Rational curves of degree 11 on a general quintic threefold

Abstract

We prove that the incidence scheme of rational curves of degree 11 on quintic threefolds is irreducible. This implies a strong form of the Clemens conjecture in degree 11. Namely, on a general quintic threefold $F$ in ${\mathbb{P}}^4$, there are only finitely many smooth rational curves of degree 11, and each curve $C$ is embedded in $F$ with normal bundle $\mathcal{O}(-1)\oplus \mathcal{O}(-1)$. Moreover, in degree 11, there are no singular, reduced, and irreducible rational curves, nor any reduced, reducible, and connected curves with rational components on $F$.

Significant reorganization; the liaison-theoretic argument handling curves contained in complete intersections of three hypercubics has been entirely rewritten.

References in corpus (1)

• Toward Clemens' Conjecture in degrees between 10 and 24

Cited by in corpus (4)

• Rational curves on general type hypersurfaces
• Dimension counts for singular rational curves via semigroups
• Kontsevich spaces of rational curves on Fano hypersurfaces
• Rational curves of degree 16 on a general heptic fourfold