Arithmetical Ranks of Ideals Associated to Symmetrical and Alternating Matrices

A note on Veronese varieties

“…) is the determinantal variety defined by the vanishing of the 2-minors of an n×n symmetric variety of indeterminates over K. Theorem 2 generalizes part of the results in [1]: there it was shown that the minimum number of equations required to define Remark 3 A general lower bound for the minimum number of equations which define a variety set-theoretically (the so-called arithmetical rank, ara) is given by the local cohomological dimension: if I = I(V ), this number is cd I = max{n ∈ N | H n I (R) = 0}, where H · I denotes local cohomology with respect to I. In the case of the Veronese variety V = V n p h , the ideal I is perfect (see [6], p. 259), so that, according to [9], Prop.…”

On a special class of simplicial toric varieties

“…The paper [3] presents an infinite class of simplicial toric varieties E-mail address: barile@dm.uniba.it. 1 of codimension 2 which are set-theoretic complete intersections only in one positive characteristic. The same property has been shown in [2] for the Veronese varieties whose degree is a prime power.…”

Local cohomology modules supported at determinantal ideals

“…In addition to determinantal ideals, our methods extend to ideals generated by Pfaffians of alternating matrices, Section 6, and minors of symmetric matrices, Section 7. For these, we use Barile's computations of arithmetic rank from [Ba2]. Section 8 deals with questions on arithmetic rank related to the vanishing theorems proved in our paper.…”