839,050 research outputs found

    F-structures and integral points on semiabelian varieties over finite fields

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    Motivated by the problem of determining the structure of integral points on subvarieties of semiabelian varieties defined over finite fields, we prove a quantifier elimination result for certain modules over finite simple extensions of the integers given together with predicates for orbits of the distinguished generator of the ring.Comment: 33 pages, correction made to authors' informatio

    Secant varieties of toric varieties

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    Let XPX_P be a smooth projective toric variety of dimension nn embedded in \PP^r using all of the lattice points of the polytope PP. We compute the dimension and degree of the secant variety \Sec X_P. We also give explicit formulas in dimensions 2 and 3 and obtain partial results for the projective varieties XAX_A embedded using a set of lattice points A \subset P\cap\ZZ^n containing the vertices of PP and their nearest neighbors.Comment: v1, AMS LaTex, 5 figures, 25 pages; v2, reference added; v3, This is a major rewrite. We have strengthened our main results to include a classification of smooth lattice polytopes P such that Sec X_P does not have the expected dimension. (See Theorems 1.4 and 1.5.) There was also a considerable amount of reorganization, and some expository material was eliminated; v4, 28 pages, minor corrections, additional and updated reference

    Most secant varieties of tangential varieties to Veronese varieties are nondefective

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    We prove a conjecture stated by Catalisano, Geramita, and Gimigliano in 2002, which claims that the secant varieties of tangential varieties to the ddth Veronese embedding of the projective nn-space Pn\mathbb{P}^n have the expected dimension, modulo a few well-known exceptions. As Bernardi, Catalisano, Gimigliano, and Id\'a demonstrated that the proof of this conjecture may be reduced to the case of cubics, i.e., d=3d=3, the main contribution of this work is the resolution of this base case. The proposed proof proceeds by induction on the dimension nn of the projective space via a specialization argument. This reduces the proof to a large number of initial cases for the induction, which were settled using a computer-assisted proof. The individual base cases were computationally challenging problems. Indeed, the largest base case required us to deal with the tangential variety to the third Veronese embedding of P79\mathbb{P}^{79} in P88559\mathbb{P}^{88559}.Comment: 25 pages, 2 figures, extended the introduction, and added a C++ code as an ancillary fil