Elimination theory has many applications, in particular, it describes explicitly an image of a complex line under rational transformation and determines the number of common zeroes of two polynomials in one variable. We generalize classical elimination theory and create elimination theory along an algebraic curve using the notion of determinantal representation of algebraic curve. This new theo...

let $r$ be a ring‎, ‎and let $n‎, ‎d$ be non-negative integers‎. ‎a right $r$-module $m$ is called $(n‎, ‎d)$-projective if $ext^{d+1}_r(m‎, ‎a)=0$ for every $n$-copresented right $r$-module $a$‎. ‎$r$ is called right $n$-cocoherent if every $n$-copresented right $r$-module is $(n+1)$-coprese-nted‎, ‎it is called a right co-$(n,d)$-ring if every right $r$-module is $(n‎, ‎d)$-projective‎. ‎$r$ ...

In this paper we consider an effective divisor on the complex projective line and associate with it the module D consisting of all the derivations θ such that θ(Ii) ⊂ Ii i for every i, where Ii is the ideal of pi. The module D is graded and free of rank 2; the degrees of its homogeneous basis, called the exponents, form an important invariant of the divisor. We prove that under certain conditio...

We prove, as an analogy of a conjecture of Artin, that if Y −→ X is a finite flat morphism between two singular reduced absolutely irreducible projective algebraic curves defined over a finite field, then the numerator of the zeta function of X divides those of Y in Z[T ]. Then, we give some interpretations of this result in terms of semi-abelian varieties.

A semi-dualizing module over a commutative noetherian ringA is a finitely generated module C with RHomA(C,C) ≃ A in the derived category D(A). We show how each such module gives rise to three new homological dimensions which we call C–Gorenstein projective, C–Gorenstein injective, and C–Gorenstein flat dimension, and investigate the properties of these dimensions.

We introduce a version of the P=W conjecture relating Borel–Moore homology stack representations fundamental group genus g Riemann surface with degree zero semistable Higgs bundles on smooth projective complex curve g. In order to state we propose construction canonical isomorphism between these groups. relate stacky original concerning cohomology moduli spaces, and PI=WI intersection groups si...

A semioval in a finite projective plane is a non-empty pointset S with the property that for every point in S there exists a unique line tP such that S ∩ tP = {P}. This line is called the tangent to S at P . Semiovals arise in several parts of finite geometries: as absolute points of a polarity (ovals, unitals), as special minimal blocking sets (vertexless triangle), in connection with cryptogr...

This study of properly or strictly convex real projective manifolds introduces notions of parabolic, horosphere and cusp. Results include a Margulis lemma and in the strictly convex case a thick-thin decomposition. Finite volume cusps are shown to be projectively equivalent to cusps of hyperbolic manifolds. This is proved using a characterization of ellipsoids in projective space. Except in dim...

In a graph Γ = (V,E) a vertex v is resolved by a vertex-set S = {v1, . . . , vn} if its (ordered) distance list with respect to S, (d(v, v1), . . . , d(v, vn)), is unique. A set A ⊂ V is resolved by S if all its elements are resolved by S. S is a resolving set in Γ if it resolves V . The metric dimension of Γ is the size of the smallest resolving set in it. In a bipartite graph a semi-resolving...

Let G be a connected complex semi-simple group, B ⊂ G a Borel subgroup, and T ⊂ B a maximal torus. We construct a class of smooth T stable subvarieties inside the flag variety G/B, each of which is an embedding of a product of projective lines.