We show that for all n â‰¥ 3 and all primes p there are infinitely many simplicial toric varieties of codimension n in the 2n-dimensional affine space whose minimum number of defining equations is equal to n in characteristic p, and lies between 2n âˆ’ 2 and 2n in all other characteristics. In particular, these are new examples of varieties which areâ€¦ (More)

We show that for the edge ideals of the graphs consisting of one cycle or two cycles of any length connected through a vertex or a path, the arithmetical rank equals the projective dimension.

We show that for every prime p, there is a class of Veronese varieties which are set-theoretic complete intersections if and only if the ground field has characteristic p.

We present a class of toric varieties V which, over any algebraically closed field of characteristic zero, are defined by codim V +1 binomial equations .

We present a class of homogeneous ideals which are generated by monomials and binomials of degree two and are set-theoretic complete intersections. This class includes certain reducible varieties of minimal degree and, in particular, the presentation ideals of the fiber cone algebras of monomial varieties of codimension two.

We describe a class of toric varieties in the N -dimensional affine space which are minimally defined by no less than N âˆ’ 2 binomial equations. Introduction The arithmetical rank (ara) of an algebraic variety is the minimum number of equations that are needed to define it set-theoretically. For every affine variety V âŠ‚ K we have that codimV â‰¤ araV â‰¤ N .â€¦ (More)