Questions tagged [algebraic-curves]

for questions on one dimensional algebraic varieties over any field, including questions of moduli, and questions about specific curves.

Filter by
Sorted by
Tagged with
3 votes
1 answer
81 views

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
52 views

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 $h^0(C,D)=s$...
  • 325
2 votes
0 answers
84 views

Genus of curves on complete intersections

Let $X$ be a smooth projective variety with $\dim X=3$ equipped with a very ample divisor $H$, and $C\subset X$ be a pure one-dimensional closed subscheme of $X$ with arithmetic genus $g$ and degree $...
  • 21
3 votes
1 answer
163 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 $\mathbb{P}^3_S$. ...
4 votes
0 answers
188 views

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 ...
  • 454
1 vote
0 answers
68 views

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 ...
2 votes
0 answers
185 views

Genus of algebraic curves

Let $X$ be an integral (possibly singular) projective algebraic curve of degree $d$ in $\mathbf{P}^n_{K}$, where $K$ is a field that is either the real numbers, the complex numbers, or a field that is ...
  • 45
4 votes
1 answer
154 views

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 ...
  • 71
0 votes
0 answers
121 views

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 $\...
  • 439
3 votes
1 answer
218 views

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 \...
  • 267
1 vote
1 answer
161 views

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 ...
  • 419
0 votes
0 answers
131 views

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 ...
4 votes
1 answer
214 views

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
69 views

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
157 views

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?
  • 1,964
2 votes
0 answers
102 views

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
335 views

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
63 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 ...
  • 643
3 votes
0 answers
62 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
78 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
128 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 ...
  • 443
1 vote
0 answers
111 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
88 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
1k 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
96 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(...
  • 29
2 votes
1 answer
156 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
673 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
109 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
429 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 ...
  • 41.7k
10 votes
0 answers
258 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
159 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
47 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 ...
  • 181
1 vote
0 answers
215 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
154 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 $\mathbb{G}_m$-...
  • 427
3 votes
0 answers
240 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 ...
  • 2,050
1 vote
0 answers
60 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 ...
  • 181
4 votes
1 answer
270 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
128 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
145 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
220 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
147 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 ...
  • 397
2 votes
1 answer
173 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 ...
  • 107
0 votes
0 answers
83 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 ...
  • 693
3 votes
0 answers
65 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
64 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 $...
  • 323
7 votes
0 answers
157 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(...
5 votes
1 answer
418 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
98 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
234 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 ...
  • 1,212
4 votes
0 answers
72 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 $U_{g,d}$...

1
2 3 4 5
19