Questions tagged [algebraic-curves]
for questions on one dimensional algebraic varieties over any field, including questions of moduli, and questions about specific curves.
985
questions
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Question about algebraic curve being birational to smooth projective curve
Let $X$ be a geometrically irreducible affine variety defined over $\mathbb{Q}$ and dimension $1$. Then it is known that $X$ is birational over $\mathbb{C}$ to a smooth projective curve $C$.
I was ...
3
votes
2
answers
218
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Writing a smooth plane quartic as the vanishing of $Q_0Q_2 - Q_1^2$ for quadratic $Q_0,Q_1,Q_2$
It seems well-known that any smooth plane quartic can be written as the vanishing of $Q_0Q_2 -Q_1^2$. Is there a good way to work out these quadratic factors $Q_0,Q_1,Q_2$? For example, given the ...
0
votes
1
answer
131
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Are maps into a smooth curve equivalent to relative effective Cartier divisors?
Let $X$ be a smooth curve and $S$ an affine scheme, both over an algebraically closed field $k$.
Given a morphism $f : S \to X$, is it true that the graph morphism $\Gamma_f : S \to X \times S$ is a ...
0
votes
0
answers
118
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Degree of the syzygy bundle of a curve of genus 3
Let X be an hyperelliptic curve of genus 3, 𝜔 its canonical sheaf, and M the syzygy bundle of 𝜔.
What is the degree of M?
1
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0
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109
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What are algebroid curves/branches and their value semigroup?
In “The moduli problem for plane branches”, by O. Zariski, the author defines a plane branch as an irreducible element $f \in \mathbb C[[x,y]]$. In the more recent article "The semigroup of a ...
4
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0
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95
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Matrix description for automorphisms of genus $2$ curve split into two copies of an elliptic curve
Consider a genus $2$ curve $C$ and its Jacobian $J$ (for simplicity, over the field $\overline{\mathbb{Q}}$ or $\mathbb{C}$). Assume that $J$ is $(2,2)$-isogenous to the direct square $E^2$ of an ...
2
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0
answers
98
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Global sections of relative characteristic of log-smooth curves
$\DeclareMathOperator\Spec{Spec}$I am currently learning about log-geometry and try to understand the theory in the example of curves with basic log-structure over a general base $S$. Especially, I am ...
5
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0
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117
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Finding an $\mathbb{F}_q$-point on one specific intersection of quadrics
Let $\mathbb{F}_q$ be a finite field of large characteristic and $a_1, a_2, \cdots, a_n \in \mathbb{F}_q$ be some pairwise different elements. I assume that $\sqrt{-1} \in \mathbb{F}_q$. Consider the ...
1
vote
1
answer
148
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Finiteness of automorphism group of finite map $f: C \to \mathbb{P}^1$
Let $C$ be a connected curve of arithmetic genus $g$
over algebraically closed field $k$ of characteristic zero having only nodes
as singularities together with finite morphism
$f: C \to \mathbb{P}^1$....
2
votes
1
answer
79
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Image of $H^0(C,\omega_C-x)$ in $G(g-1,H^0(C,\omega_C))$
Let $C$ be an algebraic curve over $\mathbb{C}$ and $\omega_C$ be its canonical bundle. We may assume that $C$ has genus $g\geq2$. Let $x\in C$ be an arbitrary point.
Question: What is the image of $...
5
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0
answers
252
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What is the algebra structure on the pushforward of the structure sheaf along a finite map to $\mathbb{P}^1$?
$\newcommand{\P}{\mathbb{P}}\newcommand{\O}{\mathcal{O}}$ Let $f : C \to \P^1$ be a ramified finite map of degree $d$ of smooth algebraic curves over an algebraically closed field $k$. How can we ...
3
votes
1
answer
281
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What do we know about the $\mathbb{F}_{q^2}$-curve $X^{q-1} + Y^{q-1} + Z^{q-1}$?
People have thoroughly studied the Hermitian $\mathbb{F}_{q^2}$-curve $H: X^{q+1} + Y^{q+1} + Z^{q+1} = 0$. What do we know about the similar $\mathbb{F}_{q^2}$-curve $C: X^{q-1} + Y^{q-1} + Z^{q-1} = ...
2
votes
1
answer
290
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Equivalent characterizations of rational normal curve
A rational normal curve $C \subset \mathbb{P}_k^d$ (assume $k= \mathbb{C}$) can be defined usually up to projective equivalence in two equivalent ways:
smooth irreducible nondegenerate curve $C \...
1
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0
answers
149
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The $H^1$ of a smooth curve and its (generalized) Jacobian variety
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 ...
1
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0
answers
95
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Is it possible to define the generalized Jacobian of a curve when the modulus $\mathfrak{m}$ is supported on points of higher degree?
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 $\...
2
votes
1
answer
237
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Motivation of Zariski–Van Kampen theorem
The Zariski–Van Kampen theorem gives the presentation of the fundamental group of the complement of the plane curve of degree $d$. But what's the motivation of this theorem? More generally, why are ...
3
votes
0
answers
61
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Vanishing odd theta characteristics on plane curves
Are there, for any $k$, smooth plane curves $C\subset\mathbb{P}^2$ of degree $d=2k$ over $\mathbb{C}$ such that the space of global sections of all odd theta characteristics on $C$ is one dimensional?
...
1
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1
answer
150
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Fibrations of curves whose singular locus on the base is not codimension $1$
Let $f : X \to B$ a relative curve meaning a flat proper map whose fibers are geometrically connected $1$-dimensional schemes. In what follows,let $B$ be a smooth variety over $\mathbb{C}$ and $f$ be ...
3
votes
1
answer
207
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If a genus 2 curve has no $k$-rational points, can it have a $k$-rational divisor class of degree 1?
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 ...
3
votes
3
answers
389
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Extension of the trivial bundle by the canonical bundle on a curve
Let $X$ be a smooth projective curve over a field $k$ and $K_X$ be its canonical line bundle. By the Serre duality, $\text{H}^1(X,K_X)$ is a one-dimensional $k$-vector space. On the other hand, $\text{...
1
vote
0
answers
81
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Density of two algebraic curves with maximum number of intersection points
Let $f$ and $g$ be two complex polynomials of degree $n$ and $m$ with two variables respectively and coefficients of these two polynomials correspond to $M=\mathbb{C}^\frac{(n+1)(n+2)+(m+1)(m+2)}{2}$ ...
1
vote
0
answers
70
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Prescribed intersection of varieties
Every variety here is complex analytic, or complex algebraic if it solves anything.
Given a germ of a (possibly singular, nor necessarily irreducible) hypersurface $(H,0)\subset(\mathbb{C}^{n+1},0)$ ...
0
votes
0
answers
107
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For curves $C$ of genus $1$, the period (or index?) of $C$ is greater than $1$ iff $C(k)$ is empty
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 ...
10
votes
3
answers
534
views
On the Klein quartic and the similar $a^2b+b^2c+c^2a$?
Given the Ramanujan theta function,
$$f(a,b) = \sum_{n=-\infty}^\infty a^{n(n+1)/2} \; b^{n(n-1)/2}$$
Let $q = e^{2\pi i \tau}$ and assume $\tau = \sqrt{-d}$.
I. Degree 5
\begin{align}
a &= q^{11/...
11
votes
2
answers
521
views
Hypersurface of singular plane cubics
In the projective space $\mathbb{P}^9 = \mathbb{P}(\mathbb{C}[x,y,z]_3)$, parametrizing plane cubics, consider the hypersurface $X\subset\mathbb{P}^9$ whose points corresponds to singular cubics. The ...
1
vote
1
answer
278
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Counterexample to purity of Brauer group for curves
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 ...
1
vote
0
answers
85
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Invariance of numerical class of a curve in Higgs-Grassmann schemes
Premise
Let $X$ be a projective variety of dimension $n\geq1$ over an algebraically closed field of characteristic $0$.
A Higgs sheaf $\mathfrak{E}$ is a pair $(E,\varphi)$ where $E$ is a $\mathcal{...
1
vote
1
answer
294
views
Short exact sequence of equivariant line bundles on $\mathbb P^1$
I have a two-dimensional vector space ${\mathbb C}^2$ with basis $e_m, f_1$ and action of ${\mathbb C}^*$ by $t \cdot e_m = t^m e_m$ and $t \cdot f_1 = f_1$ and I have the projective line ${\mathbb P}^...
4
votes
0
answers
136
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Can you determine the least degree of a morphism between algebraic curves?
I have several questions regarding the degrees of morphisms between algebraic curves.
If we have algebraic curves $X$ and $Y$ defined over some perfect field $k$, can we determine the least degree of ...
1
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0
answers
88
views
Lower bound of degree of ruled surface in $\mathbb P^n$
I have a question of Complex Algebraic surface in Beauville.
Let $S\subset\mathbb{P}^n$ be a (birationally) ruled surface of degree $d$ lying in no hyperplane.
Show that $d\geq 2 n-2$ if $S$ is not ...
1
vote
1
answer
151
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Semistable pure dimension one sheaves of rank 1 and degree 0 on a singular curve
We are working on a problem about semistable pure dimension one sheaves of rank $1$ and degree $0$ on a singular curve $C$ (for example, the Kodaira fiber of type $I_2$, i.e. $C=C_1\cup C_2$ where $...
4
votes
1
answer
316
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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
129
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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 $h^0(C,D)=s$...
4
votes
1
answer
208
views
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 $\mathbb{P}^3_S$.
...
3
votes
0
answers
251
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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
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0
answers
146
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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 ...
3
votes
1
answer
192
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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
201
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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
266
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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
183
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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
185
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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 ...
5
votes
1
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281
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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
74
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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....
-1
votes
1
answer
228
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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
155
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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
502
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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
66
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
$${x}^{3}{y}^{2}-{x}^{2}-x-1 =0 $$
$$ y^2 ...
4
votes
0
answers
80
views
The cyclic analogue of the gonality of the superelliptic curve $s^n = t^m + 1$
For naturals $n$, $m > 1$ consider the superelliptic curve $C\!: s^n = t^m + 1$, for simplicity, over an algebraically closed field of zero characteristic or large characteristic $p \nmid n$, $m$. ...
3
votes
0
answers
86
views
Quadrisecants of rational space curves via degeneration
Let $C \subset \mathbf P^3$ be a smooth rational curve of degree $d$. Cayley proved in 1863 that the number of quadrisecants to such a curve (if it is finite) is given by the formula
$$\frac{1}{12}(d-...
3
votes
0
answers
198
views
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 ...