All Questions
Tagged with curves ag.algebraic-geometry
37 questions
16
votes
4
answers
3k
views
Moduli space of genus 2 curves
Does any body know any reference in which the geometry of compactified moduli space of genus two curves ( Which is a three dimensional variety/stack/...) has been studied?
15
votes
1
answer
1k
views
Number of curves over a finite field
Let $K$ be a finite field. Is there a formula for the number of isomorphism classes of genus $g$ smooth curves over $K$?
In other words does there exists a formula for the number of rational points ...
- 8,998
10
votes
3
answers
894
views
Automorphisms of cartesian products of curves
Let $C$ be a smooth projective curve. Is it true that
$$\textrm{Aut}(C\times C)\cong S_2 \ltimes (\textrm{Aut}(C)\times \textrm{Aut}(C))$$
and in case, what would be a reference for this? Thanks.
user82886
10
votes
1
answer
294
views
Rational even polynomials maximally tangent to the unit circle
This question is motivated by College Mathematics Journal problem 1196, proposed by Ferenc Beleznay and Daniel Hwang. My solution to this problem (pre-publication version here) uses Chebyshev ...
- 11.2k
10
votes
1
answer
335
views
Shimura surfaces that do not contain a Shimura curve
Let $S$ be a Shimura surface i.e. a Shimura variety with $dimS=2$. Does $S$ necessarily contain a Shimura curve? I know that probably the answer is No, but do not have an explicit example. What is the ...
- 2,221
7
votes
1
answer
899
views
Isotrivial families with non-zero Kodaira spencer map
Let $S$ be a smooth quasi-projective curve over the complex numbers. Let $P$ be a closed point in $S$. Let $f:\mathcal X \to S$ be a polarized family of smooth projective connected varieties. To this ...
- 171
6
votes
1
answer
768
views
A regular, geometrically reduced but non-smooth curve
Can anyone give an example of a projective, regular, geometrically reduced but non-smooth curve ?
Of course, the base field should be imperfect.
In Exercise 4.3.22 of Qing Liu's book Algebraic ...
- 620
6
votes
1
answer
767
views
Rationality of the moduli space of genus g curves
I'm not an expert in this topic, so please excuse my negligence. I'd also appreciate references to the literature. Throughout, I will work over the complex numbers, although the analogous questions ...
- 5,760
6
votes
1
answer
321
views
Finite union of affinoid is affinoid in proper smooth rigid curves (unless it is everything)
In several papers I have found the surprising statement that finite unions of affinoid subspaces of a proper smooth and connected rigid curve are either the whole curve or again affinoid.
Could you ...
- 845
5
votes
2
answers
783
views
endomorphisms of the Jacobian of a curve
Let $C$ be a smooth, projective curve of genus >1 over the complex numbers and let $J(C)$ be its Jacobian. The Torelli theorem relates the automorphisms of $C$ to the automorphisms of $J(C)$. ...
- 51
5
votes
0
answers
413
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 ...
- 1,084
5
votes
0
answers
333
views
Which equation of a Butterfly?
Let $A, B$ be two points and $L$ be a line on the Euclidean Plane. Take two points $J, G$ on the line $L$ such that $JG=constant$. Let $AJ$ meet $BG$ at $P$, $AG$ meet $BJ$ at $Q$, then the locus of ...
- 477
4
votes
1
answer
544
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\...
user124234
4
votes
1
answer
578
views
A conjecture like Cayley–Bacharach theorem
Let six points $A, A', B, B', C, C'$ lie on a conic and a cubic. Let a conic through $B, B', C, C'$ and meets the cubic again at $A_1, A_2$. Let a conic through $C, C', A, A'$ and meets the cubic ...
- 1,044
4
votes
1
answer
226
views
A conjecture for a curve cuts a curve - variant Cayley-Bacharach's theorem
I propose a conjecture variant of Cayley-Bacharach's theorem.
I'm an electrical engineer, I am not a mathematician. I don't know how to prove this result. Could you give a solution or let me know ...
- 1,044
4
votes
1
answer
180
views
"Inverse problem" for the zeta function [duplicate]
Let $C$ be a smooth, projective, geometrically irreducible curve, of genus $g$, over a finite field $\mathbb{F}_q$. By the Weil conjectures, the zeta function has the shape
$$
Z_C(t)=\frac{P(t)}{(1-t)(...
- 41
4
votes
1
answer
367
views
Structure of fundamental groups arising from smooth projective morphisms
Let $f:X\to B$ be a smooth projective morphism of complex algebraic varieties.
If $f$ is of relative dimension zero, i.e., $f$ is a finite etale cover, then the image of the topological fundamental ...
- 41
4
votes
1
answer
684
views
A problem of four curves
This is a generalization of my previous question, a problem of a cubic and six conics.
Let a curve $(K)$ of degree $m$ and three curves $(C_i)$ of degree $n$, for $i=1,2,3$. Let $(C_1)$ meets $(K)$ ...
- 1,044
4
votes
0
answers
145
views
How many times do I have to blow up such a curve until it is smooth?
If I have parametrized a curve in the complex plane by $$x(t)=a_lt^l+\cdots+a_nt^n$$
$$y(t)=b_kt^k+\cdots+b_mt^m$$
and the image is reduced (there exist at least two exponents which are relatively ...
- 251
3
votes
1
answer
244
views
Etale covers of products of curves
Is a finite etale cover of a product of curves again a product of curves?
The answer is no in general. Here's one way to construct an example. Take the product $A$ of two elliptic curves and an ...
- 31
3
votes
1
answer
424
views
Are Isom-schemes geometrically connected
This question is about properties of Isom-schemes that are well-known over algebraically closed fields.
Let $K$ be a field of characteristic zero, let $C$ be a smooth projective geometrically ...
- 123
3
votes
0
answers
303
views
An algebraic proof: A line bundle on a curve with a connection must be of degree 0
Let me state it using the language of $D$-modules. Let $X$ be a smooth projective curve and $\mathcal{L}$ a line bundle on it. Assume that $\mathcal{L}$ has a left action of $\mathcal{D}_X$. Then show ...
- 1,168
3
votes
0
answers
160
views
Semistability of restrictions of a semistable vector bundle over a reducible nodal curve
Let $C$ be a reducible nodal curve over complex numbers with two smooth components $C_1$ and $C_2$ intersecting at the only node $P$. Let $E$ be a $\omega$ semistable vector bundle over $C$ of rank $r$...
- 290
3
votes
0
answers
118
views
How order of divisor with support at infinity is changed at reduction?
Jing Yu in his paper "On Arithmetic Of Hyperelliptic Curves" on page 5 asserts the following
The most interesting case is certainly the case $k = \mathbb Q$ and $D \in \mathbb Z[t]$.
To decide ...
- 424
2
votes
2
answers
287
views
Relating the toric rank of a semistable curve and the first Betti number of its reduction graph
Introduction
Let $k$ be a local field. Let $C$ be the spectrum of $\mathcal{O}_{k}$. Let $X/k$ be a smooth projective curve with a semistable model $\mathcal{X}/C$.
Let $J$ be the Jacobian of $X$. ...
- 5,504
2
votes
1
answer
306
views
Points on curves of genus 3
Let $Y$ be a smooth complex projective curve of genus two, $X$ a Galois
cover of degree two of $Y$ and $K$ the canonical divisor of $X$.
Let $i$ be the involution of $X$ over $Y$.
Can one find a point ...
- 237
2
votes
1
answer
211
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 ...
user61586
2
votes
1
answer
669
views
Kodaira dimension of the moduli space of curves
It is known that the moduli space $\overline{M}_{g}$ of genus $g$ curves is of general type for $g\geq 24$.
By Theorem 2.4 of
Logan, Adam The Kodaira dimension of moduli spaces of curves with ...
- 8,998
2
votes
1
answer
385
views
Very weak Riemann-Roch on curves (by J. Kollar)
I have a question on an unequality used in the proof of the Very weak Riemann-Roch on curves in Janos Kollar's Lecture on Resolution of Singularities (page 14):
1.13 (Very weak Riemann-Roch on curves)...
- 6,018
1
vote
1
answer
131
views
Sum of two triangles in a projective plane modulo a conic
Given a conic $C$ in the complex projective plane, say $C=\{c:=x^2+y^2+z^2=0\}$, and two “triangles” (given as zeros of products of 3 linear forms $\ell=\{ax+by+cz\}$) $\ell_1\ell_2\ell_3$, $\ell'_1\...
- 14k
1
vote
1
answer
430
views
Do negative indecomposable bundles on curves have sections?
Let $X$ be a smooth projective curve, and $E$ an indecomposable vector bundle on $X$ with $\mathrm{deg} E<0$. Is it true that $H^0(X,E)=0$?
This is true if $E$ is a line bundle, which means it is ...
- 39
1
vote
0
answers
115
views
Cokernel of map of dual of sheaves of differentials/deformations
Let $C$ be a nodal projective curve over an algebraically closed field of genus at least $2$. There are two natural "differential objects" one can consider: The sheaf of differentials $\...
- 223
1
vote
0
answers
119
views
More on points on a curve of genus 3
Let $Y$ be a smooth complex projective curve of genus two,
$X$ a Galois cover of degree two of $Y$ and $K$ the canonical
divisor of $X$. Let $i$ be the involution of $X$ over $Y$.
Can one find two ...
- 237
1
vote
0
answers
239
views
Proposition from Kollar's Rational Curves on Algebraic Varieties
$\DeclareMathOperator\Hom{Hom}$I have some questions on a proof of Proposition II.3.10 & notations from Rational Curves
on Algebraic Varieties by Janos Kollar (page 117).
We work in setting ...
- 6,018
1
vote
0
answers
516
views
support of embedded points in a curve
Let $C \subset \mathbb{P}^n$ be an one dimensional scheme. Suppose that $C$ decomposes as the union of a Cohen Macaulay reduced curve $\tilde{C}$ (in particular $\tilde{C}$ does not have embedded ...
- 103
0
votes
1
answer
141
views
meaning of $k(C)/1+\mathfrak{m}_x$ [closed]
Let $C$ be a smooth projective curve over some field $k$ and $x$ a closed point of $C$. I've seen some constructions in which people use
$k(C)^\times / 1+\mathfrak{m}_x$.
What's the meaning of that?...
-1
votes
1
answer
323
views
property of rational functions on projective curves
I have a couple of question about the proof of Lemma 1.20.5 from Janos Kolloar's Lecture on Resolution of Singularities (page 19):
Lemma 1.20.5 Let $C$ be a reduced, irreducible projective curve (=1D ...
- 6,018