Union Of Lines Research Articles

Seminormality and cohomology of projective varieties

Let X be a reduced closed subscheme of p; = Proj R = Y (k a field and R = k[X,,,..., A’,], r 2 2). Let A be the homogeneous co-ordinate ring of X. In [2, 3, 51 the various authors have attempted to determine whether A is CohenMacaulay, or seminormal (or more generally, to study the seminormalization or Cohen-Macaulification of A). The papers mentioned above work primarily with unions of straight lines. In [S, Corollary 5.91 it is proved, for example, that if X is a connected union of lines which have linearly independent directions at each intersection point, then A is seminormal if and only if A is Cohen-Macaulay. The aim of this paper is to give a better understanding of the seminormalization of A and the relationship between seminormality and CohenMacaulayness hinted at by the above mentioned Corollary 5.9. This we do in Section 2 by interpreting the seminormalization of A in terms of sheaf cohomology. We are then able to give a reasonable generalization (our Corollary 3.5) of [S, Corollary 5.91, and also to resolve several computational problems left open in [2, 31, and [4]. The Hilbert function of a graded ring A will be denoted by HA. That is, HA(i) = dim, Ai. If X= Proj A, the Hilbert polynomial of A is denoted HP(X) in some computations of Section 6.

Two characterizations of linear Baire spaces

The Wilansky-Klee conjecture is equivalent to the (unproved) conjecture that every dense, 1 1 -codimensional subspace of an arbitrary Banach space is a Baire space (second category in itself). The following two characterizations may be useful in dealing with this conjecture: (i) A topological vector space is a Baire space if and only if every absorbing, balanced, closed set is a neighborhood of some point, (ii) A topological vector space is a Baire space if and only if it cannot be covered by countably many nowhere dense sets, each of which is a union of lines ( 1 1 -dimensional subspaces). Characterization (i) has a more succinct form, using the definition of Wilansky’s text [8, p. 224]: a topological vector space is a Baire space if and only if it has the t t property.