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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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 ...
Yachen Liu's user avatar
0 votes
0 answers
93 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{...
The Thin Whistler's user avatar
14 votes
2 answers
2k views

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 ...
Vlad Matei's user avatar
3 votes
0 answers
99 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(...
John Z.'s user avatar
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2 votes
1 answer
232 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 ...
Pène Papin's user avatar
11 votes
3 answers
726 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
152 views

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 ...
Jędrzej Garnek's user avatar
9 votes
3 answers
792 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 ...
Ben Webster'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
4 votes
1 answer
198 views

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, \, ...
Francesco Polizzi's user avatar
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68 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 ...
user95246's user avatar
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1 vote
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230 views

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 ...
Francesco Polizzi's user avatar
4 votes
0 answers
188 views

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 $\mathbb{G}_m$-...
E. KOW's user avatar
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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 ...
Jose Capco's user avatar
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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 ...
T. Combot's user avatar
  • 231
4 votes
1 answer
363 views

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 ...
user267839's user avatar
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1 vote
0 answers
136 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 ...
Armando j18eos's user avatar
0 votes
0 answers
204 views

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 ...
Display Name's user avatar
2 votes
1 answer
513 views

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\...
Display Name's user avatar
1 vote
1 answer
162 views

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 ...
Puzzled's user avatar
  • 8,842
2 votes
1 answer
228 views

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 ...
babu_babu's user avatar
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0 answers
86 views

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 ...
Sky's user avatar
  • 913
3 votes
0 answers
102 views

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 $\text{Cl}(X)$ ...
Jonathan Love's user avatar
2 votes
1 answer
117 views

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 $...
hennlu's user avatar
  • 323
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
6 votes
1 answer
581 views

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: ...
Hank Scorpio's user avatar
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0 answers
101 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?
Joseph O'Rourke's user avatar
5 votes
1 answer
271 views

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 ...
Jef's user avatar
  • 949
4 votes
0 answers
78 views

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 $U_{g,d}$...
Jérémy Blanc's user avatar
4 votes
0 answers
63 views

$(n-2)$-degree curve passing through $n(n-1)/2$ midpoints

It is known that in the plane, there is an unique conic passing through given $5$ points. For any $4$ points, there is 6 segments which vertex from these points. It is known that $6$ midpoints of ...
Vu Thanh Tung's user avatar
5 votes
1 answer
177 views

Do you know explicit examples of superelliptic curves $y^{\ell} = g(x)$ (for some prime $\ell > 3$) covering some elliptic curves?

For every elliptic curve $E$ Icart in $\S 2$ of the paper explicitly constructs a superelliptic curve $S\!: y^3 = f(x)$ and a cover $\varphi\!: S \to E$. Do you know explicit examples of superelliptic ...
Dimitri Koshelev's user avatar
6 votes
2 answers
664 views

Could the Weil zeroes of curves be evenly distributed?

If $X$ is a smooth, geometrically connected, projective curve of genus $g$ over $\mathbb{F}_q$, then the zeta function of $X$ is of the form $P(s)/(1 - s)(1 - qs)$, where $P(s)$ is a polynomial of ...
LeechLattice's user avatar
  • 9,421
15 votes
3 answers
658 views

Height functions on $\mathcal{M}_g(\overline{\Bbb{Q}})$ defined via dessins d'enfants?

Belyi's theorem establishes a correspondence between smooth projective curves defined over number fields and the so called dessins d'enfants which are bipartite graphs embedded on an oriented surface ...
KhashF's user avatar
  • 2,598
4 votes
0 answers
245 views

Minimal $b_2$ in Sarkisov's construction

In the paper On the structure of conic bundles. Math. USSR, Izv., 120:355–390, 1982, Theorem 5.10, Sarkisov constructed the first example of non-rational, rationally connected $3$-fold $X$ with $H^{3}...
Nick L's user avatar
  • 6,933
1 vote
1 answer
166 views

Moduli spaces of horizontal curves

Let $f:X\rightarrow Y$ be a morphism of projective varieties. We may assume that $X$ and $Y$ are smooth, and $f$ is flat of relative dimension one. Fix an ample divisor $A$ on $X$. I would like to ask ...
Puzzled's user avatar
  • 8,842
4 votes
1 answer
366 views

Reverse residue theorem without using Serre's duality

In V. Talovikova's text "Riemann-Roch Theorem", a key part of the proof of Riemann-Roch theorem is the following proposition (4.6 in the text): Let $\{a_1, \dots,a_n\}$ be a set of points in ...
Serge the Toaster's user avatar
2 votes
1 answer
168 views

Induced action on Prym variety

Let $C$ be a smooth projective curve of genus $g$ with an involution $\iota: C \to C$. We have the quotient map $\pi: C \to C/\iota$, with $C/\iota$ a smooth curve of genus $h$. The pullback map $\pi^...
Benighted's user avatar
  • 1,701
5 votes
2 answers
494 views

Question on a constructive proof that space projective curves are the intersections of three hypersurfaces

$\newcommand\P{\mathbb P} \newcommand\C{\mathcal C}$I am a bit confused by a proof I am reading on the fact that a projective algebraic space-curve (i.e. an algebraic curve in $\P^3(k)$, where $k$ is ...
quantum's user avatar
  • 489
1 vote
0 answers
190 views

Degree and genus of projected curve

Let $C\subset\mathbb{P}^n$ be a normal curve over an algebraically closed field of characteristic $0$. Assume that $C$ is not contained in any hyperplane. We may assume that $P=[0:\cdots:0:1]$ is on $...
Li Li's user avatar
  • 383
1 vote
0 answers
183 views

Brauer-Manin obstruction and affine curves

I'm looking for references that can justify to what extent is the following statement true: Statement. Let $X$ be a smooth geometrically integral curve over a number field $k$. Then the Brauer-Manin ...
oleout's user avatar
  • 865
10 votes
1 answer
924 views

Visualizing genus-two Riemann surfaces: from the three-fold branched cover to the sphere with two handles

I am trying to visualize the genus-two Riemann surface given by the curve $$ y^3 = \frac{(x-x_1)(x-x_2)}{(x-x_3)(x-x_4)}. $$ We can regard this surface as a three-fold cover of the sphere with four ...
Holomaniac's user avatar
1 vote
0 answers
347 views

Absolute irreducibility of affine varieties

Let $V$ be an irreducible affine variety over a finite field $\mathbb{F}_q$, , given in terms of equations over $\mathbb{F}_q$, where $q$ is some prime power. Are there any methods to decide whether $...
Vanya's user avatar
  • 581
6 votes
2 answers
380 views

Nef divisors on surfaces

Let $X$ be a smooth projective rational surface over an algebraically closed field of characteristic zero, and $D$ a divisor on $X$ such that $D$ is nef and $D^2 = 0$ with the following properties: $...
Puzzled's user avatar
  • 8,842
5 votes
1 answer
424 views

Weak Mordell-Weil for EC using Chevalley-Weil theorem

I am reading the book Applications of Diophantine Approximation to Integral Points and Transcendence by Zannier and Corvaja and, after their proof of the Chevalley-Weil theorem, in Example 3.8 they ...
cartesio's user avatar
  • 233
11 votes
2 answers
2k views

Motivation for birational geometry

I'm interested in how do people that work in birational geometry view their field — specifically, what are the kinds of geometric questions (as opposed to commutative-algebraic questions) that ...
roymend's user avatar
  • 221
3 votes
0 answers
162 views

Why is a rational divisor class on $\overline{\mathcal{M}}_g$ determined by its values on families not mapping into a given subvariety?

This question is about Exercise 3.90 on page 143 of the book "Moduli of Curves" by Harris & Morrison. To avoid defining stacks, the authors define a "rational divisor class on the ...
user555203's user avatar
2 votes
1 answer
289 views

Computing $H^1$ with coefficients in a torsion-free abelian group

Let $k$ be a number field and denote by $H^i(k,-)$ the Galois cohomology functor $H^i(\mathrm{Gal}(\bar{k}/k),-)$. Let $X$ be a smooth geometrically integral curve over $k$. One can easily show that ...
oleout's user avatar
  • 865
1 vote
0 answers
83 views

Varieties swept out by Linear Spaces nondegenerated

We working over complex numbers $\mathbb{C}$ keeping our constructions as geometric as possible. Let $\Lambda_1, ..., \Lambda_m \cong \mathbb{P}^{n-2} \subset \mathbb{P}^{n} $ be pairwise distinct, ...
user267839's user avatar
  • 5,948
2 votes
0 answers
204 views

Synthetic construction of rational normal curve

We consider the so called 'Synthetic or Steiner construction', which can be found e.g. in this script or Joe Harris' Algebraic Geometry on page 14 which should finally be recognized as rational normal ...
user267839's user avatar
  • 5,948
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
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