# Questions tagged [algebraic-curves]

for questions on one dimensional algebraic varieties over any field, including questions of moduli, and questions about specific curves.

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### Compactified Jacobian of a rational curve whose normalization is a set-theoretic bijection

Suppose $C$ is a (singular) rational curve whose normalization $p: \mathbb P^1 \to C$ is a set-theoretic bijection. Can one understand how the compactified Jacobian of $C$ looks like? For example, the ...
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### Dimension of the determinantal variety

Let $C$ be a smooth projective curve of genus $g$ over the complex field, embedded into $\mathbb{P}^r$ via a line bundle $L$ of degree $n>>0$. Let $D$ be a divisor of degree $d$ with $h^0(C,D)=s$...
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### How to construct explicitly defining polynomials of an morphism between smooth irreducible curves?

Let $\phi\!: C_1 \to C_2$ be a separable morphism of smooth irreducible curves embedded as projectively normal models by invertible sheaves $\mathcal{L}_1$ and $\mathcal{L}_2$ respectively. Theorem 4....
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### Can quasi affine varieties contain projective curves [closed]

Can a regular quasi affine variety (i.e. open subscheme of an affine variety) contain a (possibly singular) projective curve?
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### Injectivity of the Abel map away from singularity

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 ...
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### Curves and semi-abelian varieties

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 ...
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### The dualising sheaf of a nodal curve by Grothendieck duality

I am trying to use Grothendieck duality (Duality) to prove that the dualising sheaf $\omega_X$ of a nodal curve $X$ can be described as the pushforward sheaf of the sheaf of differential forms on the ...
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### Blow up singularities on curves

Let $p$ be a prime number and let $\bar{\mathbb F}_p$ be an algebraic closure of $\mathbb F_p$. Let $C$ be an irreducible singular projective curve over $\bar{\mathbb F}_p$. Let $P$ be a singularity ...
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### Are the zeroes of the finite characteristic zeta functions dense in $\left\{s\in\mathbb{C}\mid\mathfrak{Re}(s)=\frac{1}{2}\right\}$?

If $p$ is a prime, $n\in\mathbb{N}$ is a natural number and $C$ is a nonsingular curve over $\mathbb{F}_{p^{n}}$, the $\zeta$ function associated to $C\mid_{\mathbb{F}_{p^{n}}}$ is defined as \begin{...
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### Polynomial values are powers of two

The initial question comes from Komal in 1999. Namely it asks to show that for infinitely many $n$ there is a polynomial $f\in\mathbb{Q}[X]$ of degree $n$ such that $f(0),f(1),\dotsc,f(n+1)$ are ...
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