863,233 research outputs found
F-structures and integral points on semiabelian varieties over finite fields
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
Let be a smooth projective toric variety of dimension embedded in
\PP^r using all of the lattice points of the polytope . 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 embedded using a set of lattice points A \subset P\cap\ZZ^n
containing the vertices of 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
We prove a conjecture stated by Catalisano, Geramita, and Gimigliano in 2002,
which claims that the secant varieties of tangential varieties to the th
Veronese embedding of the projective -space 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., , the main contribution of this work
is the resolution of this base case. The proposed proof proceeds by induction
on the dimension 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 in .Comment: 25 pages, 2 figures, extended the introduction, and added a C++ code
as an ancillary fil
- …