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### Hyperelliptic Curve [closed]

Consider the curve given by $z^{2g-2}y^2=\displaystyle\prod_{i=1}^{2g}(x-a_iz)$. This is a hyperelliptic curve and has genus $g-1$. At the same time it is a curve defined by an equation of $d=2g$ and ...

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### S genus of quadratic forms

Let $f$ be a non-degenerate quadratic form with integral coefficients. The genus of $f$ is the set of quadratic forms up to integral equivalence which are equivalent to $f$ over the $p$-adic integers ...

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### Rationality of curve does not depend on base change

By a curve I mean an integral one-dimensional scheme of finite type over a spectrum of a field.
Let $C$ be a curve over an arbitrary field $k$. It's probably a very well known fact, that $C$ is ...

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### Properties of subvarieties of a simple abelian variety

Let $A$ be a simple abelian variety over a field $k$. (For simplicity, we assume char $k =0$.)
Let $X$ be a smooth projective geometrically connected variety over $k$ of positive dimension.
Suppose ...

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### Minimal genus, adjunction inequality

Let's consider closed simply-connected 4-manifold $M$ and some $a\in H^2(M)$. It is very natural question to estimate minimal $g$ that $a$ can be presented as embedded surface of genus $g$.
As I know ...

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### smooth curves of genus 3 over an algebraic closed field

Is there a way to "easily" compute and describe the Moduli space of smooth curves of genus 3 without stacks and stable curves?
In Hartshorne's Algebraic Geometry there is a nice excercise (Chapter ...

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### Genus and Spinor genus of a lattice

Hi, I'm looking for a motivation for the names genus and spinor genus of a lattice (and spinor norm of an isometry).
Is there any relation between the genus of a lattice and the genus of an algebraic ...

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### Riemann-Roch for ARBITRARY Function Fields

I know that on an algebraic function field in one variable over any base field, there is a good divisor theory for it and a Riemann-Roch Theorem; in particular, there is a 'good' notion of 'genus'. ...

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### How do you see the genus of a curve, just looking at its function field?

Yuhao asked in the 20-questions seminar:
The genus of a curve is a birational invariant; the function field of a curve determines it up to birational equivelance.
How do you see the genus directly ...