From non-defectivity to identifiability

Tangential weak defectiveness and generic identifiability

“…Proof. If π X g : sec g (X) → P N is of fiber type then τ g−1 is of fiber type, see for instance [CM,Lemma16 (i)], and we conclude, by Theorem 24, that X is not identifiable.…”

“…Proof. The variety X is non defective and not (g − 1)-twd Therefore, by [CM,Theorem 18] the tangential projection τ g−2 , from {x 2 , . .…”

Almost all subgeneric third-order Chow decompositions are identifiable

“…Theorem 2.1 (Casarotti and Mella [10]). Let V ⊂ P n be an irreducible, nondegenerate projective variety that is not 1-twd.…”

“…for all n. The main contribution of this paper consists of leveraging the main results of [10,15,37,51] to tackle the second question on the number of Chow decompositions (CD) for cubics. In particular, we prove that for almost all subgeneric ranks r ≤ r gen there exists a Zariski-open subset of σ r (C 3,n ) such that the cubics in that set admit a unique expression as in (CD), up to the order of the summands.…”

Almost all subgeneric third-order Chow decompositions are identifiable

“…Before trying to answer these questions, recall that the Chow decomposition is a generalization of another famous polynomial (or, equivalently, symmetric tensor) decomposition: by taking L 0,j = • • • = L d−1,j in (CD) we obtain the Waring [30,34] or symmetric tensor rank decomposition [18], which was already studied by Clebsch, Sylvester, Palatini, and Terracini in the 19th and first half of the 20th century; see [8] for historical remarks. After a century-long journey in projective algebraic geometry starting in earnest with Palatini's 1903 paper [38], the necessary tools, such as those in [3,10,[12][13][14]48], were developed to study foregoing questions. For the Waring decomposition, the first question was answered by Alexander and Hirschowitz [3] in 1995, and the second was completely resolved by 2019 through the combined works of Ballico [5], Chiantini, Ottaviani, and Vannieuwenhoven [17], and Galuppi and Mella [24].…”