Waring Rank of Symmetric Tensors, and Singularities of Some Projective Hypersurfaces
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Waring Rank of Symmetric Tensors, and Singularities of Some Projective Hypersurfaces Alexandru Dimca and Gabriel Sticlaru Abstract. We show that if a homogeneous polynomial f in n variables has Waring rank n + 1, then the corresponding projective hypersurface f = 0 has at most isolated singularities, and the type of these singularities is completely determined by the combinatorics of a hyperplane arrangement naturally associated with the Waring decomposition of f . We also discuss the relation between the Waring rank and the type of singularities on a plane curve, when this curve is defined by the suspension of a binary form, or when the Waring rank is 5. Mathematics Subject Classification. Primary 14J70; Secondary 14B05, 32S05, 32S22. Keywords. Waring decomposition, Waring rank, Projective hypersurface, Isolated singularity, Hyperplane arrangement.
1. Introduction For the general question of symmetric tensor decomposition we refer to [2, 4,6,8,13,14,16,20–23], as well as to the extensive literature quoted at the references in [2] and [14]. We describe first a possibly new general approach to tensor decompositions, and then illustrate this approach by a number of very simple situations. Consider the graded polynomial ring S = C[x1 , . . . , xn ], let f ∈ Sd be a homogeneous polynomial of degree d, such that the corresponding hypersurface V = V (f ) : f = 0 1
(1.1)
This work has been partially supported by the French government, through the UCAJEDI Investments in the Future project managed by the National Research Agency (ANR) with the reference number ANR-15-IDEX-01 and by the Romanian Ministry of Research and Innovation, CNCS-UEFISCDI, grant PN-III-P4-ID-PCE-2016-0030, within PNCDI III.
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A.Dimca and G. Sticlaru
MJOM
in the complex projective space Pn−1 is reduced. We consider the Waring decomposition (D)
f = d1 + · · · + dr ,
(1.2)
where j ∈ S1 are linear forms, and r is minimal, in other words r = rank f is the Waring rank of f . We assume in the sequel that the form f essentially involves the n variables x1 , . . . , xn , in other words that f cannot be expressed as a polynomial in a fewer number of variables than n. This is equivalent to the fact that the linear forms j ’s span the vector space S1 , in particular we have r ≥ n. When such a decomposition is given, we will also use the notation V = VD to show that the hypersurface V comes from the decomposition (D). For a given form f ∈ Sd with Waring rank r, one can define the Waring locus of f to be the set Wf = {[1 ] ∈ P(S1 ) : there are 2 , . . . r ∈ S1 such that (1.2) holds }. (1.3) Moreover, the locus of forbidden linear forms is the complement Ff = P(S1 )\Wf , see [1,4]. Consider the linear embedding ϕD : Pn−1 → Pr−1 , x → (1 (x) : · · · : r (x)),
(1.4)
determined by the decomposition (D). In the projective space P we have two basic objects, namely the Fermat hypersurface of degree d, given by r−1
F : fF (y) = y1d + · · · + yrd = 0,
(1.5)
and the Boolean arrangement B : fB (y) = y1 y2 · · · yr
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