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
1
vote
0
answers
144
views
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
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{...
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 ...
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(...
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 ...
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 ...
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 ...
10
votes
0
answers
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 ...
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 $$H^0(C, \, ...
0
votes
0
answers
68
views
Sum of Weierstrass points on a curve of genus three
Let C be a smooth complex projective curve of genus 2 and X a non trivial Galois
cover of degree 2 of C. So X has genus three and it is hyperelliptic, hence X has 8
Weierstrass points. Can one compute ...
1
vote
0
answers
230
views
Missing generator for $H^0(C, \, \omega_C^{\otimes 2})$, with $C$ is hyperelliptic of genus $3$
This is probably very classical and well-known, but I could not find the answer in the literature, so let me ask it here.
Let $C$ be a hyperelliptic curve of genus $3$, defined over the complex ...
4
votes
0
answers
188
views
𝔾ₘ extensions vs line bundles over abelian varieties
Given a complex polarized abelian variety $V$, we can define a map $$\operatorname{Ext}^1\left(V, \mathbb{G}_m\right) \to \operatorname{Pic}\left(V\right)$$
by viewing an extension as a $\mathbb{G}_m$-...
3
votes
0
answers
313
views
On a conjecture of Hartshorne
Hartshorne has a conjecture in his book Ample Subvarieties of Algebraic Varieties. It's in page 126 Conjecture 5.16 where he writes that if $B$ is a finitely generated flat module over a regular local ...
1
vote
0
answers
71
views
Morphisms from plane curves to hyperelliptic curves
Consider a plane curve $\mathcal{C}$ of degree $d$. We know that if a morphism $\varphi$ from $\mathcal{C}$ to a curve of genus $g\geq 2$ exists, then $\deg \varphi \leq (g'-1)/(g-1)$ where $g'$ is ...
4
votes
1
answer
363
views
Deformation theoretic argument on dimension counting of naive Hurwitz scheme
I'm reading the Atanas Atanasov's course notes of Joe Harris' course Geometry of Algebraic Curves
and have a question about a suggested modification of an dimension
countinging argument applying ...
1
vote
0
answers
136
views
Koszul cohomology and nodal curves
In M. Aprodu, G. Farkas - Koszul Cohomology and Applications to Moduli, arXiv:0811.3117 [math.AG], proof of Theorem(s) 4.5 (and 4.12), the authors constructed a semistable nodal curve $C^{\prime}$ of ...
0
votes
0
answers
204
views
Are pure sheaves actually vector bundles on projective curves?
A coherent sheaf is pure if every non-trivial coherent subsheaf has the same dimension, where the dimension of a sheaf is the dimension of its support.
As in the title, I wonder if the notion of pure ...
2
votes
1
answer
513
views
Do we know anything about Harder-Narasimhan filtrations of tensor products of vector bundles?
I am interested in vector bundles over a nonsingular complete algebraic curve $C$ over $\mathbb C$. For a vector bundle $E$, its Harder-Narasimhan filtration is a filtration of subbundles
$$0=E_0\...
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 ...
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 ...
0
votes
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 ...
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 $\text{Cl}(X)$ ...
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 $...
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(...
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:
...
0
votes
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?
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 ...
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 $U_{g,d}$...
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 ...
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 ...
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 ...
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 ...
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}...
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 ...
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 ...
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^...
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 ...
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 $...
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 ...
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 ...
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 $...
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:
$...
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 ...
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 ...
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 ...
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 ...
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, ...
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 ...
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 ...