We work over any algebraically closed field F. However the applications are not vacuos only if char (F) > 0.
A finite set S in a projective space V is said to be in t-unifomn position, t an integer, if for any two subsets A , B of S with card
it is in t-uniform position for every t. 9 is called in linear general position if it is in 1-uniform position. It is known in several cases (always if char (F) = 0) that for every irreducible, nondegenerate curve C c P and for a general hyperplane H of P, C n H is in uniform position in H (see 3 1 for more details and the references). Definition 1. A non-degenerate, irreducible curve C c P" is called very strange if for a general hyperplane H , C n H is not in linear general position in H . Definition 2. A finite set S in a projective space-H is called in linenr semi-uniform position if it spans H and for any two linear subspaces L, M of H with dim (L) = dim ( H ) and L (resp. H) spanned by L n S (resp. M n S), we have card (L n S) = card ( M n S).
Since every symmetric product of an irreducible curve is irreducible, for every irreducible, nondegenerate curve C c PI, for a general hyperplane H of Pc, C n H is in linear semi-uniform position in H . In this paper we show how to control the postulation of any finite subset in linear semi-uniform position in any projective space H .
Our first result is the following one. * Theorem 0.1. For every finite d a e t S c P, S in linear aemi-unifomc position, and every integer t > 0, we have hO(P", O,(t)) -hO(Pm, J8,,,(t)) 2 min (card (S), mt + 1). Theorem 0.1 gives that [9], th. 1 (iii) and [4], th. 3.7 hold in any characteristic even for singular varieties and singular curves. If the set S t P is in linear semi-uniform position, but not in linear general po8itionJ it is possible (see 2.4, 2.5) it is possible to improve very much the bound of 0.1, obtaining better bounds for the arithmetic genus of very strange curves. The proofs are very, very, very elementary. This paper was stimulated by [9], 5 1 (see also [8], [lo], [ill, [3]), by RAT-% thesis (see [7]) and by several papera by KL-(e.g. [5]).