Indag. Mathem., N.S., 13 (l), 23-28 March 252002 The Grassmannians of secant varieties of curves are not defective by L. Chiantini and C. Ciliberto Dipavtimento di Matematica, Univevsitd di Siena, Via de1 Capitano, 15, 53100 Siena, Italia. e-mail: chiantini@unisi.it Dipartimento di Matematica, Universitri di Roma Tar Vergata, Via della Ricerca ScientiJica, 00133 Roma, Italia. e-mail: cilibert@axp.mat.uniroma2.it Communicated by Prof. J.P. Murre at the meeting of January 28,2002 1. INTRODUCTION A smooth, non degenerate n-dimensional projective variety X c Py is pro- jected isomorphically from a point P E P’ to Pl’-’ if and only if P does not belong to the secant variety S’(X) of X. Since dim(S’(X)) 5 2n + 1, we can always embed X in P2n+1; we can go further and project X isomorphically in some Pm, m < 2n + 1 if and only if S’(X) has dimension smaller than the ex- pected one. The classification of varieties for which S’ (X) has dimension less than expected has been studied by many classical authors. Zak’s book [Zak] contains a comprehensive overview of the theory. As s increases, one expects that projecting a smooth variety X c Py to smaller spaces Pres-l, points of high multiplicity or rather high secant spaces must arise for the image X’. For instance, if a linear span x of k + 1 points of X contains the center of projection, then it determines a (k + 1)-secant space for X’ of dimension less thank. So one is led to consider the Grassmannian G(k, Y) of k-planes in P’, the subset Gk(X) formed by (k + 1)-secant k-planes and the subset Gs,k(X) of G(s,P) formed by s-planes contained in some H E Gk(X): these are the bad centers of projection. If Gs,k(X) coincides with the Grass- mannian G(s, v), then (k + 1)-secant spaces of dimension smaller than k neces- sarily arise in a general projection X + P’-+‘. In analogy with the theory of secant varieties Sk(X), one may ask about the expected dimension of these Grassmannians ofsecant varieties G:;k(X) and one may hope for a classification of varieties such that G+(X) has dimension 23