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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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, ...
Dimitri Koshelev's user avatar
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
2 votes
0 answers
83 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{...
Dimitri Koshelev's user avatar
2 votes
1 answer
238 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\...
user avatar
1 vote
0 answers
174 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 ...
Li Li's user avatar
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3 votes
1 answer
296 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 ...
Asvin's user avatar
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4 votes
1 answer
177 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 ...
Paolo F.'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
  • 5,506
2 votes
1 answer
279 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 ...
Drew Armstrong's user avatar
13 votes
2 answers
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 ...
Đào Thanh Oai's user avatar
2 votes
1 answer
121 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 $\...
Hideus's user avatar
  • 21
2 votes
0 answers
204 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 ...
Alexander's user avatar
3 votes
2 answers
355 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 ...
Asvin's user avatar
  • 7,648
5 votes
1 answer
531 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 $\pi \colon \mathcal C_g \to \mathcal M_g$,...
Fabian Ruoff's user avatar
4 votes
1 answer
189 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 ...
Dimitri Koshelev's user avatar
14 votes
1 answer
279 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-...
David E Speyer's user avatar
2 votes
0 answers
99 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 $P$) ...
LoneStar's user avatar
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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
3 votes
1 answer
221 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 ...
User's user avatar
  • 285
2 votes
1 answer
146 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 ...
user avatar
2 votes
0 answers
142 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 {\...
mxjia's user avatar
  • 89
4 votes
1 answer
261 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 ...
User's user avatar
  • 285
3 votes
1 answer
153 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, ...
Fabian Ruoff's user avatar
1 vote
1 answer
170 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 ...
oleout's user avatar
  • 865
1 vote
1 answer
315 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 ...
oleout's user avatar
  • 865
2 votes
1 answer
285 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 $d$ ...
Dimitri Koshelev's user avatar
2 votes
0 answers
167 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 \...
user127776's user avatar
  • 5,831
3 votes
1 answer
457 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 ...
Dimitri Koshelev's user avatar
1 vote
0 answers
102 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 $c\in [0,\frac1{16}]$....
Max Alekseyev's user avatar
3 votes
1 answer
278 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.,...
user avatar
4 votes
1 answer
377 views

On degree and section of a line bundle on a smooth plane quintic

Let $X$ be a smooth plane projective quintic curve (over $\mathbb C$). Then we know that it has gonality $4$. Assume that it has genus $g(X)=6$. Then my question is the following: Is it necessarily ...
HARRY's user avatar
  • 267
4 votes
2 answers
262 views

When is a pair of space curves that intersect (plenty) a complete intersection?

Let there be two curves of degree $d=2 m^2$ in $\mathbb{A}^3$ having $\geq c d^2$ points of intersection, where $c>0$ is a constant. Then their union $V$ is a curve of degree $2 d = 4 m^2$. Can $V$ ...
H A Helfgott's user avatar
  • 19.3k
4 votes
1 answer
477 views

Higher order inflection points

Consider a smooth plane curve $X\subset\mathbb{P}^2$ of degree $d$. We will say that $x\in X$ is an inflection point of order $s$ if the tangent line $T_xX$, of $X$ at $x\in X$, intersects $X$ in $x\...
user avatar
2 votes
1 answer
193 views

Configuration of points on a plane curve

Let $C\subset\mathbb{P}^2$ be a smooth plane curve of degree six. On $C$ there are $21$ points given as the intersection points of two lines choosen among a set of seven lines. More precisely there ...
user avatar
3 votes
1 answer
309 views

Infinite automorphisms in the moduli of curves

Consider the moduli of smooth curves $M_{g,n}$ (genus $g$, $n$ marked points) and its Deligne-Mumford compactification $\overline{M}_{g,n}$ of stable nodal curves (genus $g$, $n$ marked points). This ...
StableCurves's user avatar
1 vote
1 answer
293 views

Flat connection of a degree zero line bundle on curve

The question is clear from the title. Suppose we have a line bundle on a compact smooth complex curve $X$, and a line bundle $\mathcal{L}=\mathcal{O}_X(p-q)$, where $p$ and $q$ are divisors, then what ...
MKR's user avatar
  • 93
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
  • 2,588
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
0 votes
0 answers
77 views

EXACT number of intersection points of two algebraic curves

As the picture shows2(the paper's link is in 1),it seems that I can use tools including Bezout's theorem to solve the EXACT number of intersection between two algebraic curves(F(x,y) is of degree two ...
BobSS's user avatar
  • 1
4 votes
1 answer
397 views

p-torsion in the Picard group of a regular projective curve

Let $K$ be a field of characteristic $p>0$ and $C$ a regular projective geometrically integral curve over $K$. If $C$ is smooth, then the connected component ${\rm Pic}^0_C$ of the Picard scheme of ...
Arno Fehm's user avatar
  • 1,989
3 votes
1 answer
273 views

Stability of syzygy bundles of smooth curves

For a very ample line bundle $L$, the kernel of the surjection $H^0(L)\otimes \mathcal{O}_X\rightarrow L \rightarrow 0$ is denoted by $M_L$ and is called syzygy bundle. In this paper authors claim in ...
user127776's user avatar
  • 5,831
1 vote
1 answer
179 views

Finite resolution by semi-stable bundles

Over a smooth algebraic curve, do all vector bundles admit a finite resolution by semi-stable bundles? Or is there a characterization of the vector bundles that do? Edit: As an example on $\mathbb{P}^...
user127776's user avatar
  • 5,831
1 vote
0 answers
148 views

Understanding sheaves on normalisation of a curve: $v_* \mathcal{O}_{\tilde{C}} / \mathcal{O}_C$

Let $(C, \mathcal{O}_C)$ be a reduced irreducible curve and $(\tilde{C},\mathcal{O}_{\tilde{C}})$ its normalisation with $v : \tilde{C} \rightarrow C$. Then we have an imoprtant skyscraper sheaf $v_* \...
XT Chen's user avatar
  • 1,064
5 votes
0 answers
369 views

Most divisors on a curve aren't special?

I have a generic smooth curve $C$ of genus $g$ and fixed multiplicities $a_1, \dots, a_n \geq 0$ with $\sum a_i = g+1$. Q1 : For generic marked points $p_1, \dots, p_n \in C$, must $\sum a_i p_i$ be a ...
Leo Herr's user avatar
  • 1,084
1 vote
1 answer
328 views

An automorphism of a function field

I am looking for an explicit example of a function field other than rational with an automorphism which fixes places of different degrees. Also, is there a counterexample, namely a function field with ...
Engin Şenel's user avatar
1 vote
0 answers
207 views

Intersection points of plane curves

Let $F,G\in\Bbb{C}[X,Y,Z]$ be coprime homogeneous polynomials of degrees $m$ and $n$ respectively that do not have multiple factors. By Bezout's Theorem, the number of intersection points of plane ...
KhashF's user avatar
  • 2,588
0 votes
0 answers
120 views

Sections of vector bundles interpreted as sections of line bundles

Let $X$ be a smooth projective curve of genus $g$ over $\mathbb{C}$, $K_{X}$ be a cononical sheaf on $X$ and $\mathcal{E}$ be a locally free sheaf on $X$ s.t. $H^{0}(X,\mathcal{E}^{*})=\operatorname{...
Aoki's user avatar
  • 297
2 votes
0 answers
86 views

Map from the stack of coherent sheaves on a curve to the Grothendieck group

Let $X$ be a smooth, projective curve. We let $Coh(X)$ be the stack of coherent sheaves on $X$. Its Grothendieck group is $Pic(X)\times\mathbf{Z}$. Is the map $$ Coh(X)\rightarrow Pic(X)\times \mathbf{...
hennlu's user avatar
  • 323
2 votes
1 answer
158 views

What are the possible Clifford functions of a curve?

Let $C$ be some smooth proper curve of genus $g$ over an algebraically closed field $k$. In order to understand special divisors on C one may consider the following function c(r), which I will call ...
Nuno Hultberg's user avatar

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