Algebraic Barth-Lefschetz theorems

We shall work over a fixed algebraically closed field k of arbitrary characteristic. By an algebraic variety over k we shall mean a reduced algebraic scheme over k. Fix a positive integer n and e = (e0, el,…, en ) a system of n + 1 weights (i.e. n + 1 positive integers e0, el,…, en ). If k[T0, Tl ,…, Tn] is the polynomial k-algebra in n + 1 variables, graded by the conditions deg(T i) = ei i = 0, 1,…, n, denote by Pn(e) = Proj(k[T0, T1,…, Tn]) the n-dimensional weighted projective space over k of weights e. We refer the reader to [3] for the basic properties of weighted projective spaces.