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Fix $k$ a number field, and let $C$ be a smooth geometrically integral affine curve over $k$. We can "associate" to $C$ a semi-abelian variety in the following way: One knows that $C$ is a finite numb …

Let $C$ be a smooth projective curve of genus $\geq 1$ over a number field $k$ with a $k$-rational divisor of degree $1$ inducing the embedding $C \hookrightarrow J$, where $J$ is the Jacobian variety …

The classical book of Walker on algebraic curves described a method of computing the genus of curves with non-ordinary singularities by transforming such a curve into a birational one with only ordina …

It is known that a smooth projective curve $C$ of genus $\geq 1$ over $\mathbb{C}$ embeds into its Jacobian $J(C)$, via the isomorphism $J(C) \cong \mathrm{Pic}^0(C)$. Question 1. Is this embedding st …

This question is related to my post Interpretation of some maps involving cohomology groups. $C$ is a smooth geometrically integral affine curve over a number field $k$, and $C_1$ is its smooth comple …

Let $C$ be a smooth affine geometrically integral curve of genus $\geq 1$ over an algebraically closed field $k$, and let $\iota: C \rightarrow C'$ denote the inclusion into its smooth compactificatio …

The purity of Brauer group states that for a smooth (quasi-)projective variety $X$ over a field $k$, removing a closed subscheme $Z \subset X$ of codimension at least $2$ ensures that the restriction …

As the title suggests, does anyone have a reference for the proof of this fact? Actually, I can't remember where I've seen it before, or if I even remembered the statement correctly. Here are some con …

Let $C$ be a smooth projective geometrically connected curve of genus 2 defined over a number field $k$. Here are some definitions: The index $I$ of a curve $C$ is the greatest common divisor of all e …

The book Algebraic groups and class fields by Serre explains a lot about the construction of the generalized Jacobian $J_\mathfrak{m}$ of a smooth projective curve $X$ with respect to the modulus $\ma …

Let $C$ be a smooth geometrically integral affine curve. This question concerns the smooth completion $C_1$ of $C$, both defined over a number field $k$. We know that given any smooth projective geome …

Let $k$ be a number field and denote by $H^i(k,-)$ the Galois cohomology functor $H^i(\mathrm{Gal}(\bar{k}/k),-)$. Let $X$ be a smooth geometrically integral curve over $k$. One can easily show that t …

I'm looking for references that can justify to what extent is the following statement true: Statement. Let $X$ be a smooth geometrically integral curve over a number field $k$. Then the Brauer-Manin o …