Rational curves on general projective hypersurfaces

Abstract

In this article, we study the geometry of k k -dimensional subvarieties with geometric genus zero of a general projective hypersurface X d ⊂ P n X_d\subset \mathbf {P}^n of degree d = 2 n − 2 − k d=2n-2-k , where k k is an integer such that 1 ≤ k ≤ n − 5 1\leq k\leq n-5 . As a corollary of our main result, we obtain that the only rational curves lying on the general hypersuface X 2 n − 3 ⊂ P n X_{2n-3}\subset \mathbf {P}^n , for n ≥ 6 , n\geq 6, are the lines.