Non-existence of smooth rational curves of degree d = 13, 14, 15 con-tained in a general quintic hypersurface of P4 and in some quadric hypersurface


Let W \subset \mathbb {P}^4 be a general quintic hypersurface. We prove that W contains no smooth rational curve C\subset \mathbb {P}^4 with degree d\in \{13,14,15\}, h^0(\mathcal {I} _C(1)) =0 and h^0(\mathcal {I} _C(2)) >0.

DOI Code: 10.1285/i15900932v36n2p77

Keywords: General quintic $3$-fold; rational curve; Clemens' conjecture

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