# Questions tagged [algebraic-curves]

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

906
questions

3
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1
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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 ...

2
votes

0
answers

52
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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 $h^0(C,D)=s$...

2
votes

0
answers

84
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### Genus of curves on complete intersections

Let $X$ be a smooth projective variety with $\dim X=3$ equipped with a very ample divisor $H$, and $C\subset X$ be a pure one-dimensional closed subscheme of $X$ with arithmetic genus $g$ and degree $...

3
votes

1
answer

163
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### Does a smooth relative curve $X/S$ embed into $\mathbb{P}^3_S$?

Suppose that $\pi:X\to S$ is a smooth projective morphism of relative dimension 1. If $S$ is the spectrum of an algebraically closed field, then it is known that $X$ embeds into $\mathbb{P}^3_S$.
...

4
votes

0
answers

188
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### On the normal crossing divisor of $\overline{\mathcal M}_g$

Let $g\geq 2$ be an integer. Let $\overline{\mathcal M}_g$ denote the DM stack of stable curves of genus $g$. It is well-known that the moduli stack is smooth and has a natural normal-crossing divisor ...

1
vote

0
answers

68
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### Derivation for genus-degree formula from algebraic functions field theory

This is a copy of my question from math.stackexchange: https://math.stackexchange.com/questions/4517289/derivation-for-genus-degree-formula-from-algebraic-functions-field-theory. I didn't get any ...

2
votes

0
answers

185
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### Genus of algebraic curves

Let $X$ be an integral (possibly singular) projective algebraic curve of degree $d$ in $\mathbf{P}^n_{K}$, where $K$ is a field that is either the real numbers, the complex numbers, or a field that is ...

4
votes

1
answer

154
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### Varieties connected by curves in projective spaces of small dimension

Let $X\subset\mathbb{P}^N$ be an irreducible complex variety. Fix an integer $a\geq 2$ and call $P_a$ the following property: given $x_1,\dots,x_a\in X$ general points there exists an irreducible ...

0
votes

0
answers

121
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### The genus of hyperplane sections

Let $S$ be a connected smooth projective surface over $\mathbb{C}$.
Let $\Sigma$ be the complete linear system of a very ample divisor $D$ on $S$, and let $d=\dim(\Sigma)$ be the dimension of $\...

3
votes

1
answer

218
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### Is the theta characteristic attached to an etale double cover of a plane quintic arising from a cubic threefold even in characteristic two?

We work over an algebraically closed field $k$ of characteristic $2$. Let $X$ be a cubic threefold realized as a conic bundle via $f: X \to \mathbb{P}^2$ (after blowing up some line). Let $C \to \...

1
vote

1
answer

161
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### A question on curves on a hypersurface

Let $X$ be a hypersurface of degree $r$ in $\mathbb{P}^n$, and $Z\subset X$ be a closed subscheme of pure dim 1. Let $g(Z):=1-\chi(\mathcal{O}_Z)$ and $d(Z)$ be its degree. I'm wondering that is there ...

0
votes

0
answers

131
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### Cohomology map induced by inclusion of curves

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 ...

4
votes

1
answer

214
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### Some questions about the (projectivized cotangent bundle of the) symmetric square of a genus $3$ curve

Let $C$ be a smooth, non-hyperelliptic curve of genus $3$ and $X:= \mathrm{Sym}^2{C}$ its symmetric square. Then $X$ is a smooth, minimal surface of general type with $p_g=q=3, \, K^2=6$.
Calling $\...

1
vote

0
answers

69
views

### 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....

-1
votes

1
answer

157
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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?

2
votes

0
answers

102
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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 ...

4
votes

1
answer

335
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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 ...

2
votes

0
answers

63
views

### Gluing genus 0 Bessel curve to get genus 1 curve

I was attending some seminar where the following was mentioned, I never understood things in deep. So I ask the community to give me reference or explain.
Let
$${x}^{3}{y}^{2}-{x}^{2}-x-1 =0 $$
$$ y^2 ...

3
votes

0
answers

62
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### The cyclic analogue of the gonality of the superelliptic curve $s^n = t^m + 1$

For naturals $n$, $m > 1$ consider the superelliptic curve $C\!: s^n = t^m + 1$, for simplicity, over an algebraically closed field of zero characteristic or large characteristic $p \nmid n$, $m$. ...

3
votes

0
answers

78
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### Quadrisecants of rational space curves via degeneration

Let $C \subset \mathbf P^3$ be a smooth rational curve of degree $d$. Cayley proved in 1863 that the number of quadrisecants to such a curve (if it is finite) is given by the formula
$$\frac{1}{12}(d-...

3
votes

0
answers

128
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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 ...

1
vote

0
answers

111
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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 ...

0
votes

0
answers

88
views

### 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{...

14
votes

2
answers

1k
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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 ...

3
votes

0
answers

96
views

### Rationality of plane curves with a certain property

Let $C\subset\mathbb{C}^2$ be an irreducible algebraic curve defined over a number field $F.$ Suppose that for any $(z, w)\in C, z\in \mathbb{\overline{Q}}, w\in\mathbb{\overline{Q}},$
either $z\in F(...

2
votes

1
answer

156
views

### Curves having only one linear system realizing its gonality

$\DeclareMathOperator\gon{gon}$Let $C$ be a smooth irreducible projective curve defined over complex numbers. Recall that the gonality of $C$, $\gon(C)$, is defined to be the minimal possible degree ...

11
votes

3
answers

673
views

### Explicit triples of isomorphic Riemann surfaces

Inspired by a discussion with Neil Strickland I am very interested to hear of explicit examples (one per answer, please), as follows.
A compact Riemann surface can be presented in many different ways....

3
votes

0
answers

109
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### Lifting morphism of smooth projective curves

I'm wondering about the following question.
Let $R$ be a complete DVR with residue field $k$. Let $f_k : X \to Y$
be a finite, separable morphism of smooth projective curves over $k$.
Suppose we are ...

9
votes

3
answers

429
views

### Is there a simple explanation for why rational plane curves of degree $>2$ are singular?

Its a well-known result that smooth projective plane curves of degree $d$ have genus $(d-1)(d-2)/2$, so in particular, smooth curves of degree $1$ and $2$ are genus 0, and those of higher degree have ...

10
votes

0
answers

258
views

### Is every finite group the automorphism group of a smooth projective curve?

$\DeclareMathOperator\Aut{Aut}$Let $G$ be a finite group and let $k$ be a field with algebraic closure $K$. Is there a smooth projective curve $C$ defined over $k$ such that $\Aut_k(C)=\Aut_K(C)$ is ...

4
votes

1
answer

159
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### Looking for a curve with a special, free $\mathbb{Z}/2$-action

I am looking for a smooth curve $C$ of genus $g=2k+1 \geq 5$ over the complex numbers, endowed with a free $\mathbb{Z}/2$-action such that the following condition is satisfied: denoting by $$H^0(C, \, ...

0
votes

0
answers

47
views

### Sum of Weierstrass points on a curve of genus three

Let C be a smooth complex projective curve of genus 2 and X a non trivial Galois
cover of degree 2 of C. So X has genus three and it is hyperelliptic, hence X has 8
Weierstrass points. Can one compute ...

1
vote

0
answers

215
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### Missing generator for $H^0(C, \, \omega_C^{\otimes 2})$, with $C$ is hyperelliptic of genus $3$

This is probably very classical and well-known, but I could not find the answer in the literature, so let me ask it here.
Let $C$ be a hyperelliptic curve of genus $3$, defined over the complex ...

4
votes

0
answers

154
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### 𝔾ₘ extensions vs line bundles over abelian varieties

Given a complex polarized abelian variety $V$, we can define a map $$\operatorname{Ext}^1\left(V, \mathbb{G}_m\right) \to \operatorname{Pic}\left(V\right)$$
by viewing an extension as a $\mathbb{G}_m$-...

3
votes

0
answers

240
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### On a conjecture of Hartshorne

Hartshorne has a conjecture in his book Ample Subvarieties of Algebraic Varieties. It's in page 126 Conjecture 5.16 where he writes that if $B$ is a finitely generated flat module over a regular local ...

1
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0
answers

60
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### Morphisms from plane curves to hyperelliptic curves

Consider a plane curve $\mathcal{C}$ of degree $d$. We know that if a morphism $\varphi$ from $\mathcal{C}$ to a curve of genus $g\geq 2$ exists, then $\deg \varphi \leq (g'-1)/(g-1)$ where $g'$ is ...

4
votes

1
answer

270
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### Deformation theoretic argument on dimension counting of naive Hurwitz scheme

I'm reading the Atanas Atanasov's course notes of Joe Harris' course Geometry of Algebraic Curves
and have a question about a suggested modification of an dimension
countinging argument applying ...

1
vote

0
answers

128
views

### Koszul cohomology and nodal curves

In M. Aprodu, G. Farkas - Koszul Cohomology and Applications to Moduli, arXiv:0811.3117 [math.AG], proof of Theorem(s) 4.5 (and 4.12), the authors constructed a semistable nodal curve $C^{\prime}$ of ...

0
votes

0
answers

145
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### Are pure sheaves actually vector bundles on projective curves?

A coherent sheaf is pure if every non-trivial coherent subsheaf has the same dimension, where the dimension of a sheaf is the dimension of its support.
As in the title, I wonder if the notion of pure ...

2
votes

1
answer

220
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### Do we know anything about Harder-Narasimhan filtrations of tensor products of vector bundles?

I am interested in vector bundles over a nonsingular complete algebraic curve $C$ over $\mathbb C$. For a vector bundle $E$, its Harder-Narasimhan filtration is a filtration of subbundles
$$0=E_0\...

1
vote

1
answer

147
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### Space of rational conics

Let $K$ be a field of characteristic different from two. Conics over $K$ (that is curves of degree two in $\mathbb{P}^2_K$) are parametrized by $\mathbb{P}(k[x,y,z]_2) = \mathbb{P}^5_K$.
Conisider the ...

2
votes

1
answer

173
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### Strict henselianization and branches of explicit curve at singularity

Let $A$ be a local ring, which we can assume is reduced. Let $k$ be the residue field of $A$.
In the Stacks project (https://stacks.math.columbia.edu/tag/06DT), I have learned some notion of the ...

0
votes

0
answers

83
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### Computational tool for checking the existence of non-trivial rational zero of a cubic form

Suppose we consider a arbitrary cubic homogeneous form $f$ in in four or five variables over the rational field. Is there any computational tool or algorithm to check whether this cubic homogeneous ...

3
votes

0
answers

65
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### Detecting non-principal Weil divisors on normal varieties using curves

Let $X$ be a normal projective variety over an algebraically closed field $k$. Given any morphism $f:Y\to X$, there is a pullback homomorphism $f^*:\text{Cl}(X)\to\text{Cl}(Y)$, where $\text{Cl}(X)$ ...

2
votes

1
answer

64
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### Grassmannian of line subbundle of a stable rank 2 vector bundle on a smooth projective curve

Let $X$ be a smooth projective curve of genus $g\geq 2$. Given a rank two, degree $d=0$ vector bundle $\mathcal{F}$ on $X$, we consider the grassmannian of sub-line bundles of $\mathcal{F}$ of degree $...

7
votes

0
answers

157
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### Maximum number of linearly independent quadrics containing a curve in $\mathbb{P}^4$ not contained in a hyperplane?

Consider everything over $\mathbb{C}$. My question is:
What is the maximal number $k$ of linearly independent homogeneous quadratic forms $Q_1,\dots,Q_k$ in $5$ variables such that the intersection $V(...

5
votes

1
answer

418
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### Hartshorne's proof of Halphen's theorem

Apologies if this is not quite at the level of MathOverflow, but it has already been asked at MSE and gone unresolved for several years despite a bounty.
Hartshorne states the theorem as follows:
...

0
votes

0
answers

98
views

### Knots with everywhere positive curvature

A naive question that my searches have not resolved:
Q. Can every knot be realized by a curve in $\mathbb{R}^3$ with strictly positive
curvature at every point?

5
votes

1
answer

234
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### First cohomology of tangent sheaf of rational curve

Let $C$ be a reduced, connected, projective and purely one-dimensional scheme of finite type over a field $k$.
Suppose that $C$ is rational, i.e. that the normalisation of $C$ is a disjoint union of ...

4
votes

0
answers

72
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### Curves not invariant by non-trivial projective automorphisms

Let $g\ge 0$, $d\ge 1$ be integers. We consider the Moduli space $H_{g,d}$ parametrising smooth irreducible closed curves $C\subset \mathbb{P}^3$ of degree $d$ and genus $g$. Let us denote by $U_{g,d}$...