In this short note we prove that in many cases the failure of a variety to be separably rationally connected is caused by the instability of the tangent sheaf (if there are no other obvious reasons). A simple application of the results proves that a smooth Fano complete intersection is separably rationally connected if and only if it is separably uniruled. In particular, a general such Fano com...

A word w is a fixed point of a nontrival morphism h if w = h(w) and h is not the identity on the alphabet of w. The paper presents the first polynomial algorithm deciding whether a given word is such a fixed point. The algorithm also constructs the corresponding morphism, which has the smallest possible number of non-erased letters.

We introduce the concept of firm classes of morphisms as basis for the axiomatic study of completions of objects in arbitrary categories. Results on objects injective with respect to given morphism classes are included. In a finitely well-complete category, firm classes are precisely the coessential first factors of morphism factorization structures.

We characterize the (effective) E-descent morphisms in the category Cat of small categories, when E is the class of discrete fibrations or the one of discrete cofibrations, and prove that every effective global-descent morphism is an effective E-descent morphism while its converse fails.

Let k be a field of characteristic zero. For every smooth, projective k-variety Y of dimension n which admits a connected, proper morphism f : Y → S of relative dimension one, we construct idempotent correspondences (projectors) πij(Y ) ∈ CH (Y × Y,Q) generalizing a construction of Murre. If n = 3 and the transcendental cohomology group H tr(Y ) has the property that H tr(Y,C) = f H tr(S,C) + I...

We consider algebraic manifolds Y of dimension 3 over C with H(Y,ΩjY ) = 0 for all j ≥ 0 and i > 0. Let X be a smooth completion of Y with D = X − Y , an effective divisor on X with normal crossings. If the D-dimension of X is not zero, then Y is a fibre space over a smooth affine curve C (i.e., we have a surjective morphism from Y to C such that the general fibre is smooth and irreducible) suc...

We provide the main results of a deformation theory of smooth formal schemes as defined in [2]. Smoothness is defined by the local existence of infinitesimal liftings. Our first result is the existence of an obstruction in a certain Ext group whose vanishing guarantees the existence of global liftings of morphisms. Next, given a smooth morphism f0 : X0 → Y0 of noetherian formal schemes and a cl...

Let X be a smooth projective variety over an algebraically closed field k of characteristic zero and Y ⊂ X a smooth ample hyperplane section. The Weak Lefschetz conjecture for Chow groups states that the natural restriction map CH(X)Q → CH(Y )Q is an isomorphism for all p < dim(Y )/2. In this note, we revisit a strategy introduced by Grothendieck to attack this problem by using the Bloch-Quille...

Problem 1. Let R be a ring and let M be a left R-module. (a) Let I be a right ideal in R and let IM be the abelian subgroup of M generated by the elements rm, r ∈ I, m ∈M . Then R/I ⊗RM ∼= M/IM as an abelian group. (b) If I is a two-sided ideal, then the isomorphism in (a) is that of left R-modules. (c) Let R be commutative, I, J ⊂ R be ideals. Then R/I ⊗R R/J ∼= R/(I + J) as an R-module. (d) L...