On birational properties of smooth codimension two determinantal varieties (Q953129)

Let \(X\) be a general arithmetically Cohen-Macaulay (ACM) variety of codimension \(2\) in \(\mathbb{P}^n\). \(X\) is the base locus scheme of a Cremona transformation of \(\mathbb{P}^n\) and then is not a complete intersection. Moreover, \(X\) is smooth if and only if \(3\leq n \leq 5\). These possibilities for \(n\) are those studied in the article under review. The author proves the existence of a birational morphism \(\eta: X\to Y\) where \(Y\) is a hypersurface of degree \(n+1\) in \(\mathbb{P}^{n-1}\) which is an isomorphism if \(n=3,4\) and an isomorphism in codimension \(1\) if \(n=5\). As a consequence, if \(X\) and \(X'\) are ACM varities of codimension \(2\) in \(\mathbb{P}^n\) for \(3\leq n \leq 5\), any birational map \(X\dasharrow X'\) is in fact an isomorphism.