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
3
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0
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137
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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, ...
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 ...
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{...
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\...
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 ...
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 ...
4
votes
1
answer
177
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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
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 ...
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 ...
13
votes
2
answers
2k
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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
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 $\...
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 ...
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 ...
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 $\pi \colon \mathcal C_g \to \mathcal M_g$,...
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 ...
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-...
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 $P$) ...
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 ...
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 ...
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 ...
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 {\...
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 ...
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, ...
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 ...
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 ...
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$ ...
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 \...
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 ...
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}]$....
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.,...
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 ...
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$ ...
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\...
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 ...
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 ...
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 ...
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)$. ...
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 ...
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$...
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 ...
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 ...
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 ...
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}^...
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_* \...
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 ...
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 ...
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 ...
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{...
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{...
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 ...