Rational curves on general projective hypersurfaces

Abstract

Abstract. In this article, we study the geometry of k-dimensional subvarieties with geometric genus zero of a general projective hyper-where k is an integer such that 1 ≤ k ≤ n − 5. As a corollary of our main result we obtain that the only rational curves lying on the general hypersuface X 2n−3 ⊂ P n , for n ≥ 6, are the lines.

Rational curves on hypersurfaces

“…Remark The corollary for the case n ≥ 5 is also a consequence of Voisin's theorem in [7] (see [6]), even though her approach is quite different from ours. The corollary for n = 4 is one of Calabi-Yau cases above, which also is the Clemens' conjecture.…”

“…Pacienza in ( [6]) obtained this result for n ≥ 6 ; (3) if h is in ( 3n 2 − 1, 2n − 3), the inequality (4.5) implies that X 0 does not contain rational curves other than lines. This is the known result of Clemens and Ran ( [3]), which is implied by their bound of twisted genus; (4) if h = 3n 2 − 1, the inequality (4.5) implies that X 0 contains no rational curves other than lines and quadratic curves.…”

“…Later Voisin and Pancienza borrowed Ein's original idea of using adjunction formula and pushed it further to obtain the minimum genus of subvaries in hypersurfaces [6], [7]. Their immediate application of it is for rational curves.…”

“…(2) (Pancienza) A generic hypersurface of P n of degree 2n − 3 for n ≥ 6 does not contain rational curves of degree larger than 1 (see [6]). …”

Gonality of curves on general hypersurfaces

“…[5,9]), and if n 5, lines are the only rational curves on a very general hypersurface of degree d = 2n − 1 (cf. [16]).…”