Toward Clemens' Conjecture in Degrees between 10 and 24

1 Supported in part by the Norwegian Research Council for Science and the Humanities. It is a pleasure for this author to thank the Department of Mathematics of the University of Sofia for organizing the remarkable conference in Zlatograd during the period August 28-September 2, 1995. It is also a pleasure to thank the M.I.T. Department of Mathematics for its hospitality from January 1 to July 31, 1993, when this work was started. 2Supported in part by NSF grant 9400918-DMS.

We introduce and study a likely condition that implies the following form of Clemens’ conjecture in degrees d between 10 and 24: given a general quintic threefold F in complex P^4, the Hilbert scheme of rational, smooth and irreducible curves C of degree d on F is finite, nonempty, and reduced; moreover, each C is embedded in F with balanced normal sheaf O(−1) ⊕ O(−1), and in P^4 with maximal rank.