On the non-defectivity of Segre–Veronese embeddings

We prove a theorem which implies that all Segre–Veronese varieties of multidegree \((d_1,\dots ,d_k)\) and format \((n_1,\dots ,n_k)\) with \(n_1\ge \cdots \ge n_k>0\) are not defective if \(d_1\ge 3\), \(d_2\ge 3\) and \(d_i\ge 2\) for all \(i>2\). As a particular case we prove the non-defectivity of any Segre–Veronese variety with at least 2 factors and \(d_i\ge 3\) for all i, extending to the case \(k>2\) a theorem of Galuppi and Oneto. Our general result also shows that many Segre–Veronese varieties with 2 factors are not secant defective if they are embedded in bidegree (x, 2), \(x\ge 4\).