On a special class of simplicial toric varieties

Abstract

We show that for all n ⩾ 3 and all primes p there are infinitely many simplicial toric varieties of codimension n in the 2 n-dimensional affine space whose minimum number of defining equations is equal to n in characteristic p, and lies between 2 n − 2 and 2 n in all other characteristics. In particular, these are new examples of varieties which are set-theoretic complete intersections only in one positive characteristic. Moreover, we show that the minimum number of binomial equations which define these varieties in all characteristics is 4 for n = 3 and 2 n − 2 + ( n − 2 2 ) whenever n ⩾ 4 .