Rational Curves of Degree 10 on a General Quintic Threefold

Abstract

We prove the "strong form" of the Clemens conjecture in degree 10. Namely, on a general quintic threefold F in 4 , there are only, finitely, many smooth rational curves of degree 10, and each curve C is embedded in F with normal bundle −1 ⊕ −1 . Moreover, in degree 10, there are no singular, reduced, and irreducible rational curves, nor any reduced, reducible, and connected curves with rational components on F.

Finiteness of rational curves of degree $12$ on a general quintic threefold

“…We remark that the strong form of Clemens' conjecture (as proved by Cotterill in [5] and [6] for d = 10, 11, characterizing also singular irreducible rational curves on the general quintic threefold) cannot be achieved by our methods.…”

“…We point out that the cases d ≤ 11 have been previously addressed in [14] (d ≤ 7), [17] and [13] (d = 8, 9), [5] (d = 10), [6] and [7] (d = 11), and we recall the general set-up.…”

“…Now we are going to apply all of the dimension-counting remarks and lemmas above and to use liaison in order to show that degenerate rational curves which are sufficiently generic (with respect to the properties described in the remarks and lemmas) must in fact have h 1 (I C (5)) < 11, contradiction. Our argument hinges on a careful case-by-case analysis involving the types of divisors that that arise as components of certain residuals C T to C inside of complete intersections of type (5,5).…”