Clemens' conjecture for octic and nonic curves (Q1897546)

Clemens conjectured that on a general quintic threefold \(F\) in \(\mathbb{P}^4\) there are only finitely many smooth rational curves of a given degree \(d\) [\textit{H. Clemens} in: Topics in transcendental algebraic geometry, Ann. Math. Stud. 106, 289-304 (1984; Zbl 0575.14007), p. 300]. \textit{S. Katz} published a proof for \(d\leq 7\) [Compos. Math. 60, 151-162 (1986; Zbl 0606.14039); \S 1.2]. Here we will consider the degrees \(d=8\) and 9. Main theorem. The space of smooth rational curves of degree \(\leq 9\) on a general quintic in \(\mathbb{P}^4\) is finite, nonempty and reduced.