We prove that vector bundles give equations of cactus varieties. We derive from it that equations coming from vector bundles are not enough to define secant varieties of Veronese varieties in general.
We generalize methods to compute various kinds of rank to the case of a toric variety X embedded into projective space using a very ample line bundle L. We use this to compute rank, border rank, and cactus rank of monomials in H 0 (X, L) * when X is P 1 × P 1 , the Hirzebruch surface F1, the weighted projective plane P(1, 1, 4), or a fake projective plane.
In this paper we study homogeneous polynomials and vector subspaces of a polynomial ring that are divisible by a large power of a linear form. We compute their cactus and border cactus ranks. We show that for d ≥ 5, the component of the cactus variety κ14(ν d (P 6 )) other than the secant variety σ14(ν d (P 6 )) consists of degree d polynomials divisible by (d − 3)-th power of a linear form. For d ≥ 6 we present an algorithm for deciding whether a point in the cactus variety κ14(ν d (P 6 )) belongs to the secant variety σ14(ν d (P 6 )). Analogously, we show that for d ≥ 5, the component of the Grassmann cactus variety κ8,3(ν d (P 4 )) other than the Grassmann secant variety σ8,3(ν d (P 4 )) consists of subspaces divisible by (d − 2)-th power of a linear form. Finally, for d ≥ 5 we present an algorithm for deciding whether a point in the Grassmann cactus variety κ8,3(ν d (P 4 )) belongs to the Grassmann secant variety σ8,3(ν d (P 4 )).
We generalize methods to compute various kinds of rank to the case of a toric variety 𝑋 embedded into projective space using a very ample line bundle . We find an upper bound on the cactus rank. We use this to compute rank, border rank, and cactus rank of monomials in 𝐻 0 (𝑋, ) * when 𝑋 is ℙ 1 × ℙ 1 , the Hirzebruch surface 𝔽 1 , or the weighted projective plane ℙ(1, 1, 4).
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