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Symmetric tensor rank with a tangent vector: a generic uniqueness theorem
Ist Teil von
Proceedings of the American Mathematical Society, 2012-10, Vol.140 (10), p.3377-3384
Ort / Verlag
American Mathematical Society
Erscheinungsjahr
2012
Link zum Volltext
Quelle
Alma/SFX Local Collection
Beschreibungen/Notizen
Let ${X_{m,d}} \subset {ℙ^N},N: = \left( {\begin{array}{*{20}{c}}{m + d}\\m\\\end{array} } \right)-1$ , be the order d Veronese embedding of ℙ m . Let τ(X m,d ) ᑕ ℙ N be the tangent developable of X m,d . For each integert ≥ 2 let τ(X m,d t) ⊆ ℙ N be the join of τ(X m,d ) and t — 2 copies of X m,d . Here we prove that if m ≥ 2, d ≥ 7 and $t \leqslant 1 + \left\lfloor {\left( {\begin{array}{*{20}{c}} {m + d - 2} \\ m \\ \end{array} } \right)/\left( {m + 1} \right)} \right\rfloor $ , then for a general P ∈ τ(X m,d, t) there are uniquely determined P₁ ,..., P t ₋₂ ∈ X m,d and a unique tangent vector v of X m,d such that P is in the linear span of v ᑌ {P₁, · · · , P t ₋₂}; i. e. a degree d linear form f (a symmetric tensor T of order d) associated to P may be written as $f = L\begin{array}{*{20}{c}} {d - 1} \\ {t - 1} \\ \end{array} {L_t} + \sum\limits_{i = 1}^{t - 2} {L_i^d,} \left( {T = \upsilon _{t - 1}^{ \otimes \left( {d - 1} \right)}\upsilon t + \sum\limits_{i = 1}^{t - 2} {\upsilon _i^{ \otimes d}} } \right)$ with L i linear forms on ℙ m (v i vectors over a vector field of dimension m + 1 respectively), 1 < i < t, that are uniquely determined (up to a constant).