Even sets of nodes are bundle symmetric

Let k be an algebraically closed field of characteristic p different from 2, and let F be a nodal surface of degree d in the projective 3-space P over k (i.e. the singularities of F are only ordinary quadratic, nodes for short). Let N be a subset of the set of nodes of F: then N is said to be n/2-even for n=0,1 if the following condition (*) holds. Namely, let F' be a minimal resolution of the singularities of F, let N' be the inverse image of N in F', let H be the inverse image of a plane section of F. Then N is n/2 if: (*) the class of N'+nH is 2-divisible in Pic(F'). In this paper we prove the following characterization of even sets, which has been conjectured by W. Barth and the second author in 1979. Let F be as above. Then the n/2-even set of nodes N on F are exactly the bundle symmetric sets, i.e. the degeneracy loci of symmetric maps s:E*(-d-n)->E of a suitable locally free sheaf E over P (i.e. F is the locus where rk(s)<rkE, N is the locus where rk(s)=rkE-2). We also give a classification of even sets in degree d=4,5.