Analysis on some infinite modules, inner projection, and applications (Q2844730)

Let \(X\) be a nondegenerate reduced closed subscheme in a projective space \(\mathbb{P}^N\) over an algebrically closed field \(k\). Let \(I\subseteq k[x_0,\cdots,x_N]\) be a homogeneous ideal generated by homogeneous polynomials of degree \(2\) such that \(\tilde{I}=\mathfrak{I}_X\). The authors call \(X\) to satisfy condition \textit{\(N_{2,p}\) scheme-theoretically} if \(I\) has linear syzygies at least up to the first \(p\)th steps. This definition generalizes the notation \(N_p\) for normal subschemes in [\textit{M. Green} and \textit{R. Lazarsfeld}, Compos. Math. 67, No. 3, 301--314 (1988; Zbl 0671.14010)].NEWLINENEWLINEOne of the purposes of the paper is to see how the property \(N_{2,p}\) behaves under projection from a linear subvariety. Reducing to projection from a point, the authors consider the \textit{\(x_0\)-elimination ideal} of \(I\) in \(S=k[x_1,\cdots,x_N]\) which is \(J=I\cap S\). To study \(J\), the authors consider the \(S-\)module structure of \(I\) (which is an infinitely generated module) and apply the \textit{elimination mapping cone} construction as in [\textit{J. Ahn} and \textit{S. Kwak}, J. Algebra 331, No. 1, 243--262 (2011; Zbl 1232.14035)]. It is shown in Corollary 3.4 that the inner projection from a smooth point satisfies \(N_{2,p-1}\), provided \(X\) satisfies \(N_{2,p}\). Also Theorem 4.1 shows that the arithmetic depth is preserved under the inner projection from a smooth point. These results enable the authors to characterize varieties which satisfy condition \(N_{2,p}\) in some extremal cases in Theorem 4.3, namely if \(X\) is \(r\)-dimensional variety, \(N=r+e\), \(p=e\) or \(p=e-1\).NEWLINENEWLINEAnother interesting result of the paper is that the ideal \(I\), defined at the beginning, must have at least \(\text{codim}(X).p-\frac{p(p-1)}{2}\) generators, if \(X\) satisfies \(N_{2,p}\).NEWLINENEWLINEThe paper ends with some examples and open questions.