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Unique decomposition for a polynomial of low rank

Edoardo Ballico, Alessandra Bernardi (2013)

Annales Polonici Mathematici

Let F be a homogeneous polynomial of degree d in m + 1 variables defined over an algebraically closed field of characteristic 0 and suppose that F belongs to the sth secant variety of the d-uple Veronese embedding of m into m + d d - 1 but that its minimal decomposition as a sum of dth powers of linear forms requires more than s summands. We show that if s ≤ d then F can be uniquely written as F = M d + + M t d + Q , where M , . . . , M t are linear forms with t ≤ (d-1)/2, and Q is a binary form such that Q = i = 1 q l i d - d i m i with l i ’s linear forms and m i ’s forms...

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Todor Milanov, Yefeng Shen (0)

Annales de l’institut Fourier

Varieties of minimal rational tangents of codimension 1

Jun-Muk Hwang (2013)

Annales scientifiques de l'École Normale Supérieure

Let  X be a uniruled projective manifold and let  x be a general point. The main result of [2] says that if the ( - K X ) -degrees (i.e., the degrees with respect to the anti-canonical bundle of  X ) of all rational curves through x are at least dim X + 1 , then X is a projective space. In this paper, we study the structure of  X when the ( - K X ) -degrees of all rational curves through x are at least dim X . Our study uses the projective variety 𝒞 x T x ( X ) , called the VMRT at  x , defined as the union of tangent directions to the rational curves...

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