scholarly journals Syzygies of quadratic Veronese embedding

2017 ◽  
Vol 208 (2) ◽  
pp. 200-222
Author(s):  
I V Netay
Keyword(s):  
Veronese Embedding

scholarly journals Reconstruction of a homogeneous polynomial from its additive decompositions when identifiability fails

2020 ◽  
Vol ahead-of-print (ahead-of-print) ◽  
Author(s):  
E. Ballico
Keyword(s):  
Homogeneous Polynomial ◽  
Linear Span ◽  
Veronese Embedding ◽  
Complex Variety ◽  
Uniqueness Set

Let X⊂ℙr be an integral and non-degenerate complex variety. For any q∈ℙr let rX(q) be its X-rank and S(X,q) the set of all finite subsets of X such that |S|=rX(q) and q   ∈  〈S〉, where 〈〉 denotes the linear span. We consider the case |S(X,q)|>1 (i.e. when q is not X -identifiable) and study the set W(X)q:=∩ S∈S(X,q)〈S〉, which we call the non-uniqueness set of q. We study the case dimX=1 and the case X a Veronese embedding of ℙn. We conclude the paper with a few remarks concerning this problem over the reals.

scholarly journals Resolution of veronese embedding of plane curves

2012 ◽  
Vol 43 (2) ◽  
pp. 155-178
Author(s):  
Aaloka Kanhere
Keyword(s):  
Plane Curves ◽  
Veronese Embedding

scholarly journals DEFORMATIONS OF THE VERONESE EMBEDDING AND FINSLER -SPHERES OF CONSTANT CURVATURE

2021 ◽  
pp. 1-32
Author(s):  
Christian Lange ◽  
Thomas Mettler
Keyword(s):  
Ricci Curvature ◽  
Constant Curvature ◽  
Projective Spaces ◽  
Positive Definite ◽  
The Other ◽  
Duality Result ◽  
Veronese Embedding ◽  
Other Hand ◽  
The One ◽  
Weyl Connections

Abstract We establish a one-to-one correspondence between, on the one hand, Finsler structures on the $2$ -sphere with constant curvature $1$ and all geodesics closed, and on the other hand, Weyl connections on certain spindle orbifolds whose symmetric Ricci curvature is positive definite and whose geodesics are all closed. As an application of our duality result, we show that suitable holomorphic deformations of the Veronese embedding $\mathbb {CP}(a_1,a_2)\rightarrow \mathbb {CP}(a_1,(a_1+a_2)/2,a_2)$ of weighted projective spaces provide examples of Finsler $2$ -spheres of constant curvature whose geodesics are all closed.

scholarly journals On syzygies of Veronese embedding of arbitrary projective varieties

2009 ◽  
Vol 322 (1) ◽  
pp. 108-121 ◽  
Author(s):  
Euisung Park
Keyword(s):  
Projective Varieties ◽  
Veronese Embedding

scholarly journals Set Evincing the Ranks with Respect to an Embedded Variety (Symmetric Tensor Rank and Tensor Rank

Mathematics ◽  
2018 ◽  
Vol 6 (8) ◽  
pp. 140
Author(s):  
Edoardo Ballico
Keyword(s):  
Linear Span ◽  
Symmetric Tensor ◽  
Tensor Rank ◽  
General Point ◽  
Symmetric Tensor Rank ◽  
Maximum Rank ◽  
Veronese Embedding ◽  
Finite Set

Let X ⊂ P r be an integral and non-degenerate variety. We study when a finite set S ⊂ X evinces the X-rank of the general point of the linear span of S. We give a criterion when X is the order d Veronese embedding X n , d of P n and | S | ≤ ( n + ⌊ d / 2 ⌋ n ) . For the tensor rank, we describe the cases with | S | ≤ 3 . For X n , d , we raise some questions of the maximum rank for d ≫ 0 (for a fixed n) and for n ≫ 0 (for a fixed d).

Veronese embedding and two–character sets

2006 ◽  
Vol 42 (1) ◽  
pp. 103-107 ◽  
Author(s):  
Antonio Cossidente ◽  
Giuseppe Marino
Keyword(s):  
Veronese Embedding

scholarly journals Rank 1 Decompositions of Symmetric Tensors Outside a Fixed Support

ISRN Geometry ◽  
2013 ◽  
Vol 2013 ◽  
pp. 1-4
Author(s):  
E. Ballico
Keyword(s):  
Linear Form ◽  
Homogeneous Polynomial ◽  
Linear Subspace ◽  
Symmetric Tensor ◽  
Minimal Cardinality ◽  
Symmetric Tensors ◽  
Veronese Embedding ◽  
Integral Variety

Let νd:ℙm→ℙn, n:=(n+dn)-1, denote the degree d Veronese embedding of ℙm. For any P∈ℙn, let sr(P) be the minimal cardinality of S⊂νd(ℙm) such that P∈〈S〉. Identifying P with a homogeneous polynomial q (or a symmetric tensor), S corresponds to writing q as a sum of ♯(S) powers Ld with L a linear form (or as a sum of ♯(S) d-powers of vectors). Here we fix an integral variety T⊊ℙm and P∈〈νd(T)〉 and study a similar decomposition with S⊈T and ♯(S) minimal. For instance, if T is a linear subspace, then we prove that ♯(S)≥♯(S∩T)+d+1 and classify all (S,P) such that ♯(S)-♯(S∩T)≤2d-1.

scholarly journals The $2$-Hessian and sextactic points on plane algebraic curves

2019 ◽  
Vol 125 (1) ◽  
pp. 13-38
Author(s):  
Paul Aleksander Maugesten ◽  
Torgunn Karoline Moe
Keyword(s):  
Algebraic Curve ◽  
Algebraic Curves ◽  
Rational Curves ◽  
Veronese Embedding ◽  
Plane Algebraic Curve ◽  
Special Case

In an article from 1865, Arthur Cayley claims that given a plane algebraic curve there exists an associated $2$-Hessian curve that intersects it in its sextactic points. In this paper we fix an error in Cayley's calculations and provide the correct defining polynomial for the $2$-Hessian. In addition, we present a formula for the number of sextactic points on cuspidal curves and tie this formula to the $2$-Hessian. Lastly, we consider the special case of rational curves, where the sextactic points appear as zeros of the Wronski determinant of the 2nd Veronese embedding of the curve.

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