Weakly defective varieties

By Luca Chiantini, C. Ciliberto. 2001

221 citations recorded by the source corpus.

Subfields: Geometry and Topology, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/ffe9b11a7b5432d9cfcbf77497452f61a3b71de8

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Citations in the corpus

Varieties with minimal secant degree and linear systems of maximal dimension on surfaces (2004, background): “4 of [14] that we partially recall here.”
Geometric Aspects of Polynomial Interpolation in More Variables and of Waring’s Problem (2001, result): “A full classification of weakly defective surfaces, which extends previous partial results by Terracini [84], has been recently obtained by Chiantini-Ciliberto [20].”
On the Alexander–Hirschowitz theorem (2007, background): “It is worth to remark that Campbell proved in the same paper that the only Veronese surfaces which are weakly defective (in the modern notation, according to [14]) are given by the linear systems |O(d)| with d = 2, 4 or 6.”
The Hitchhiker Guide to: Secant Varieties and Tensor Decomposition (2018, background): “The above conjectures are based on results that can be found in [123, 131, 93, 128, 99, 122, 126, 130, 112, 129, 134, 125, 124, 132, 133, 127, 121].”
Decomposition of homogeneous polynomials with low rank (2010, background): “By [6], Lemma 2.1.5, or [3], Proposition 11, there is a smoothable 0-dimensional scheme E ⊂ Xm,d such that deg(E) ≤ sbr(P ) and P ∈ 〈E〉.”
Higher secant varieties of Segre-Veronese varieties (2003, methodology): “s defective, and dimV s = 3s−2 (its defectivness is 1). This theorem could be proved by methods similar to those used in [CGG3], [CGG4], but we omit the lenghty and tedious proof here. Notice that in [ChCi] a total classiﬁcation of all the surfaces with some defective secant variety can be found and”
Equations for secant varieties of Veronese and other varieties (2011, background): “[10, 36, 1] The k-weakly defective varieties vd(P n) are the triples (k, d, n): (i) the k-defective varieties, namely (k, 2, n), k = 2, .”
On the Concept of k‐Secant Order of a Variety (2006, background): “As the classification of k-weakly defective surfaces is known [4], the classification of such surfaces X with s(X) = 3k + 2 < r and μk(X) = 1 will give, by exclusion, the classification of surfaces X ′ with s(X ′) = 3k + 2 < r and μk(X ′) > 1.”
On generic identifiability of symmetric tensors of subgeneric rank (2015, background): “Actually, Theorem 6.1.2 of [26] only intended to collect previous results by Ballico [4], Ciliberto and Chiantini [8], and Mella [27,28], which are individually correct.”
IRREDUCIBLE SYMPLECTIC 4-FOLDS NUMERICALLY EQUIVALENT TO (K3)[2] (2008, background): “Since curves are never defective (see [4]) there exists a 3-secant plane of W containing p, call it Ω.”