In this paper we consider the Segre-Veronese varieties, i.e., the embeddings of products of t projective spaces in the projective space PN via divisors of multi-degree (d1, . . . , dt), and we study the dimension of their higher secant varieties. We give the dimensions of all the higher secant varieties of P1 x P1 embedded by divisors of any bi-degree (d1, d2). We find that Pr x Pk, embedded by divisors of bi-degree (k + 1, 1), has no deficient higher secant varieties, and we give several examples of defective and Grassmann defective Segre-Veronese varieties.
Higher Secant Varieties of Segre-Veronese varieties
CATALISANO, MARIA VIRGINIA;GERAMITA, ANTHONY VITO;
2005-01-01
Abstract
In this paper we consider the Segre-Veronese varieties, i.e., the embeddings of products of t projective spaces in the projective space PN via divisors of multi-degree (d1, . . . , dt), and we study the dimension of their higher secant varieties. We give the dimensions of all the higher secant varieties of P1 x P1 embedded by divisors of any bi-degree (d1, d2). We find that Pr x Pk, embedded by divisors of bi-degree (k + 1, 1), has no deficient higher secant varieties, and we give several examples of defective and Grassmann defective Segre-Veronese varieties.
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