We explore the interlacing between model category structures attained to classes of modules of finite X -dimension, for certain classes of modules X . As an application we give a model structure approach to the Finitistic Dimension Conjectures and present a new conceptual framework in which these conjectures can be studied. Let Λ be a finite dimensional algebra over a field k (or more generally...

In dimension 1 and 2 and over an algebraically closed field of characteristic 0, unirationality implies rationality (hence a fortiori stable rationality). In the surface case, this follows from the Castenuovo characterization of smooth projective rational surfaces as those smooth projective surfaces X satisfying q(X) = p2(X) = 0 (see [5]). Starting from dimension 3, the answer to the Lüroth pro...

We describe new classes of noetherian local rings R whose finitely generated modules M have the property that ToriR(M,M)=0 for i≫0 implies has finite projective dimension, or ExtRi(M,M)=0 dimension injective dimension.

We prove that every projective variety of dimension n over a field of positive characteristic admits a morphism to projective n-space, étale away from the hyperplane H at infinity, which maps a chosen divisor into H and a chosen smooth point not on the divisor to some point not in H .

We generalize Horrocks’ criterion for the splitting of vector bundles on projective space. We establish an analogous splitting criterion for vector bundles on arbitrary smooth complex projective varieties of dimension ≥ 4, which asserts that a vector bundle E on X splits iff its restriction E|Y to an ample smooth codimension 1 subvariety Y ⊂ X splits.

A new method of fractal measurement, the projective covering method (PCM), is proposed in this Letter. Based on the measurements by means of a laser profilometer, the fractal dimension D E [2,3) of a fracture surface is directly estimated. The projective covering probability function is introduced to systematically analyze the multifractal behavior of the fracture surfaces. @ 1998 Elsevier Scie...

It is shown that a complex normal projective variety has non-positive Kodaira dimension if it admits a non-isomorphic quasi-polarized endomorphism. The geometric structure of the variety is described by methods of equivariant lifting and fibrations. Endomorphisms of the projective spaces are also discussed and some results on invariant subvarieties under the pullback of the endomorphism are obt...

We generalize the well-known numerical criterion for projective spaces by Cho, Miyaoka and Shepherd-Barron to varieties with at worst quotient singularities. Let X be a normal projective variety of dimension n ≥ 3 with at most quotient singularities. Our result asserts that if C · (−KX) ≥ n + 1 for every curve C ⊂ X, then X ∼= P .

A projective algebraic surface which is homeomorphic to a ruled surface over a curve of genus g ≥ 1 is itself a ruled surface over a curve of genus g. In this note, we prove the analogous result for projective algebraic manifolds of dimension 4 in case g ≥ 2.

An (r,M, 2δ; k)q constant–dimension subspace code, δ > 1, is a collection C of (k−1)–dimensional projective subspaces of PG(r−1, q) such that every (k−δ)–dimensional projective subspace of PG(r−1, q) is contained in at most a member of C. Constant–dimension subspace codes gained recently lot of interest due to the work by Koetter and Kschischang [18], where they presented an application of such...