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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3
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0answers
110 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}...
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Seeking reference on monomial curves

I am looking for survey articles, lecture notes, or any other material that could be the starting point to explore known results and open questions in the theory related to monomial curves. Thank you.
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1answer
146 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 ...
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106 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
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1answer
126 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^...
2
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1answer
184 views

Question on a constructive proof of space projective curve are the intersection 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 ...
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97 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 $...
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0answers
105 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 ...
9
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1answer
239 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 ...
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0answers
235 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
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1answer
204 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: $...
4
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1answer
302 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 ...
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2answers
778 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 ...
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0answers
124 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
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1answer
179 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 ...
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67 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, ...
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0answers
145 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 ...
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160 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 ...
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82 views

Richelot isogenies in characteristic $2$

I am interested in Richelot isogenies between ordinary abelian surfaces in characteristic $2$. If I am not mistaken, the corresponding theory is developed in Article "J.-B. Bost, J.-F. Mestre, ...
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179 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 ...
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0answers
71 views

Are there non-trivial $\mathbb{F}_q$-covers of the j-invariant 0 elliptic curve by a hyperelliptic or cyclic trigonal curve?

Consider the ordinary elliptic curve $E\!: y^2 = x^3 + b$ (of $j$-invariant $0$) over a finite field $\mathbb{F}_q$ such that $\sqrt{b}, \sqrt[3]{b} \not\in \mathbb{F}_q$. Also, for any $n \in \mathbb{...
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1answer
226 views

Limit along the category of all algebraic curves over a field

Let $k$ be algebraically closed field of charactersistic zero and $\mathcal C$ be the category of irreducible smooth projective curves over $k$ and non-constant maps between them. I have a functor $F\...
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0answers
126 views

Minimal degree of the morphism from a curve to $\mathbb{P}^n$

We do all the things in an algebraically closed field $k$ of characteristic $0$. Let $C$ be a projective curve over $k$. We have been familiar with the notion "gonality", which is the ...
3
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1answer
201 views

What is the involution on the moduli space of genus 3 curves induced by the Torelli map

Let $M_g$ be the moduli space of genus $g$ curves, $A_g$ be the moduli space of principally polarized dimension $g$ abelian varieties. They have dimensions $3g-3,g(g+1)/2$ respectively. The Torelli ...
4
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1answer
125 views

Link at infinity of a complex algebraic curve transverse to S^3 and non-singular in D^4

I am currently working on the following paper by Lee Rudolph: https://arxiv.org/abs/math/9307233 Using Kronheimer-Mrowka's theorem, he proves in page 6 that the slice Euler characteristic of a given ...
12
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277 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 ...
2
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1answer
144 views

Existence of complete intersections of codimension 2

The following excerpt is from page 147 of Dieudonne's History of Algebraic Geometry: Can anyone provide a reference for this result? Is it difficult to find explicit equations for the two ...
13
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2answers
2k views

Is it a new discovery on conic section?

I discovered a problem in plane geometry (there are some nice special cases) as follows: Let $ABC$ be a triangle and $\Omega$ be arbitrary circumconic. Let two points $A_b, A_c \in BC$, $B_c, B_a \in ...
2
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1answer
109 views

Curves sharing points over finite fields, and their mutual divisibility

Consider in $\mathbb{A}^2(\mathbb{F}_q)$ two $\mathbb{F}_q$-rational curves $\mathcal{X}:f(x,y)=0$ and $\mathcal{Y}:g(x,y)=0$, and let $\mathcal{Y}$ be absolutely irreducible. Suppose also that $\...
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0answers
101 views

Cartier operator and logarithmic differentials

Let $k$ be an algebraically closed field of characteristic $p$, let $C$ be a curve over $k$ and let $\omega$ be a meromorphic differential form on $C$. If $\omega$ gets mapped to itself by the Cartier ...
2
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2answers
180 views

Galois stable elements of the Picard group of a curve and the rational divisors

Let $C$ be a (smooth,proper) curve over a field $k$. Let $\operatorname{Div}_C(k)$ be the free abelian group generated by the closed points of $C/k$ and $k(C)^\times$ be the group of rational ...
5
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1answer
252 views

Tangent Space of the Hodge bundle on the moduli space of curves

Let $k$ be an algebraically closed field and $\mathcal M_g$ denote the moduli space (stack) of smooth curves of genus $g$ over $k$. Using the universal curve $\pi \colon \mathcal C_g \to \mathcal M_g$,...
4
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1answer
164 views

Jacobians $\mathbb{F}_q$-isogenous to the direct square of an ordinary elliptic $\mathbb{F}_q$-curve of $j$-invariant $0$

Consider an ordinary elliptic curve $E_b\!: y^2 = x^3 + b$, of $j$-invariant $0$ over a finite field $\mathbb{F}_q$, such that $\sqrt{b} \not\in \mathbb{F}_q$. Question. What are some examples of ...
14
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1answer
251 views

Lower bounds for class number of function fields with fixed $q$, growing $g$

Let $X$ be a smooth project curve of genus $g$ over the finite field with $q$ elements. Let $h$ be $\# \mathrm{Pic}^0(X)(\mathbb{F}_q)$. Weil showed that $h \geq (\sqrt{q}-1)^{2g}$. Lachaud and Martin-...
2
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0answers
74 views

Weierstrass points in Artin-Schreier extensions of the rational function field in characteristic $p$

I encountered a problem when proving a result regarding the orders at non-Weierstrass points ($n$ is an order of the point $P$ if there is some holomorphic differential $\omega$ with order $n$ at $P$) ...
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0answers
38 views

Name for a curve defined by detours

define the length of the detour of going from $A$ to $B$ via the intermediate point $C$ as $\left\|C-A\right\| + \left\|B-C\right\| -\left\|B-A\right\|$ Let now $C=(0,0)$ be located in the origin, $\...
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0answers
94 views

Some possible implication of existence of a $g^r_d$ on a smooth plane curve

Let $X$ be a smooth plane curve of degree $d$ and genus $g$ (over complex numbers). For example we can take for the time-being $d=6$ and $g=10$. Let's also assume that there exists a divisor on $X$ ...
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115 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 ...
3
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1answer
182 views

On emptiness of certain $G^r_d(X)$ on a smooth plane curve

Let $X$ be a smooth plane projective curve of degree $6$ and genus $10$ (over complex numbers). Then my question is the following : Question : Is it possible that there exists a special divisor $D$ of ...
3
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1answer
138 views

Degenerations of hyperelliptic coverings

Take six distinct points $p_1,\dots,p_6\in\mathbb{P}^1$ and consider the double covering $f:C\rightarrow \mathbb{P}^1$ ramified over $p_1,\dots,p_6\in\mathbb{P}^1$. Then $C$ is a smooth curve of genus ...
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0answers
113 views

The genus of the following algebraic curve(tetragonal curve)

$$\mathcal F(\lambda, y) = {y}^{4}- \left( 2 {\lambda}^{8}+{\lambda}^{4}+2 \right) {y}^{3}+ \left( 2 {\lambda}^{16}+4 {\lambda}^{14}+3 {\lambda}^{12} +4 {\lambda}^{2}+2 \right) {y}^{2} \\-4 {\...
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1answer
204 views

Special divisors on smooth plane curves

Let $X$ be a smooth, plane projective curve of degree $6$ and genus $10$ (over complex numbers). Question : Is it possible that there exists a special divisor $\Delta$ of degree $10$ on $X$ such ...
3
votes
1answer
122 views

Topological properties of differentials with prescribed zeroes on an algebraic curve

Let $C$ be an algebraic curve (one dimensional projective regular connected scheme of finite type) of genus $g$ over an algebraically closed field $k$ with structure morphism $\pi$. By Riemann-Roch, ...
1
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1answer
145 views

The groups $H^i(k,\mathbb{Z})$ for $i=1,2$

This question is related to my post Interpretation of some maps involving cohomology groups. $C$ is a smooth geometrically integral affine curve over a number field $k$, and $C_1$ is its smooth ...
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1answer
170 views

The smooth completion of a curve

Let $C$ be a smooth geometrically integral affine curve. This question concerns the smooth completion $C_1$ of $C$, both defined over a number field $k$. We know that given any smooth projective ...
2
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1answer
188 views

Is the reduction of an absolutely irreducible plane curve still irreducible except for the finite number of cases?

Help me please. Let $k$ be an algebraically closed field (I am mainly interested in $k = \overline{\mathbb{Q}}, \overline{\mathbb{F}_q}$). Consider a plane curve $C \subset \mathbb{A}^2$ of degree $d$ ...
2
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0answers
152 views

Covering abelian varieties over finite fields with the product of curves

Question. Given an $n$-dimensional abelian variety $X$ over a finite field, is it possible to find smooth projective curves $C_1,\ldots, C_n$ such that there exists a finite regular map $C_1\times \...
3
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1answer
329 views

Under what conditions is the polynomial of degree $6$ irreducible?

Let $k$ be a perfect field of characteristic $p \neq 2,3$ such that $\omega := \sqrt[3]{1} \in k$, where $\omega \neq 1$. Consider an absolutely irreducible (not necessarily homogenous) quadratic ...
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0answers
77 views

Length of isoline $x(1-x)y(1-y)=c$

For the integral appearing in this answer, it may be beneficial to derive the length $L(c)$ of the isoline: $$x(1-x)y(1-y) = c,$$ where $x,y$ are ranging in $[0,1]$, and constant $c\in [0,\frac1{16}]$....
3
votes
1answer
120 views

3-secant lines of a projective curve

Consider a smooth projective curve $C\subset\mathbb{P}^n$. Let $G(1,n)$ the Grassmannian of lines of $\mathbb{P}^n$. The variety $S_2(C)\subset G(1,n)$ parametrizing lines that are secant to $C$ (i.e.,...

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