Frobenius semisimplicity for convolution morphisms

Rendiconti del Seminario Matematico della Università di Padova, 1996

For every projective scheme X and projective embedding e : X → P, there is a reduced scheme C_n(X,e) parametrizing all cycles of pure dimension n with support in X. The construction is carried over in terms of the given embedding, so that in particular the space of cycles C_n(X,e) comes out equipped with a partition into disjoint subschemes C_{n,k}(X,e), each endowed with a natural projective embedding, for all degrees k of cycles in P. The idea goes back to Cayley, but the effective construction is due to Chow. In recent times new important applications appeared which motivate the present contribution to a long standing problem of setting suitable foundations. Over the complex field, there is a theorem of Barlet saying that different projective embeddings e do always give rise to isomorphic schemes C_n(X,e). This is not true in positive characteristics as shown by Nagata. We therefore confine ourselves to complex schemes. The question arises how to describe morphisms T → C_n(X,e) intrinsically by means of incidence correspondences (cycles) Z in T \times X, without reference to the embedding. This would be an “explicit" universal property for C_n(X,e), for the essential independence of e is already an “implicit" one. In general, a (relative) incidence cycle Z only induces a regular map on the smooth locus T_{sm} → C_n(X,e). The geometrically relevant point is therefore to determine when the induced map on T_{sm} admits a continuous extension f_Z : T → C_n(X,e), what we call a "semi-regular" (rational) map. This requires a careful study of the limit behaviour of fibre cycles, when approaching a singular point moving along the smooth locus in the parameter variety T. We prove that there is a unique "limit cycle" along any path contained within a given branch of T at the singular point. The "regular" cycles Z, those which correspond to semi-regular maps into C_n(X,e), are therefore characterized by the property that the limit cycle is the same one along every branch. This is the main result of the present paper, which is missing in the existing literature. More precisely, we prove that the space of cycles C_n(X,e) represents a certain functor of regular relative cycles on the category of reduced schemes and semi-regular maps. This implies the earlier result of Andreotti-Norguet, that the semi-normalization of C_n(X,e) is essentially independent of the embedding. Finally, once a (regular) cycle Z is known to determine a semi-regular map f_Z, then the property of f_Z being everywhere regular is essentially independent of the embedding e, according to the theorem of Barlet.

This file, provided as bibliographical information for the benefit of contributors, contains the table of contents and editors' foreword from the proceedings of the July 1996 Warwick EuroConference on Algebraic Geometry.

2011

Two quasi-projective varieties are called piecewise isomorphic if they can be stratified into pairwise isomorphic strata. We show that the m-th symmetric power S m (C n) of the complex affine space C n is piecewise isomorphic to C mn and the m-th symmetric power S m (CP ∞) of the infinite dimensional complex projective space is piecewise isomorphic to the infinite dimensional Grassmannian Gr(m, ∞). Let K 0 (V C) be the Grothendieck ring of complex quasi-projective varieties. This is the Abelian group generated by the classes [X] of all complex quasi-projective varieties X modulo the relations: 1) [X] = [Y ] for isomorphic X and Y ; 2) [X] = [Y ] + [X \ Y ] when Y is a Zariski closed subvariety of X. The multiplication in K 0 (V C) is defined by the Cartesian product of varieties: [X 1 ] • [X 2 ] = [X 1 × X 2 ]. The class [A 1 C ] ∈ K 0 (V C) of the complex affine line is denoted by L.

arXiv: Algebraic Geometry, 2019

Let S be a locally noetherian scheme and consider two extensions G_1 and G_2 of abelian S-schemes by S-tori. In this note we prove that the fppf-sheaf Corr _S(G_1,G_2) of divisorial correspondences between G_1 and G_2 is representable. Moreover, using divisorial correspondences, we show that line bundles on an extension G of an abelian scheme by a torus define group homomorphisms between G and Pic_{ G/S}.

1995

A criterion for a functor between derived categories of coherent sheaves to be full and faithful is given. A semiorthogonal decomposition for the derived category of coherent sheaves on the intersection of two even dimensional quadrics is obtained. The behaviour of derived categories with respect to birational transformations is investigated.

Bulletin des Sciences Mathématiques, 2012

Let k be an algebraically closed field of characteristic p > 0, W the ring of Witt vectors over k and R the integral closure of W in the algebraic closure K of K := F rac(W); let moreover X be a smooth, connected and projective scheme over W and H a relatively very ample line bundle over X. We prove that when dim(X/W) ≥ 2 there exists an integer d 0 , depending only on X, such that for any d ≥ d 0 , any Y ∈ |H ⊗d | connected and smooth over W and any y ∈ Y (W) the natural R-morphism of fundamental group schemes π 1 (Y R , y R) → π 1 (X R , y R) is faithfully flat, X R , Y R , y R being respectively the pull back of X, Y , y over Spec(R). If moreover dim(X/W) ≥ 3 then there exists an integer d 1 , depending only on X, such that for any d ≥ d 1 , any Y ∈ |H ⊗d | connected and smooth over W and any section y ∈ Y (W) the morphism π 1 (Y R , y R) → π 1 (X R , y R) is an isomorphism.

2015

on the occasion of his 60th birthday Abstract. We prove that for any open polynomial mapping f: Cn → Cm, m ≥ 3, there is a linear change of coordinates α: Cm → Cm such that for each component fi of α ◦ f, every fibre of fi is irreducible. This is a generalization of the Kaliman result to the multidimensional case. Introduction. Let X, Y be irreducible algebraic varieties over C in the sense of Weil-Serre (algebraic spaces in terms of [5], VII, §17) and f: X → Y be a regular dominating mapping. We call f primitive if the generic fibre of f is irreducible and we call f totally primitive if each fibre of f is irreducible. In the case that f: Cn → C is a polynomial, the primitivity of f is equivalent to

Inventiones Mathematicae, 1992

Springer eBooks, 1983

2012

This is a survey of the language of polyhedral divisors describing T-varieties. This language is explained in parallel to the well established theory of toric varieties. In addition to basic constructions, subjects touched on include singularities, separatedness and properness, divisors and intersection theory, cohomology, Cox rings, polarizations, and equivariant deformations, among others.