Questions tagged [algebraic-curves]
for questions on one dimensional algebraic varieties over any field, including questions of moduli, and questions about specific curves.
344
questions with no upvoted or accepted answers
54
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Uniformization over finite fields?
The following is a question I've been asking people on and off for a few years, mostly out of idle curiosity, though I think it's pretty interesting. Since I've made more or less no progress, I ...
22
votes
0
answers
693
views
Knots realized as algebraic curves
Two questions:
Q1. Have researchers worked out minimum-degree
real algebraic curves in $\mathbb{R}^3$ realizing specific knots?
Some work on the trefoil is reported in this MSE question.
&...
14
votes
0
answers
545
views
Map of the Klein quartic from $CP^2$ to $R^3$
The Klein quartic $\mathcal{Q}$ is cut out of $\mathbb{CP}^2$ by the homogeneous equation $$x^3 y + y^3 z + z^3 x = 0.$$ It has 168 orientation preserving automorphisms and includes several copies of ...
14
votes
0
answers
877
views
Degrees of maps from curves to $\mathbb P^1$
Let $a$ and $b$ be two relatively prime natural numbers. What is the largest number $c$ such that there is a curve with maps to $\mathbb P^1$ of degree $a$ and $b$ but no map to $\mathbb P^1$ of ...
13
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0
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332
views
An example of curves with the same Jacobian, but different Jacobian automorphism groups (wrt their respective canonical principal polarizations)?
I am trying to understand examples of differing curves with the same Jacobian, and the quirks of the Jacobian.
Here is my question: What is an example where $Aut(Jac(X, a))$ and $Aut(Jac(X', a'))$ ...
12
votes
0
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341
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Quivers as noncommutative curves
I've heard that an idea behind noncommutative geometry (in dim 1) is to study "noncommutative" analogues of $\text{Coh}(\text{curve})$, rather than the curve directly. Apparently the ...
12
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0
answers
207
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Totally real points on curves
Let $X$ be a smooth, projective (geometrically integral) curve defined over $\mathbb{Q}$ with genus $g \geq 3$. Suppose that $X(\mathbb{R}) \neq \emptyset$. Does $X$ have a point defined over a ...
12
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0
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440
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Where is the representability of the moduli of curves with framed points proved?
There is a variant of the Knudsen-Mumford moduli problem $\mathcal{M}_{g,n}$ of pointed curves, where one endows the $n$ marked points with non-zero tangent vectors. It shows up in the theory of ...
11
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0
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264
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What is the lowest-weight non-cyclotomic Galois representation in $\overline{\mathcal M}_{g,n}$?
I want to know about low-weight Galois representations in $H^i(\overline{\mathcal M}_{g,n}, \overline{\mathbb Q}_\ell)$ that aren't cyclotomic. This should be equivalent to finding $p,q$ such that $H^{...
10
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325
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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 ...
10
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232
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On tangent space to the fundamental group scheme
Let $X$ be a smooth, projective complex curve of genus at least $2$. If I understand correctly, after choosing a base point, one can associate to $X$, a fundamental group scheme $\pi$. I am trying to ...
10
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0
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398
views
Is there an algorithm which determines if a curve has good reduction outside a given set of primes
Fix a number field $K/\mathbf{Q}$, a finite set of places $S$ in $K$, an integer $g$ and a curve $X$ over $K$ of genus $g$.
Is there an algorithm which tells you if $X$ has good reduction outside $S$?...
9
votes
0
answers
155
views
Is there a classification of non-simple Jacobians?
An abelian variety in the interior of the Torelli locus is non-decomposable, but it could possibly be non-simple (i.e. isogenous to a product of abelian varieties with lower dimension). For certain ...
9
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0
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207
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Kronecker's theorem in higher dimension
Recall the following classical theorem of Kronecker: if $P(x) \in \mathbb{Z}[x]$ is a monic irreducible polynomial with all roots on the unit circle $S^1$, then $P(x)$ is a cyclotomic polynomial (and ...
9
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0
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546
views
Maximum number of connected components of a real affine curve
Harnack's curve theorem tells us that the maximum number of connected components of an algebraic curve of degree $d$ in the real projective plane is $1 + (d-1)(d-2)/2$ (and this bound is sharp).
What ...
9
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0
answers
616
views
Motivic fundamental group of the moduli space of curves?
Suppose I have a smooth projective family of varieties of varieties over $\mathcal M_g$ - i.e. a universal functor, commuting with deformations, from curves to smooth projective varieties. Can I ...
9
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0
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553
views
Function fields of characteristic p modular curves, and mod p reductions of the classical modular equation
Let l and p be distinct primes, l>2. There are "characteristic p modular curves" X_0(l) and X(l), defined over an algebraic closure, K, of Z/p, solving moduli problems for elliptic curves with some ...
8
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197
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Belyi-like results over function fields of characteristic zero
In the paper Unifying themes suggested by Belyi's theorem from 2011, the following question is raised:
Let $X$ be a projective non-singular curve over the function field $K:=\overline{\Bbb{Q}}(t)$. ...
8
votes
0
answers
166
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On a smooth curve $C$, when is $K_C \sim_\mathbb{Q} (2g-2)P$?
Let $C$ be a smooth curve of genus $g$ over $\mathbb{C}$. I am interested in the following property:
There exists a point $P \in C$ such that $K_C \sim_\mathbb{Q} (2g-2)P$. Equivalently, $K_C - (2g-2)...
8
votes
0
answers
644
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Non-hyperelliptic families of curves with trivial Ceresa class (or Gross-Schoen class)
Suppose X/K is a curve over a field K, which we want to think of as non-algebraically closed, and let x be a point of X(K). The Ceresa cycle is defined as follows; you can embed X in Jac(X) by sending ...
8
votes
0
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429
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The curve $(x+y+z)^3=27xyz$
Can someone point me to literature about the curve defined by $F(x,y,z):=(x+y+z)^3-27xyz$? I'm sure this curve must be well-studied, due to the remarkable property that
$$
F(x^3,y^3,z^3) = \prod_{\...
8
votes
0
answers
500
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Deligne-Mumford moduli spaces and compactification of symmetric matrices
The real Deligne-Mumford moduli space $\bar M_{0,n+1}(\mathbb R)$ of stable genus
zero curves with $n+1$ marked points is a compactification of the space of
configurations of $n$ distinct ordered ...
8
votes
0
answers
846
views
Elementary proof of the Hurwitz formula
I am aware of two forms of the Hurwitz formula. The first is more common, and deals only with the degrees. So if $f:X \rightarrow Y$ is a non-constant map of degree $n$ between two projective non-...
7
votes
0
answers
257
views
Complete curves in $\mathcal{M}_g$ all of whose Jacobians have trivial endomorpism ring
I'm trying to construct complete smooth curves $C$ in $\overline{M}_g$ such that for all points $S \in C \cap \mathcal{M}_g$, its Jacobian $\text{Jac}(S)$ satisfies $\text{End}(\text{Jac}(S)) = \...
7
votes
0
answers
174
views
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(...
7
votes
0
answers
161
views
Is there an algorithm to determine if there exists a dominant map between two curves?
Suppose I am given two smooth projective curves $C_1$ and $C_2$ over a field $k$ I want to know if there is an algorithm to decide whether there exists a nonconstant (and thus dominant) map $f : C_1 \...
7
votes
0
answers
308
views
What is the kernel of $i^*:H^*(\overline{\mathcal{M}}_{g,n},\mathbb{Q}) \to H^*(\overline{\mathcal{M}}_{g-1,n+2},\mathbb{Q})$?
Let $\overline{\mathcal{M}}_{g,n}$ be the Deligne-Mumford moduli space of stable algebraic curves. I would like to know which rational cohomology classes or at least which tautological classes on $\...
7
votes
0
answers
590
views
Automorphisms of semistable $G$-bundles
Let $X$ be a projective smooth curve over an algebraically closed field $\mathbb k$ of any characteristic. Let also $G$ be a reductive group over $\mathbb k$. (Probably) following Ramanathan, a $G$-...
7
votes
0
answers
347
views
Are curves over imperfect fields defined over a smaller field?
Let $C$ be regular projective curve defined over a field $K$. Let $K/L$ be a totally inseparable finite extension. Does there exist a regular projective curve $C'$ over $L$ such that that the pullback ...
7
votes
0
answers
206
views
Unicritical rational functions on curves in characteristic $p$
Let $k$ be an algebraically closed field of positive characteristic $p$, and let $X_{/k}$ be a smooth projective connected curve. Let $x_0$ be a point of $X(k)$.
How precisely can one describe the ...
7
votes
0
answers
311
views
Relations among Hodge classes?
Let $\pi : C_g \to M_g$ be the universal curve over the moduli stack of genus $g$ curves. Let $\omega_\pi$ be the relative canonical bundle. Then $\mathbb{H} := \pi_\ast \omega_\pi$ is a rank $g$ ...
7
votes
0
answers
431
views
Defining equations for hyperelliptic Jacobians in a neighbourhood of the identity
Let $X$ be a hyperelliptic curve of genus $g \ge 2$ over a field $k$ (of characteristic not 2, 3 or 5, if you like, but could be positive in general). Let $J$ be the Jacobian of $X$, thought of as $\...
7
votes
0
answers
203
views
sheaves on thickened nodal cubics
Suppose F is an algebraically closed field (of any characteristic) and that h in F[x,y,z]
is an irreducible cubic form defining a plane curve C with a node. A lot is known about
sheaves on C; for ...
6
votes
0
answers
155
views
Does there exist a plane curve such that it has the heart curve as catacaustic?
Given a curve $C$ and a fixed point $L$ (the light source), the catacaustic of $C$ with respect to $L$ is the envelope of light rays coming from $L$ and reflected from the curve $C$.
The catacaustic ...
6
votes
0
answers
417
views
Global sections of canonical line bundle on projective curve with everywhere vanishing derivative
Let $k$ be an algebraically closed field of positive characteristic $p$, $C$ be a curve (projective, non-singular, connected) of genus $g\geq 2$ over $k$ and $\omega \in H^0(C, \Omega_C)$ be a regular ...
6
votes
0
answers
122
views
For which (g,q) does there exist a supersingular curve?
We say a curve over a finite field $\mathbb F_q$ is supersingular if it's Frobenius eigenvalues (on $H^1(X,\mathbb Z_\ell)$) are of the form $q^{1/2}\alpha$ for $\alpha$ a root of unity.
As far as I ...
6
votes
0
answers
265
views
*Why* is Bombieri-Pila uniform?
I am about to give a couple of lectures on Bombieri-Pila/the determinant method. Bombieri-Pila gives a bound $$|C(\mathbb{Q}) \cap B \cap \mathbb{Z}^2| \ll_{d,\epsilon} N^{1/d+\epsilon}$$ for $C$ a ...
6
votes
0
answers
180
views
Čech-to-cohomology spectral sequence for fppf cohomology of $\mathbb G_m$
The Čech-to-cohomology spectral sequence is fundamental in proving foundational results on cohomology of sheaves, and is not only used for Zariski coverings.
If $f: X \rightarrow Y$ is a finite ...
6
votes
0
answers
153
views
Descent via an explicit isogeny (genus 2)
This question is related to a previous question posted by me here answered by Prof. M. Stoll. 5-Descent or ($\sqrt{5}$-Descent?) on certain genus 2 Jacobians.
Here I ask some technicalities of a ...
6
votes
0
answers
279
views
Is there a finite number of supersingular genus 2 curves?
Consider an algebraically closed field $k$ of prime characteristic. It is widely known that up to $k$-isomorphism there is a finite number of supersingular elliptic curves over $k$. Let $C$ be a ...
6
votes
0
answers
251
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Concavity of a function implicitly defined by a polynomial
Consider the following system of $n$ equations:
\begin{equation}f_j^2 = x_j^2\sum_{i=1}^n A_{ij} f_i
\tag{$\star$}
\end{equation}
where $A_{ij}\geq 0$ are known constants and where $x_j>0$ for ...
6
votes
0
answers
862
views
Kähler differentials, intuition behind $\text{div}(\omega)$, canonical divisor on algebraic curves?
See my two previous questions here: Intuition for thinking about R-module of Kähler differentials, universal receptacles, derivations? and Kähler differentials, define valuation? for background.
If $\...
6
votes
0
answers
483
views
Global sections for a locally free sheaf over curves
Let $B$ be a complete algbraic curve of genus $g$, and $\mathcal{E}$ be a semi-stable locally free sheaf of rank $r$ over $B$. Assume that the slope of $\mathcal{E}$ is $\mu(\mathcal E):=\frac{\deg \...
6
votes
0
answers
662
views
Bezout Theorem in $\mathbb P^3$
For a $\mathbb C P^2$ is known a result: if through the generic points $p_1,p_2,\dots p_n$ with multiplicities $m_1,m_2\dots, m_n$ correspondingly a degree $d$ curve passes then $d^2\geq m_1^2+m_2^2+\...
6
votes
0
answers
236
views
Counting plane curves over various fields
Fix two integers $d$ and $g$. The number of genus $g$ and degree $d$ curves passing through $3d+g-1$ generic points on the complex projective plane is finite and doesn't depend on the choice of points....
5
votes
0
answers
117
views
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 ...
5
votes
0
answers
252
views
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 ...
5
votes
0
answers
211
views
Belyi functions with prescribed image of a given point
$\newcommand{\bP}{\mathbb{P}}\newcommand{\bQ}{\mathbb{Q}}$Definition. A Belyi function is a non-constant rational function $f:\bP_{\bQ}^1\to \bP^1_{\bQ}$ such that the image of any of its critical ...
5
votes
0
answers
151
views
The image of a curve under the multiplication endomorphism of its Jacobian
Let $X$ be a complex smooth projective curve of genus $g\geq 2$. Embed $X$ in its Jacobian
${\rm{J}}(X)\cong{\rm{Div_0}}(X)/{\rm{Div_p}}(X)$ where ${\rm{Div_0}}(X)$ is the group of degree zero ...
5
votes
0
answers
153
views
Curves of genus 0 over a DVR determined by fibers?
Closely related is this question.
Suppose $R$ is a DVR with fraction field $K$ and residue field $k$ (say finite), and $S = \mathrm{Spec}(R)$.
I am interested in regular, proper, flat schemes $X \to S$...