The kernel of the Gysin homomorphism for smooth projective curves
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866911009780793344 |
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| author | Schoemann, Claudia |
| author_facet | Schoemann, Claudia |
| contents | 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 $ϕ_Σ: S \hookrightarrow \mathbb{P}^{d}$ the closed embedding of $S$ into $\mathbb{P}^{d}$, induced by $Σ$. For any closed point $t\inΣ\cong\mathbb{P}^{d^*}$, let $C_t$ be the corresponding hyperplane section on $S$, and let $ r_t:C_t\hookrightarrow S$ be the closed embedding of the curve $C_t$ into $S$. Let $Δ:= \{t \in Σ: C_t \text{ is singular}\}$ be the discriminant locus of $Σ$ and let $U :=Σ\setminus Δ$.
For $t \in U$, the kernel of the Gysin homomorphism of the Chow groups of $0$-cycles of degree zero, from $CH_0(C_t)_{deg=0}$ to $CH_0(S)_{deg=0}$ is the countable union of shifts of a certain abelian subvariety $A_t$ inside $J(C_t)$, the Jacobian of the curve $C_t$ (\cite{PS24} for $k \cong \mathbb{C}$, \cite{SW25} for $k \cong \overline{\mathbb{F}_q((t))}$). We prove that for every closed point $t \in U$ either $A_t$ coincides with the abelian variety $B_t$ inside $J(C_t)$ corresponding to the vanishing cohomology $H^1(C_t, k')_{van}$, where $k'$ is the minimal field of definition of $k$, and then the Gysin kernel is a countable union of shifts of $B_t$, or $A_t = 0$, 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 $U_i, i \in I,$ of $U$, where $I$ is a countable set and by applying a convergence argument. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2506_14528 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | The kernel of the Gysin homomorphism for smooth projective curves Schoemann, Claudia Algebraic Geometry 14A20, 14C25, 14D05 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 $ϕ_Σ: S \hookrightarrow \mathbb{P}^{d}$ the closed embedding of $S$ into $\mathbb{P}^{d}$, induced by $Σ$. For any closed point $t\inΣ\cong\mathbb{P}^{d^*}$, let $C_t$ be the corresponding hyperplane section on $S$, and let $ r_t:C_t\hookrightarrow S$ be the closed embedding of the curve $C_t$ into $S$. Let $Δ:= \{t \in Σ: C_t \text{ is singular}\}$ be the discriminant locus of $Σ$ and let $U :=Σ\setminus Δ$. For $t \in U$, the kernel of the Gysin homomorphism of the Chow groups of $0$-cycles of degree zero, from $CH_0(C_t)_{deg=0}$ to $CH_0(S)_{deg=0}$ is the countable union of shifts of a certain abelian subvariety $A_t$ inside $J(C_t)$, the Jacobian of the curve $C_t$ (\cite{PS24} for $k \cong \mathbb{C}$, \cite{SW25} for $k \cong \overline{\mathbb{F}_q((t))}$). We prove that for every closed point $t \in U$ either $A_t$ coincides with the abelian variety $B_t$ inside $J(C_t)$ corresponding to the vanishing cohomology $H^1(C_t, k')_{van}$, where $k'$ is the minimal field of definition of $k$, and then the Gysin kernel is a countable union of shifts of $B_t$, or $A_t = 0$, 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 $U_i, i \in I,$ of $U$, where $I$ is a countable set and by applying a convergence argument. |
| title | The kernel of the Gysin homomorphism for smooth projective curves |
| topic | Algebraic Geometry 14A20, 14C25, 14D05 |
| url | https://arxiv.org/abs/2506.14528 |