (see [6]) is simply the variety Lx parameterizing lines contained in X through x. The simplest case of rational connectedness to study, in the embedded setting, is the existence of a line through two generic points; clearly this is not interesting because a variety with this property is necessarily a linear space. A more interesting case is given by considering the next one, i.e. when two general points can be connected by a conic; varieties with this property are called conic-connected, CC for short. Such property is mostly studied in the context of covered by lines, secant defective, QEL, and LQEL varieties. Consider a smooth irreducible n-dimensional complex variety X ⊂ PN (we will denote by c its codimension), its secant variety SX is the closure of the locus of its secant lines. The secant defect of X is the number δ (X) = 2n + 1 − dim(SX); the variety is called secant defective if δ (X) ≥ 1. The locus determined on X by the cone of secant lines through a general point z ∈ SX is called the entry locus of X and denoted by Ez , note that Ez is a purely δ -dimensional subvariety of X. The variety X is said to be QEL (quadratic entry locus) if Ez is a quadric, while X is a LQEL variety (local quadratic entry locus) if for x, y ∈ X general points there is a quadric Qx,y ⊂ X through x, y. These classes of varieties have been widely studied by Ionescu and Russo in [13], [10] and [8]. In section 1 we will give preliminary notions and some notation and in section 2 we will recall two basic theorems due to Zak and Ein, see [15] and [4]. In section 3 we concentrate on non trivial relations between these classes of COVERED BY LINES AND CONIC CONNECTED VARIETIES 139 varieties, dual defective and prime Fano varieties, mainly using Zak’s Theorem on tangencies, Ein’s classification of dual defective varieties and properties of Lx . Indeed X inherits significant properties from the geometry of Lx ; see [9] for a discussion on this issue. In particular we show that if a ≥ n − c holds, where a := dim(Lx ), we also have a ≤ n+c−3 2 . Moreover if the last bound is an equality, well known varieties naturally arise such as the Grassmannian G(1, 4) ⊂ P9 (lines in P4 ), and the Spinor variety S10 ⊂ P15 . We highlight a relation between varieties covered by lines such that a ≥ n − c and the Hartshorne conjecture on complete intersections (Conjecture 3.4). In section 4 we study conics on X and get the following result (Theorem 4.3). Theorem 0.1. Let X ⊂ PN be a variety set theoretically defined by homogeneous polynomials Gi of degree di , for i = 1, . . . , m. If m ∑ di ≤ i=1 N +m , 2 then X is connected by singular conics. Assume X to be smooth and the equations Gi ’s to be scheme theoretical equations for X and in decreasing order of degrees. If c ∑ di ≤ i=1 N +c , 2 where c = N − n, then X is conic-connected by smooth conics also. This result is closely related to a result obtained by Bonavero and Höring in [2], which gives a numerical criterion for conic-connectedness. However while Bonavero and Höring only consider schematic smooth complete intersections we allow X to be singular and give a condition ensuring the existence of a singular conic through two general points. Furthermore, in Remark 4.7, we show that our inequality is sharp considering a smooth cubic hypersurface in P4 . 1. Notation and Preliminaries We work over the complex field. We mainly follow notation and definitions of [9]. Throughout this paper we denote by X ⊂ PN a smooth irreducible variety of dimension n ≥ 1. We assume X to be non-degenerate of codimension c, so that N = n + c. If x ∈ X, we write Tx X for the projective closure of the embedded Zariski tangent space of X at x. 140 SIMONE MARCHESI - ALEX MASSARENTI - SAEED TAFAZOLIAN The Secant Variety Let X ⊂ PN be a closed, irreducible subvariety of dimension n. Consider the following incidence variety, SX , called the abstract secant variety of X: SX = {(x, x0 ,t)| x, x0 ∈ X, x 6= x0 ,t ∈ hx, x0 i ⊂ PN } ⊂ X × X × PN , with SX irreducible, of dimension 2n + 1. Definition 1.1. Let X ⊂ PN be an irreducible variety. Its secant variety, denoted by SX, is the image of SX in PN via the natural projection. The dimension of SX may be smaller than 2n + 1. In this case we say that X is secant defective and introduce the secant defect of X to be δ := 2n + 1 − dim(SX) ≥ 0. As SX ⊆ PN , we have that dim(SX) ≤ N, which implies δ ≥ n−c+1, where c is the codimension of X in PN . QEL, LQEL, and CC Varieties Let x, y ∈ X be two general points, and let z ∈ lx,y be a general point on the secant line lx,y (...truncated)