Multi-secant lemma

Abstract

We present a new generalization of the classical trisecant lemma. Our approach is quite different from previous generalizations [8, 10, 1, 2, 4, 7]. Let X be an equidimensional projective variety of dimension d. For a given k ≤ d + 1, we are interested in the study of the variety of k-secants. The classical trisecant lemma just considers the case where k = 3 while in [10] the case k = d + 2 is considered. Secants of order from 4 to d + 1 provide service for our main result. In this paper, we prove that if the variety of k-secants (k ≤ d +1) satisfies the following three conditions: (i) through every point in X, there passes at least one k-secant, (ii) the variety of k-secants satisfies a strong connectivity property that we define in the sequel, (iii) every k-secant is also a (k +1)-secant; then the variety X can be embedded into ℙd+1. The new assumption, introduced here, that we call strong connectivity, is essential because a naive generalization that does not incorporate this assumption fails, as we show in an example. The paper concludes with some conjectures concerning the essence of the strong connectivity assumption.