Almost set-theoretic complete intersections in characteristic zero (Q2642841)

A classical problem in algebraic geometry is to determine the minimum number of equations needed to define an algebraic variety \(V\) set-theoretically. This number is always bounded below by the codimension of \(V\). If equality holds, the variety \(V\) is called set-theoretic complete intersection. Let \(K\) be an algebraically closed field and \(V\) an affine toric variety of codimension \(r \geq 2\) over \(K\). The variety \(V\) is called simplicial when it has a parametrization of the form \(x_1 = u_1^{c}, \ldots, x_n = u_n^{c}, \ldots,y_i = u_1^{a_{i1}} \ldots u_n^{a_{in}}, \,i = 1, \ldots, r,\) where \(c\) is a natural number and, for all \(i=1,\ldots,r\), \((a_{i1},\ldots,a_{in})\) are non-zero integer vectors with non-negative entries. It was proven by \textit{M. Barile, M. Morales} and \textit{A. Thoma} [J. Algebra 226, No. 2, 880--892 (2000; Zbl 0971.14041)] that every simplicial toric affine variety of codimension two is defined by three binomial equations. Moreover they showed that any simplicial toric affine variety with full parametrization, i.e. all \(a_{jk}\) are positive, is defined by \((r+1)\) binomial equations. In the paper under review the author presents a class of simplicial toric varieties, which does not have a full parametrization, defined by \((r+1)\) binomial equations in any characteristic \(p\). When \(p>0\), it is shown that the above class is set-theoretic complete intersection on binomials.