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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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 ...
Daniel Litt's user avatar
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22 votes
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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.   &...
Joseph O'Rourke's user avatar
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 ...
Henry Segerman's user avatar
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 ...
Will Sawin's user avatar
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13 votes
0 answers
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 $Aut(Jac(X', a'))$ ...
Catherine Ray's user avatar
12 votes
0 answers
341 views

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 ...
Pulcinella's user avatar
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12 votes
0 answers
207 views

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 ...
John Voight's user avatar
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12 votes
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440 views

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 ...
S. Carnahan's user avatar
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11 votes
0 answers
264 views

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^{...
Will Sawin's user avatar
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10 votes
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325 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 ...
Jérémy Blanc's user avatar
10 votes
0 answers
232 views

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 ...
Ron's user avatar
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10 votes
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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 $S$?...
Ali's user avatar
  • 153
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 ...
Ryan Keast's user avatar
9 votes
0 answers
207 views

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 ...
François Brunault's user avatar
9 votes
0 answers
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 ...
Timothy Chow's user avatar
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9 votes
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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 ...
Will Sawin's user avatar
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9 votes
0 answers
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 ...
paul Monsky's user avatar
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8 votes
0 answers
197 views

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 $K:=\overline{\Bbb{Q}}(t)$. ...
KhashF's user avatar
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8 votes
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166 views

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)...
Stefano's user avatar
  • 625
8 votes
0 answers
644 views

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 ...
JSE's user avatar
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8 votes
0 answers
429 views

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_{\...
Michael Zieve's user avatar
8 votes
0 answers
500 views

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 ...
Konrad Schöbel's user avatar
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-...
Tait's user avatar
  • 485
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)) = \...
Gina's user avatar
  • 131
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(...
Sameera Vemulapalli's user avatar
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 \...
Ben C's user avatar
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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 $\...
issoroloap's user avatar
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 $G$-...
user42024's user avatar
  • 790
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 ...
Will Sawin's user avatar
  • 135k
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 ...
Xander Faber's user avatar
  • 1,169
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 $g$ ...
Kevin H. Lin's user avatar
  • 20.7k
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 $\...
Hamish's user avatar
  • 231
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 ...
paul Monsky's user avatar
  • 5,412
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 ...
zemora's user avatar
  • 545
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 ...
Fabian Ruoff's user avatar
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 ...
Asvin's user avatar
  • 7,648
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 ...
H A Helfgott's user avatar
  • 19.3k
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 ...
sawdada's user avatar
  • 6,148
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 ...
Eduardo R. Duarte's user avatar
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 ...
Dimitri Koshelev's user avatar
6 votes
0 answers
251 views

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 ...
user_lambda's user avatar
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 $\...
user avatar
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 \...
Pyramid's user avatar
  • 288
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+\...
Nikita Kalinin's user avatar
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....
Misha's user avatar
  • 143
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 ...
Dimitri Koshelev's user avatar
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 ...
C.D.'s user avatar
  • 545
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 ...
SashaP's user avatar
  • 7,027
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 ...
KhashF's user avatar
  • 2,588
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 $X \to S$...
PrimeRibeyeDeal's user avatar

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