Linear determinantal equations for all projective schemes
Several classically studied projective varieties (rational normal curves, degree 2 Veronese embeddings of \({\mathbb P}^r \), 2-factors Segre varieties) have their homogeneous ideal generated by 2-minors of matrices of linear forms. This fact allows to get descriptions of minimal resolutions of such ideal, and also equations for higher secant varieties. Mumford and Eisembud suggested that for every projective variety the embeddings given by a sufficiently large multiple of the embedding line bundle should have this property.NEWLINENEWLINEIn this paper criteria for linear bundles on certain varieties are given, in order to have embeddings with this property.NEWLINENEWLINELet \(X\subseteq {\mathbb P}^r \) be a projective scheme, embedded by a linear system \(|L|\), where \(L\) is a line bundle on \(X\). \(L\) is said to be determinantally presented if \(I_{X|{\mathbb P}^r}\) is generated by the 2-minors of a 1-generic matrix of linear forms (i.e. no conjugate matrix has a zero entry). We will say that a property holds for a sufficiently ample line bundle on \(X\) if there exist a line bundle \(A\) on \(X\) such that the property holds for all \(L\) such that \(L\otimes A^{-1}\) is ample.NEWLINENEWLINEThe main result in this paper states that if \(X\) is a connected projective scheme, then every sufficiently ample line bundle is determinantally generated.NEWLINENEWLINEFor smooth schemes a better description is given: if dim\(X=n\), \(A\) is very ample on \(X\) with \((X,A) \neq ({\mathbb P}^n , {\mathcal O}_{{\mathbb P}^r}(1))\) and \(B\) is a nef line bundle, then each \(L:= K_X^2\otimes A^j\otimes B\) is determinantally presented when \(j\geq 2n+2\) (this holds also if \(X\) is a Gorenstein toric variety, not necessarily smooth).NEWLINENEWLINEThe case of Segre-Veronese varieties is also studied, showing that on \(X = {\mathbb P}^{n_1}\otimes \dots \otimes {\mathbb P}^{n_l}\), the line bundle \({\mathcal O}_{X}({\mathbf m}))\), \({\mathbf m} = (m_1,\dots,m_l)\), \(m_i>0\), is determinantally presented whenever at least \(l-2\) among the \(m_i\)'s are \(\geq 2\).
scientific article; zbMATH DE number 6054526