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 $\IP^4$, 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
$\O(-1)\oplus\O(-1)$, and in $\IP^4$ with maximal rank.Comment: Plain Tex, This eleven page paper is a joint manuscript, produced in
  connection with the first author's participation in the conference "Geometry
  and Physics", Zlatograd, Bulgaria, August 28 - Sept.2, 1995." This version
  contains a small change in Remark (3.3); the hope expressed there has been
  refine

Johnsen, Trygve

Kleiman, Steven L.

English

arXiv

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

Kleiman, Steven

Bulgarian Digital Mathematics Library at IMI-BAS

Toward Clemens' Conjecture in Degrees between 10 and 24

http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.251.5423

Toward Clemens' Conjecture in degrees between 10 and 24

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