Intersections Of Codimension Research Articles

Minimal Resolutions Over Codimension 2 Complete Intersections

We construct an explicit free resolution T for a maximal Cohen-Macaulay module M over a local complete intersection of codimension 2 with infinite residue field. The resolution is minimal when the module M is a sufficiently high syzygy. Our starting point is a layered free resolution L, described in [7], of length 2 over a regular local ring. We provide explicit formulas for the differential in T in terms of the differential and homotopies on the finite resolution L. One application of our construction is to describe Ulrich modules over a codimension 2 quadratic complete intersection.

Some Characterizations of Singular Components of Springer Fibers in the Two-Column Case

Let u be a nilpotent endomorphism of a finite dimensional ℂ-vector space. The set \({\mathcal F}_u\) of u-stable complete flags is a projective algebraic variety called a Springer fiber. Its irreducible components are parameterized by a set of standard tableaux. We provide three characterizations of the singular components of \({\mathcal F}_u\) in the case u2 = 0. First, we give the combinatorial description of standard tableaux corresponding to singular components. Second, we prove that a component is singular if and only if its Poincare polynomial is not palindromic. Third, we show that a component is singular when it has too many intersections of codimension one with other components. Finally, relying on the second criterion, we infer that, for u general, whenever \({\mathcal F}_u\) has a singular component, it admits a component whose Poincare polynomial is not palindromic. This work relies on a previous criterion of singularity for components of \({\mathcal F}_u\) in the case u2 = 0 by the first author and on the description of the B-orbit decomposition of orbital varieties of nilpotent order two by the second author.

Intersecting hypersurfaces in anti-de Sitter and Lovelock gravity

Colliding and intersecting hypersurfaces filled with matter (membranes) are studied in the Lovelock higher order curvature theory of gravity. Lovelock terms couple hypersurfaces of different dimensionalities, extending the range of possible intersection configurations. We restrict the study to constant curvature membranes in constant curvature anti-de Sitter (AdS) and dS background and consider their general intersections. This illustrates some key features which make the theory different from the Einstein gravity. Higher co-dimension membranes may lie at the intersection of co-dimension one hypersurfaces in Lovelock gravity; the hypersurfaces are located at the discontinuities of the first derivative of the metric, and they need not carry matter. The example of colliding membranes shows that general solutions can only be supported by (spacelike) matter at the collision surface, thus naturally conflicting with the dominant energy condition (DEC). The imposition of the DEC gives selection rules on the types of collision allowed. When the hypersurfaces do not carry matter, one gets a solitonlike configuration. Then, at the intersection one has a co-dimension two or higher membrane standing alone in AdS-vacuum space–time without conical singularities. Another result is that if the number of intersecting hypersurfaces goes to infinity the limiting space–time is free of curvature singularities if the intersection is put at the boundary of each AdS bulk.

ON TWO SIMPLE CRITERIA FOR RECOGNIZING COMPLETE INTERSECTIONS IN CODIMENSION 2

Developing a previous idea of Faltings, we characterize the complete intersections of codimension 2 in P n , n ≥ 3, over an algebraically closed field of any characteristic, among l.c.i. X, as those that are subcanonical and scheme-theoretically defined by p ≤ n − 1 equations. Moreover, we give some other results assuming that the normal bundle of X extends to a numerically split bundle on P n and p ≤ n. Finally, using our characterization, we give a (partial) answer to a question posed recently by Franco, Kleiman and Lascu ([5]) on self-linking and complete intersections in positive characteristic.