‎Let $C$ be a semidualizing module‎. ‎We first investigate the properties of‎ ‎finitely generated $G_C$-projective modules‎. ‎Then‎, ‎relative to $C$‎, ‎we introduce and study the rings over which‎ ‎every submodule of a projective (flat) module is $G_C$-projective (flat)‎, ‎which we call $C$-Gorenstein (semi)hereditary rings‎. ‎It is proved that every $C$-Gorenstein hereditary ring is both cohe...

Let $R$ be a commutative Noetherian ring. We prove that  over a local ring $R$ every finitely generated $R$-module $M$ of finite Gorenstein projective dimension has a Gorenstein projective cover$varphi:C rightarrow M$ such that $C$ is finitely generated and the projective dimension of $Kervarphi$ is finite and $varphi$ is surjective.

In [1] and [2] we defined new invariants of both bipartite and multipartite quantum systems under local unitary transformations via algebraic-geometry of determinantal varieties, and gave a new separability criterion based on our new invariants. The purpose of this note is to show how Schmidt number of pure states in bipartite systems, a classical invariant, can be read out from algebraic-geome...

Recent results in geometric invariant theory (GIT) for non-reductive linear algebraic group actions allow us to stratify quotient stacks of the form [X/H ], where X is a projective scheme and H is a linear algebraic group with internally graded unipotent radical acting linearly onX , in such a way that each stratum [S/H ] has a geometric quotient S/H . This leads to stratifications of moduli st...

Parallax displacement is used to determine a relative 3D conic projective structure. This value is invariant between any number of views in time if the conic is not moving with respect to the plane of the homography. It can be used to determine conic correspondence between three simultaneous views. The corresponding conics may then be used to determine the epipolar geometry. This method of dete...

This paper shows how the computation of the singularity locus of a 3R robot can be reduced to the analysis of the relative position of two coplanar ellipses. Since the relative position of two conics is a projective invariant, and the basic projective geometric invariants are determinants, it is not surprising that, using Distance Geometry, the computation of the singularity locus of a 3R robot...

Raphaël Rouquier introduced an invariant of triangulated categories which is known as dimension. Orlov conjectured that for any smooth quasi-projective variety X the dimension $$D^{\textrm{b}}_{\textrm{coh}}(X)$$ equal to $$\textrm{dim}\, X$$ . In this note we show some blow-ups projective spaces satisfy Orlov’s conjecture. This includes a blow-up $${\mathbb {P}}^2$$ in nine arbitrary distinct ...

We discuss inequalities between the values of \emph{homotopical and cohomological Poincar\'e polynomials} self-products rationally elliptic spaces. For quasi-projective varieties, we prove generating functions for ranks graded pieces weight Hodge filtrations canonical mixed structures on homotopy cohomology groups. Several examples such polynomials related algebraic varieties are presented. One...

Serre [Se 64] and Abelson [Ab 74] have produced examples of conjugate algebraic varieties which are not homeomorphic. We show that if the field of definition of a polarized projective variety coincides with its field of moduli then all of its conjugates have the same topological type. This immediately extends the class of varietie s known to possess invariant topological type to all canonically...

This article is a survey of recent work [15, 6, 7, 13] developing a new approach to quantization based on the equivariance with respect to some Lie group of symmetries. Examples are provided by conformal and projective differential geometry: given a smooth manifold M endowed with a flat conformal/projective structure, we establish a canonical isomorphism between the space of symmetric contravar...