Projective varieties of maximal sectional regularity (Q308147)

Let \(X\subset{\mathbb P}^r\) be a nondegenerate irreducible projective variety of dimension \(n\), codimension \(c\geq 2\) and degree \(d\) over an algebraically closed field \(\mathbb K\). If \(X\) is \(m\)-regular, then any \((m+1)\)-secant line to \(X\) should be contained in \(X.\)NEWLINENEWLINEA conjecture by \textit{D. Eisenbud} and \textit{S. Goto} [J. Algebra 88, 89--133 (1984; Zbl 0531.13015)] says that NEWLINE\[NEWLINE\text{reg}\,X \leq d-c+1,NEWLINE\]NEWLINE where \(\text{reg}\,X\) is the Castelnuovo-Mumford regularity of \(X.\) In particular this conjecture implies that NEWLINE\[NEWLINEX \text{ has no proper }k\text{-secant line if }k > d-c+1.NEWLINE\]NEWLINENEWLINENEWLINEThe main subject of the paper is to study the geometry of proper \((d-c+1)\)-secant lines to a projective variety.NEWLINENEWLINEOne of the main results is the classification of all varieties of \textit{maximal sectional regularity}, i.e. varieties whose general linear curve section is of maximal regularity.