From a defective Segre-Veronese embedding to a non-defective one adding a factor

Fix x N, a multiprojective space Y and a very ample line bundle L on Y .We say that (Y, L) satisfies x -non-defectivity if the s-secant variety of (Y, L) has the expected dimension if eitherNatural examples arise when L is the Segre line bundle and all factors have the same dimension (Abo -Ottaviani -Peterson).We take integers r > 0, t 2 and set X := Y P r .Let L[t] be the line bundle on X coming from L and O P r (t).Under certain assumptions on x, dim Y , h 0 (L), r and t we prove that L[t] is not secant defective.Two of the main results are for r 2. In particular we extend a recent result by Ballico, Bernardi and Ma duz on the non-defectivity of Segre-Veronese embeddings of multidegree (t 1 , . . ., t k ) of (P 2 ) k , k 3, to the case in which t i = 1 for y > 0 integers i: we require y 9.