Abstract
We prove a base point freeness result for linear systems of forms vanishing at general double points of the projective plane. For tensors we study the uniqueness problem for the representation of a tensor as a sum of terms corresponding to points and tangent vectors of the Segre variety associated with the format of the tensor. We give complete results for unions of one point and one tangent vector.
Highlights
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There is a connected zero-dimensional scheme Z ⊂ P such that Z is a flat limit of a family of z pairwise disjoint double points, h1 (I Z (d)) = 0 and |I Z (d)| has no base points outside Zred
Tensors associated with an arrow are exactly the tensors contained in the tangential variety of the Segre variety related to the format of the tensor
Summary
Uniqueness of Decompositions and Double Points for Veronese and Segre Varieties. Let Xreg denote the set of all smooth points of X. The linear span h2Si has the expected dimension if and only if all unions of s arrows, each of them with as its support a different point of S, are linearly independent. H1 (I Z (d)) = 0 and I Z (d) has no base points outside Zred ), where Z is a general union of x double points of Pn These integers may be expressed with the geometric language used for the additive decomposition of degree d forms in the following way. Unless otherwise stated we work over an algebraically closed field K of characteristic 0
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