Let Vt=P1x … x P1 be the product of t copies of the 1-dimensional projective space P1, embedded in the N-dimensional projective space PN via the Segre embedding. Let (Vt)^s be the s-secant varieties of Vt, that is, the subvariety of PN which is the closure of the union of all the (s-1)-dimensional projective space s-secant to Vt. The expected dimension of (Vt)^s is min { st + (s-1), N }. This is not the case for (V4)^3, which we conjecture is the only defective example in this infinite family. We show that all the higher secant varieties (Vt)^s have the expected dimension—except, possibly, for one higher secant variety for each such t. Moreover, whenever t +1 is a power of 2, (Vt)^s has the expected dimension for every s.
Higher Secant Varieties of the Segre Varieties P^1 x ... x P^1
CATALISANO, MARIA VIRGINIA;GERAMITA, ANTHONY VITO;
2005-01-01
Abstract
Let Vt=P1x … x P1 be the product of t copies of the 1-dimensional projective space P1, embedded in the N-dimensional projective space PN via the Segre embedding. Let (Vt)^s be the s-secant varieties of Vt, that is, the subvariety of PN which is the closure of the union of all the (s-1)-dimensional projective space s-secant to Vt. The expected dimension of (Vt)^s is min { st + (s-1), N }. This is not the case for (V4)^3, which we conjecture is the only defective example in this infinite family. We show that all the higher secant varieties (Vt)^s have the expected dimension—except, possibly, for one higher secant variety for each such t. Moreover, whenever t +1 is a power of 2, (Vt)^s has the expected dimension for every s.
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