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Adv. Pure Appl. Math. 4 (2013), 215–250 DOI 10.1515/apam-2013-0015 © de Gruyter 2013 Stratification of the fourth secant variety of Veronese varieties via the symmetric rank Edoardo Ballico and Alessandra Bernardi Abstract. IfX Pn is a projective non-degenerate variety, theX -rank of a point P 2 Pn is defined to be the minimum integer r such thatP belongs to the span of r points ofX . We describe the complete stratification of the fourth secant variety of any Veronese variety X via the X -rank. This result has an equivalent translation in terms either of

Higher secant varieties of Segre-Veronese varieties M.V. Catalisano, A.V. Geramita and A. Gimigliano Abstract. In this paper we consider the Segre-Veronese varieties, i.e., the embeddings of Pn1 × · · · × Pnt in the projective space PN via divisors of multi-degree (d1, . . . , dt), (N = Π di+ni ni ), and we study the dimension of their higher secant varieties. We give the dimensions of all the higher secant varieties of P1 × P1 embedded by divisors of any bi-degree (d1, d2). We find that P r × Pk, embedded by divisors of bi-degree (k + 1, 1), has no deficient

Characterization of Veronese varieties via projection in Grassmannians Enrique Arrondo and Raffaella Paoletti A Giacomo Paoletti, amico e padre Abstract. We characterize for any d the d-uple Veronese embedding of Pn as the only variety that, under certain general conditions, can be projected from the Grassmannian of (d − 1)-planes in Pnd+d−1 to the Grassmannian of (d − 1)-planes in Pn+2d−3 in such a way that two (d − 1)-planes meet at most in one point. We also study the relation of this problem with the Steiner bundles over Pn. 2000 Mathematics Subject

c© de Gruyter 2008 J. Math. Crypt. 2 (2008), 63–107 DOI 10.1515 / JMC.2008.004 A geometric view of cryptographic equation solving S. Murphy and M. B. Paterson Communicated by Hideki Imai Abstract. This paper considers the geometric properties of the Relinearisation algorithm and of the XL algorithm used in cryptology for equation solving. We give a formal description of each algo- rithm in terms of projective geometry, making particular use of the Veronese variety. We establish the fundamental geometrical connection between the two algorithms and show how both

surface 𝒱 {\mathcal{V}} (for further details about the Veronese surface over finite fields see [ 11 , Chapter 25]). If we use the so-called determinantal representation of the Veronese variety of degree 2 (see [ 10 , Example 2.6]), then PG ⁡ ( 5 , q ) = PG ⁡ ( Sym 3 ⁡ ( q ) , q ) {\operatorname{PG}(5,q)=\operatorname{PG}(\operatorname{Sym}_{3}(q),q)} , where Sym 3 ⁡ ( q ) {\operatorname{Sym}_{3}(q)} is the space of symmetric matrices of order 3, and the points of 𝒱 {\mathcal{V}} correspond to the matrices of rank 1, i.e. P ∈ 𝒱 {P\in\mathcal{V}} if and only

Abstract

Fix integers n, x, k such that n≥3, k>0, x≥4, (n, x)≠(3, 4) and k(n+1)<(nn+x). Here we prove that the order x Veronese embedding ofP n is not weakly (k−1)-defective, i.e. for a general S⊃P n such that #(S) = k+1 the projective space | I 2S (x)| of all degree t hypersurfaces ofP n singular at each point of S has dimension (n/n+x )−1− k(n+1) (proved by Alexander and Hirschowitz) and a general F∈| I 2S (x)| has an ordinary double point at each P∈ S and Sing (F)=S.

[1] B. Ådlandsvik: “Joins and higher secant varieties”, Math. Scand., Vol. 61, (1987), pp. 213–222. [2] J. Alexander and A. Hirschowitz: “Polynomial interpolation in several variables”, J. of Alg. Geom., Vol. 4 (1995), pp. 201–222. [3] J. Alexander and A. Hirschowitz: “An asymptotic vanishing theorem for generic unions of multiple points”, Invent. Math., Vol. 140, No. 2, (2000), pp. 303–325. http://dx.doi.org/10.1007/s002220000053 [4] E. Ballico: “On the secant varieties to the tangent developable of a Veronese variety”, preprint, (2003). [5] M.V. Catalisano

Contents Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . v E. Arrondo, R. Paoletti Characterization of Veronese varieties via projections in Grassmannians . . . . . . . 1 E. Ballico, C. Fontanari Birational geometry of defective varieties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .13 M. Beltrametti, M.L. Fania Fano Threefolds as hyperplane sections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 C

.V., Gimigliano A., Erratum to: “Ranks of tensors, secant varieties of Segre varieties and fat points” [Linear Algebra Appl. 355 (2002) 263–285], Linear Algebra Appl., 2003, 367, 347–348 http://dx.doi.org/10.1016/S0024-3795(03)00455-5 [9] Catalisano M.V., Geramita A.V., Gimigliano A., Higher secant varieties of Segre-Veronese varieties, In: Projective Varieties with Unexpected Properties, Walter de Gruyter, Berlin, 2005, 81–107 [10] Catalisano M.V., Geramita A.V., Gimigliano A., Secant varieties of Grassmann varieties, Proc. Amer. Math. Soc., 2005, 133(3), 633–642 http

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. T. CHUNG Multiple solutions for a class of p.x/-Kirchhoff type problems with Neumann boundary conditions 165 H. BOR A further application of power increasing sequences 179 S. A. PRASAD Regularity of fractal interpolation functions via wavelet transform 189 G. ZHONG, X. MA, L. YANG, J. XIA Finite groups with some weakly s-semipermutable subgroups 203 E. BALLICO, A. BERNARDI Stratification of the fourth secant variety of Veronese varieties via the symmetric rank 215