On Varieties Which Are Uniruled by Lines

Using the ♯-minimal model program of uniruled varieties we show that for any pair (X,H) consisting of a reduced and irreducible variety X of dimension k ≥ 3 and a globally generated big line bundle H on X with d := H and n := h(X,H) − 1 such that d < 2(n− k)− 4, then X is uniruled of H-degree one, except if (k, d, n) = (3, 27, 19) and a ♯-minimal model of (X,H) is (P3,OP3(3)). We also show that the bound is optimal for threefolds. 0. Introduction It is well-known that an irreducible nondegenerate variety X ⊆ Pn of degree d satisfies d ≥ n− dimX +1. Varieties for which equality is obtained are the well-known varieties of minimal degree, which are completely classified. Varieties for which d is “small” compared to n have been the objects of intensive study throughout the years, see e.g. [Ha, Ba, F1, F2, F3, Is, Io, Ho, Re2, Me2]. One of the common features is that such varieties are covered by rational curves. More generally one can study pairs (X,H) where X is an irreducible variety (possibly with some additional assumptions on its singularities) and H a line bundle on X which is sufficiently “positive” (e.g. ample or (birationally) very ample or big and nef). Naturally we set d := HdimX and n := dim |H|. The difference between d and n is measured by the ∆-genus: ∆(X,H) := d+dimX−n−1, introduced by Fujita (cf. [F1] and [F2]) and Fujita in facts shows that ∆(X,H) ≥ 0 for X smooth and H ample. The cases with ∆(X,H) = 0 are the varieties of minimal degree and the cases with ∆(X,H) = 1 have been classified by Fujita [F1, F2, F3] and Iskovskih [Is]. The notion of being covered by rational curves is incorporated in the concept of a variety being uniruled. Roughly speaking a variety is uniruled if through any point there passes a rational curve. In many ways these are the natural generalizations to higher dimensions of ruled surfaces. In the Mori program they play an important role, because like in the case of ruled surfaces these are the varieties for which the program does not yield a minimal model, but a Mori fiber space. Uniruled varieties can also be considered to be the natural generalizations to higher dimensions of surfaces of negative Kodaira dimension: in fact it is conjectured that a (smooth) variety is uniruled if and only if its Kodaira dimension is negative. The conjecture has been established for threefolds by Miyaoka [Mi]. With the evolution of a structure theory for higher dimensional varieties in the past decades, namely the Mori program, the geometry of rational curves on varieties has gained new importance. The main idea is to obtain information about varieties by studying the rational curves on them (cf. e.g. [Ko]). 2000 Mathematics Subject Classification: Primary: 14E30, 14J30, 14J40, 14N25; Secondary: 14C20, 14H45.