arXiv:2007.12789v1 [math.AG] 24 Jul 2020 GEOMETRY OF TENSOR EIGENSCHEMES VALENTINA BEORCHIA, FRANCESCO GALUPPI AND LORENZO VENTURELLO Abstract. We study projective schemes arising from eigenvectors of tensors, called eigenschemes. After some general results, we give a birational description of the variety parametrizing eigenschemes of general ternary symmetric tensors and we compute its dimension. Moreover, we characterize the locus of triples of homogeneous polynomials defining the eigenscheme of a ternary symmetric tensor. Finally, we give a geometric characterization of all reduced zero-dimensional eigenschemes. The techniques used rely both on classical and modern complex projective algebraic geometry. 1. Introduction Tensors are natural generalizations of matrices in higher dimension. Just as matrices are crucial in linear algebra, tensors play the same role in multilinear algebra, and find applications in several branches of mathematics, as well as many applied sciences. And, just as for matrices, it is possible to give a notion of eigenvectors and eigenvalues for tensors, as introduced independently in 2005 by Lim [15] and Qi [21]. As we shall soon see, when dealing with eigenvectors it is not restrictive to focus on partially symmetric tensors. In this work, a tensor of order d is considered partially symmetric if it is symmetric with respect to the first d − 1 indices. Hence we identify these with tuples of homogeneous polynomials of degree d − 1. Some of our results will concern the subspace of symmetric tensors which is canonically isomorphic to the space of homogeneous polynomials. An eigenvector of a partially symmetric tensor T =(g 0 ,...,g n ) is a vector v such that T (v)= (g 0 (v),...,g n (v)) = λv for some constant λ. Since the property of being an eigenvector is preserved under scalar multiplication, it is natural to think about the map T : P n  P n as a rational map defined by T (P )=(g 0 (P ): ... : g n (P )), and regard eigenvectors as points in P n ; hence the name eigenpoints instead of eigenvectors. In the symmetric case we define the eigenpoints of a homogeneous polynomial f as the fixed points of the polar map, or equivalently the eigenpoints of the partially symmetric tensor ∇f =(∂ 0 f,...,∂ n f ). Tensor eigenpoints appear naturally in optimization. As an example, consider the problem of maxi- mizing a polynomial function f over the unit sphere in R n+1 . Eigenvectors of the symmetric tensor f are critical points of this optimization problem. Another interesting framework in which eigenvectors of symmetric tensors arise is the variational context: indeed, by Lim’s Variational Principle [15], given a symmetric tensor f , the critical rank one symmetric tensors for f are exactly of the form v d , where v is an eigenvector of f . This has applications in low-rank approximation of tensors (see [20]) as well as maximum likelihood estimation in algebraic statistics. Finally, in [17] Oeding and Ottaviani employ eigenvectors of tensors to produce an algorithm to compute Waring decompositions of homogeneous polynomials. Since eigenschemes are not GL(n +1)-invariant, one might argue that the right setting to study eigenvectors is affine Euclidean geometry over the real numbers Nevertheless, we will study them through the lenses of complex projective geometry. One of the reasons is that being an eigenvector of a tensor is an algebraic condition, described by the vanishing of minors of suitable matrices. 2010 Mathematics Subject Classification. 13P25, 14M12, 14C21, 15A69, 15A72. 1