scholarly journals On blowup of secant varieties of curves

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Lawrence Ein ◽  
Wenbo Niu ◽  
Jinhyung Park
Keyword(s):  
Positive Integer ◽  
Projective Curve ◽  
Secant Varieties ◽  
Secant Variety

<p style='text-indent:20px;'>In this paper, we show that for a nonsingular projective curve and a positive integer <inline-formula><tex-math id="M1">\begin{document}$ k $\end{document}</tex-math></inline-formula>, the <inline-formula><tex-math id="M2">\begin{document}$ k $\end{document}</tex-math></inline-formula>-th secant bundle is the blowup of the <inline-formula><tex-math id="M3">\begin{document}$ k $\end{document}</tex-math></inline-formula>-th secant variety along the <inline-formula><tex-math id="M4">\begin{document}$ (k-1) $\end{document}</tex-math></inline-formula>-th secant variety. This answers a question raised in the recent paper of the authors on secant varieties of curves.</p>

A Degree Formula for Secant Varieties of Curves

2013 ◽  
Vol 57 (2) ◽  
pp. 305-322 ◽  
Author(s):  
Rüdiger Achilles ◽  
Mirella Manaresi ◽  
Peter Schenzel
Keyword(s):  
Plane Curve ◽  
Natural Generalization ◽  
Singular Points ◽  
The Self ◽  
Hilbert Polynomial ◽  
Intersection Numbers ◽  
Secant Varieties ◽  
Secant Variety ◽  
Intersection Multiplicity ◽  
Polar Curve

AbstractUsing the Stückrad–Vogel self-intersection cycle of an irreducible and reduced curve in pro-jective space, we obtain a formula that relates the degree of the secant variety, the degree and the genus of the curve and the self-intersection numbers, the multiplicities and the number of branches of the curve at its singular points. From this formula we deduce an expression for the difference between the genera of the curve. This result shows that the self-intersection multiplicity of a curve in projectiveN-space at a singular point is a natural generalization of the intersection multiplicity of a plane curve with its generic polar curve. In this approach, the degree of the secant variety (up to a factor 2), the self-intersection numbers and the multiplicities of the singular points are leading coefficients of a bivariate Hilbert polynomial, which can be computed by computer algebra systems.

scholarly journals ON DIFFERENCES BETWEEN THE BORDER RANK AND THE SMOOTHABLE RANK OF A POLYNOMIAL

2014 ◽  
Vol 57 (2) ◽  
pp. 401-413 ◽  
Author(s):  
WERONIKA BUCZYŃSKA ◽  
JAROSŁAW BUCZYŃSKI
Keyword(s):  
General Point ◽  
Secant Varieties ◽  
Secant Variety ◽  
Veronese Variety ◽  
Border Rank ◽  
Veronese Varieties ◽  
Special Points

AbstractWe consider higher secant varieties to Veronese varieties. Most points on the rth secant variety are represented by a finite scheme of length r contained in the Veronese variety – in fact, for a general point, the scheme is just a union of r distinct points. A modern way to phrase it is: the smoothable rank is equal to the border rank for most polynomials. This property is very useful for studying secant varieties, especially, whenever the smoothable rank is equal to the border rank for all points of the secant variety in question. In this note, we investigate those special points for which the smoothable rank is not equal to the border rank. In particular, we show an explicit example of a cubic in five variables with border rank 5 and smoothable rank 6. We also prove that all cubics in at most four variables have the smoothable rank equal to the border rank.

scholarly journals Tropical secant graphs of monomial curves

2010 ◽  
Vol DMTCS Proceedings vol. AN,... (Proceedings) ◽  
Author(s):  
María Angélica Cueto ◽  
Shaowei Lin
Keyword(s):  
Tropical Geometry ◽  
Newton Polytope ◽  
Projective Curve ◽  
Secant Variety ◽  
Monomial Curves ◽  
Surface Graph ◽  
International Audience ◽  
Monomial Curve

International audience We construct and study an embedded weighted balanced graph in $\mathbb{R}^{n+1}$ parametrized by a strictly increasing sequence of $n$ coprime numbers $\{ i_1, \ldots, i_n\}$, called the $\textit{tropical secant surface graph}$. We identify it with the tropicalization of a surface in $\mathbb{C}^{n+1}$ parametrized by binomials. Using this graph, we construct the tropicalization of the first secant variety of a monomial projective curve with exponent vector $(0, i_1, \ldots, i_n)$, which can be described by a balanced graph called the $\textit{tropical secant graph}$. The combinatorics involved in computing the degree of this classical secant variety is non-trivial. One earlier approach to this is due to K. Ranestad. Using techniques from tropical geometry, we give algorithms to effectively compute this degree (as well as its multidegree) and the Newton polytope of the first secant variety of any given monomial curve in $\mathbb{P}^4$. On construit et on étude un graphe plongé dans $\mathbb{R}^{n+1}$ paramétrisé par une suite strictement croissante de $n$ nombres entiers $\{ i_1, \ldots, i_n\}$, premiers entre eux. Ce graphe s'appelle $\textit{graphe tropical surface sécante}$. On montre que ce graphe est la tropicalisation d'une surface dans $\mathbb{C}^{n+1}$ paramétrisé par des binômes. On utilise ce graphe pour construire la tropicalisation de la première sécante d'une courbe monomiale ayant comme vecteur d'exponents $(0, i_1, \ldots, i_n)$. On représente cette variété tropicale pour un graphe balancé (le $\textit{graphe tropical sécante}$). La combinatoire qu'on utilise pour le calcul du degré de ces variétés sécantes classiques n'est pas triviale, et a été developpée par K. Ranestad. En utilisant des techniques de la géométrie tropicale, on donne des algorithmes qui calculent le degré (même le multidegré) et le polytope de Newton de la première sécante d'une courbe monomiale de $\mathbb{P}^4$.

About the Defectivity of Certain Segre–Veronese Varieties

2008 ◽  
Vol 60 (5) ◽  
pp. 961-974 ◽  
Author(s):  
Silvia Abrescia
Keyword(s):  
Secant Varieties ◽  
Secant Variety ◽  
Determinantal Equation ◽  
Veronese Varieties

AbstractWe study the regularity of the higher secant varieties of ℙ1 × ℙn, embedded with divisors of type (d, 2) and (d, 3). We produce, for the highest defective cases, a “determinantal” equation of the secant variety. As a corollary, we prove that the Veronese triple embedding of ℙn is not Grassmann defective.

scholarly journals A Characterization of Secant Varieties of Severi Varieties Among Cubic Hypersurfaces

2020 ◽  
Author(s):  
Baohua Fu ◽  
Yewon Jeong ◽  
Fyodor L Zak
Keyword(s):  
Lie Algebra ◽  
Singular Locus ◽  
Secant Varieties ◽  
Secant Variety ◽  
Severi Varieties ◽  
Severi Variety ◽  
Cubic Hypersurfaces

Abstract It is shown that an irreducible cubic hypersurface with nonzero Hessian and smooth singular locus is the secant variety of a Severi variety if and only if its Lie algebra of infinitesimal linear automorphisms admits a nonzero prolongation.

scholarly journals Symmetric Tensor Rank and Scheme Rank: An Upper Bound in terms of Secant Varieties

Geometry ◽  
2013 ◽  
Vol 2013 ◽  
pp. 1-3 ◽  
Author(s):  
E. Ballico
Keyword(s):  
Upper Bound ◽  
Linear Span ◽  
Symmetric Tensor ◽  
Tensor Rank ◽  
Secant Varieties ◽  
Secant Variety ◽  
Symmetric Tensor Rank ◽  
Tangent Vectors ◽  
Minimal Integer

Let X⊂ℙr be an integral and nondegenerate variety. Let c be the minimal integer such that ℙr is the c-secant variety of X, that is, the minimal integer c such that for a general O∈ℙr there is S⊂X with #(S)=c and O∈〈S〉, where 〈 〉 is the linear span. Here we prove that for every P∈ℙr there is a zero-dimensional scheme Z⊂X such that P∈〈Z〉 and deg(Z)≤2c; we may take Z as union of points and tangent vectors of Xreg.

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