On the linearly general position of a general hyperplane section of nonreduced curves (Q1921899)

A zero dimensional subscheme \(Z\) of a projective space \(\Pi\) is said to be in linearly general position if every subscheme \(W\subseteq Z\) spans a linear subspace of \(\Pi\) of dimension \[ \min \{\dim(\Pi), \text{length} (W)-1\}. \] Let \(D\subset \mathbb{P}^n\) be a smooth, connected curve and \(T\) a one-dimensional locally Cohen-Macaulay scheme supported on \(D\). Following \textit{K. A. Chandler} [Trans. Am. Math. Soc. 347, No. 3, 767-784 (1995; Zbl 0830.14020)], \(T\) is said to be an \(h\)-rope if \(\deg T-h \cdot \deg D\) and \(T\) is contained in the first infinitesimal neighborhood of \(D\). [See also the paper by the reviewer, \textit{C. Peterson} and \textit{Y. Pitteloud}, ``Ropes in projective space'', J. Math. Kyoto Univ. 36, No. 2, 251-278 (1996)], for basic results about ropes.] Let \(H\) be a general hyperplane of \(\mathbb{P}^n\). The question of when \(T\cap H\) is in linearly general position was raised by \textit{D. Eisenbud} and \textit{J. Harris} [J. Algebr. Geom. 1, No. 1, 15-30 (1992; Zbl 0804.14002)]. They considered the case of a ribbon (i.e. \(h=2)\), and this case was also considered by the author in an earlier paper [J. Algebra 159, No. 1, 265-274 (1993; Zbl 0812.14018)]. Chandler considered this question for \(h\)-ropes over the complex numbers. In the current paper, the author extends this notion to a generalized \(h\)-rope, by removing the assumption that \(D\) is smooth and assuming only integral, and by allowing arbitrary characteristic. His main results give conditions which guarantee that the general hyperplane section of \(T\) will be in linear general position. The author also shows, as a corollary, that under certain mild assumptions, \(T\) satisfies the Castelnuovo bound for the arithmetic genus of an integral non-degenerate degree \(hd\) curve in \(\mathbb{P}^n\).