Nonrational nodal quartic threefolds

Ivan Cheltsov

Vol. 226 (2006), No. 1, 65–81

DOI: 10.2140/pjm.2006.226.65

Abstract

It is known that the ℚ-factoriality of a nodal quartic 3-fold in ℙ⁴ implies its nonrationality. We prove that a nodal quartic 3-fold with at most 8 nodes is ℚ-factorial, while one with 9 nodes is not ℚ-factorial if and only if it contains a plane. There are nonrational non-ℚ-factorial nodal quartic 3-folds. In particular, we prove the nonrationality of a general non-ℚ-factorial nodal quartic 3-fold that contains either a plane or a smooth del Pezzo surface of degree 4.

Keywords

quartic threefold, nodal variety, Fano variety, del Pezzo surface, ℚ-factorial

Mathematical Subject Classification 2000

Primary: 14E08, 14J30, 14J45, 14J70, 14M20

Milestones

Received: 10 August 2004

Revised: 21 November 2004

Accepted: 21 November 2004

Published: 1 July 2006