On birational properties of smooth codimension two determinantal varieties

Ivan Pan

Vol. 237 (2008), No. 1, 137–150

DOI: 10.2140/pjm.2008.237.137

Abstract

We show that a smooth arithmetically Cohen–Macaulay variety X, of codimension 2 in ℙⁿ if 3 ≤ n ≤ 5 and general if n > 3, admits a morphism onto a hypersurface of degree (n + 1) in ℙⁿ⁻¹ with, at worst, double points; moreover, this morphism comes from a (global) Cremona transformation which induces, by restriction to X, an isomorphism in codimension 1. We deduce that two such varieties are birationally equivalent via a Cremona transformation if and only if they are isomorphic.

Keywords

Cremona transformation, determinantal variety, birational properties

Mathematical Subject Classification 2000

Primary: 13C40, 14E05, 14E07

Milestones

Received: 21 September 2007

Revised: 5 March 2008

Accepted: 6 March 2008

Published: 1 September 2008