2025/06/17 by Claudia Schoemann, Schoemann, Claudia
Mathematics · #14A20 #14C25 #14D05 #Advanced Algebra and Geometry #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology
paper · pdf · doi:10.48550/arxiv.2506.14528
openalex publication_date 2025/06/17 · openalex created_date 2025/10/18 · openalex updated_date 2026/07/28
Let S be a smooth projective connected surface over an algebraically closed field k and Σ the linear system of a very ample divisor D on S. Let d:=dim(Σ) be the dimension of Σ and ϕΣ: S \hookrightarrow ℙd the closed embedding of S into ℙd, induced by Σ. For any closed point t∈Σ≅ℙd^*, let Ct be the corresponding hyperplane section on S, and let rt:Ct\hookrightarrow S be the closed embedding of the curve Ct into S. Let Δ:= \t ∈ Σ: Ct is singular\ be the discriminant locus of Σ and let U :=Σ∖ Δ. For t ∈ U, the kernel of the Gysin homomorphism of the Chow groups of 0-cycles of degree zero, from CH0(Ct)deg=0 to CH0(S)deg=0 is the countable union of shifts of a certain abelian subvariety At inside J(Ct), the Jacobian of the curve Ct (\citePS24 for k ≅ ℂ, \citeSW25 for k ≅ \mathbbFq((t))). We prove that for every closed point t ∈ U either At coincides with the abelian variety Bt inside J(Ct) corresponding to the vanishing cohomology H1(Ct, k')van, where k' is the minimal field of definition of k, and then the Gysin kernel is a countable union of shifts of Bt, or At = 0, in which case the Gysin kernel is countable. Using the language of algebraic stacks as a generalisation of algebraic varieties this is done by constructing an increasing filtration of Zariski countable open substacks Ui, i ∈ I, of U, where I is a countable set and by applying a convergence argument.