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
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1
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Finding the genus of a certain algebraic curve
I asked the following question in the mathstackexchange but I did not get an answer, probably the level of this question is not appropriate for mathoverflow, so I would like to apologize in advance if ...
1
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0
answers
626
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First chern class of vector bundle w/ push / pull
(Context: I am trying to prove the Pluecker formula.)
Let $E^{k+1}_L = \pi_{2,*}(\pi_1^*L\otimes O_{C\times C}/J_\Delta^{k+1})$, where $C$ is a smooth curve of genus $g$, $L$ a line bundle of degree $...
2
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2
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516
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Points in the plane imposing independent conditions: reference request
Hello,
Does anybody know a reference for the following result: $d\ge 5$ points of $\mathbb P^2$ fail to impose independent conditions on curves of degree $d-3$ if and only if at least $d-1$ of these ...
1
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1
answer
201
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Assumption of genus at least $2$ for stable curves
In the article "The irreducibility of the space of curves of given genus" by Deligne, Mumford, the definition of stable curves start with the assumption that the genus is at least $2$. Why is this ...
1
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0
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142
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Smoothability of stable curves in mixed characteristic
Let $R$ be a complete DVR with residue field $k$ algebraically closed of characteristic $p$ and fraction field $K$ of characteristic zero. Let $C$ be a stable curve (in the sense of Mumford-Deligne) ...
2
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0
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342
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Examples of semi-stable models of curves
Let $R$ be a discrete valuation ring with fraction field $K$ of characteristic zero and residue field $k$ of characteristic $p>0$. Assume $k$ is algebraically closed. I want to produce examples of ...
6
votes
1
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805
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A Jacobian with a good reduction, which is simple : how is the reduction of the curve?
Let $C$ be a smooth genus $g>1$ curve defined over a number field (say over $\mathbb Q$). Let $J(C)$ be its jacobian.
Suppose that $J(C)$ has good reduction $J_p$ at a prime $p$ and moreover $J_p$ ...
1
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0
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103
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Degree of an isogeny in the endomorphism ring of the jacobian of a curve and self intersection index in its ring of correspondences
I hope this question is not too basic.
Let $C/\bar{k}$ be a nonsingular irreducible curve of genus $g$ and $\mathfrak{C}(C\times C)\cong \text{CH}^1(C\times C)$ be its ring of correspondences.
I am ...
5
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1
answer
247
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Degree of irrationality and hyperelliptic curves
For a variety $V$ of dimension $n$, let $Irr(V)$ denote the minimal degree of a dominant rational map $V\to \mathbb{P}^n$.
Suppose that a curve $X$ admits a dominant map from a variety $V$ with $...
18
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2
answers
5k
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What does the Jacobian of a curve tell us about the curve?
A natural object in the study of curves is the Jacobian of a curve. What are some natural geometric properties of the curve that the Jacobian encapsulates? In other words, what can the Jacobian tell ...
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3
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Klein's curve (algebraic geometry)
I can't find any information about the canonical ring of Klein's quartic curve (the one with 168 automorphisms). I would imagine there is a lot known about the structure of this ring. Can anybody help ...
7
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1
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891
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Modular curve X(2)
Let $\mathfrak{M}(2)$ be the algberaic stack over $\mathbb{Z}[1/2]$ which classifies the elliptic curves with the two level structure and let $X(2)$ be the coarse moduli space of $\mathfrak{M}(2)$ ($X(...
0
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1
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178
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Nondegenerate projective plane conics over a nonperfect field $k$ of even characteristics
Let $C_1\!: a_1x^2 + b_1y^2 + c_1z^2 + d_1xy + e_1xz + f_1yz$ and $C_2\!: a_2x^2 + b_2y^2 + c_2z^2 + d_2xy + e_2xz + f_2yz$ be nondegenerate projective plane conics over a nonperfect field $k$ of even ...
3
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2
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341
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Canonical curve of genus 8
I want to write explicit equations for a curve $C$ that allows a 2-to-1 morphism onto a curve $E$ that is not rational, actually, $E$ is an elliptic curve. In more detail, I know that $E$ is an ...
7
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2
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770
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Section of universal curve
Let $\mathcal{M}_{g,1}\to \mathcal{M}_g$ be the universal genus $g$ curve, let $K_g$ denote the funciton field of $\mathcal{M}_g$. Take the generic fiber $C\to \mathrm{Spec}{(K_g)}$, after some finite ...
6
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251
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Concavity of a function implicitly defined by a polynomial
Consider the following system of $n$ equations:
\begin{equation}f_j^2 = x_j^2\sum_{i=1}^n A_{ij} f_i
\tag{$\star$}
\end{equation}
where $A_{ij}\geq 0$ are known constants and where $x_j>0$ for ...
4
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1
answer
98
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Convexity of a set related to certain class of Laurent polynomials
For $r,s\in\mathbb{N}$, let
$$L(z):=\sum_{j=-r}^{s}a_{j}z^{j}$$
be a Laurent polynomial with real coefficients such that there exists a closed curve $\gamma$ encircling the origin, i.e., $0\in\mbox{...
3
votes
1
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382
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Modern reference for the theory of correspondences for curves
The classic theory of correspondences between smooth algebraic curves can be found in André Weil's Foundations of algebraic geometry. However, this reference works in a pre-modern algebraic geometry ...
3
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2
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704
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Curves and trisecant lines
We know that rational normal curves and elliptic normal curves have no trisecant lines. For the "next" case, this is still true. That is, a nondegenerate curve of degree $d\geq 5$ and genus $2$ in $\...
3
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0
answers
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Calculation of Cartier-Manin matrix
Let $\mathbb{F}_q$ be a finite field of characteristic $p$ and let $C$ be a plane projective nonsingular curve over $\mathbb{F}_q$ ,
with function field $K = \mathbb{F}_q(C)$. Let $K^p$ denote the ...
1
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0
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207
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A Zero divisor of an induced section
Let $f:X\rightarrow Y$ be a smooth family of complex projective curves on a smooth curve $Y,$ with singular fibers over some finite set of $Y.$ Let $\mathcal{A}$ and $\mathcal{B}$ are invertible ...
3
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0
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157
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Normalizing a family of plane curves containing a cusp
In Caporaso and Harris's paper (http://www.mat.uniroma3.it/users/caporaso/CaporasoHarrisInv.pdf page 368, right before section 3), it is mentioned that if $[X]$ is a point on the variety of plane ...
2
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0
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108
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Birational map from even to odd degree curve. What is the image of one of infinity points?
Suppose I have a curve $C_1$ determined by $C_1: y^2 = (x+a)(f_{2g+1} x^{2g+1} + \dots + f_0)$. It has even degree polynomial of $x$ on the right side. I want to consider its image under birational ...
0
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0
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262
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Dimension of the Severi Variety
Edit: In short, I know how to get a lower bound given that nodal curves are dense, and I know how to show nodal curves are dense given the lower bound, I can't find a reference that proves either fact ...
0
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2
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317
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Connection between 'Separated scheme of finite type over spec(k)' and 'Curve in $\mathbb R^n$ [closed]
is there some connection between a curve in the algebraic geometry sense, e.g.
Separated scheme of finite type over spec($k$)
for a field $k$
and a curve in the sense of a smooth map from an ...
2
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0
answers
137
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Is a supersingular Kummer surface $k$-unirational in characteristic 2?
Let $k$ be a perfect field of even characteristic. Consider the simplest example of a supersingular genus 2 curve, i.e.,
$$
C\!: y^2 + y = x^5.
$$
By the article of J. S. Müller "Explicit Kummer ...
5
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1
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350
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Do singularities of plane curves deform independently?
If $C\subset\mathbb{P}^2$ is an integral curve of degree $d$, do its singularities deform independently as we vary $C$ over degree $d$ curves? If not, what about in the case $C$ is a nodal curve or $C$...
15
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2
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976
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Original reference for Riemann's inequality
Let $D$ be a divisor on a compact Riemann surface of genus $g$. The inequality
$$
l(D)\geq {\textrm {deg}}(D)-g+1
$$
is called Riemann's inequality.
$ \phantom{aaaaaaaa}$In which of Riemann's papers ...
1
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0
answers
104
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Smoothing a map between curves
In the paper Families of Rationally Connected Varieties, the authors consider a map $X\rightarrow B$, where $X$ is obtained by taking disjoint copies of $B$ mapping isomorphically to $B$ and ...
5
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0
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244
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Is the Chow scheme of 1-cycles the space of Cohen-Macaulay curves?
Let $C\subset X$ be a smooth irreducible curve of genus $g$, embedded in a smooth projective 3-fold $X$. So its homology class $\beta=[C]\in H_2(X)$ is an irreducible class. I want to compare two ...
2
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1
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137
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Intersection multiplicity of limit linear spaces
Let $X\subset\mathbb{P}^N$ be a smooth projective variety. Let us fix a general point $q \in X$, and let $C\subseteq X$ be a smooth curve passing through $q$.
Now let $\Lambda_{\xi, q}$, with $\xi \...
5
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1
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605
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Moduli of hyperelliptic curves: odd vs even genus
I'm stumped by Exercise 2.3 in Harris-Morrison, which says:
"Show that there does not exist a universal family of curves of genus 2 over any open subset $U \subset M_2$. In general, if $H_g \subset ...
11
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1
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Is Proposition 2.6 in J. Silverman's book Arithmetic of Elliptic Curves correct?
In J. Silverman's book "Arithmetic of Elliptic Curves" Chapter 2 Proposition 2.6 (a) it is considered a non constant morphism $\Phi:C_{1}→C_{2}$ between two smooth curves defined over a perfect field $...
3
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1
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350
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Are there integer solutions to $3y^2 = 4x^3-1$ other than $(1,1)$ and $(1,-1)$?
Let $x,y \in \mathbb{Z}$ satisfying $3y^2 = 4x^3 - 1$. Does it follow
that $x = 1$ and $y = \pm 1$?
Wolfram Alpha says that the answer is positive, but I am not so satisfied with an answer by a ...
15
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1
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1k
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Is a one-dimensional compact complex analytic space necessarily projective?
Let $X$ be a compact complex analytic space with singular locus $X^{\mathrm{sing}}$. Suppose that $X\setminus X^{\mathrm{sing}}$ is a Riemann surface. If $X^{\mathrm{sing}} = \emptyset$, then $X$ is ...
2
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0
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Is it clear that $y^3=f(x)$ has bad reduction at $3$?
Bad reduction is defined as 'nonexistence' of a model where the curve has good reduction. So let's take the curve $C$ which is affinely given by
$$y^3 = f(x)$$
(absolutely irred, $f$ no multiple roots)...
2
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0
answers
185
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singular points of $\alpha_{p}$-torsor and $\mu_{p}$-torsor of curves
Let $X$ be a smooth projective curve over an algebraically closed field of characteristic $p>0$. Suppose that the genus of $X$ is >2. Let $Y$ be a non-trivial $\alpha_{p}$-torsor or $\mu_{p}$-...
8
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818
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Lattice points near a curve
Bombieri and Pila had a well known bound for the count of lattice points on an algebraic curve in the plane. Does it generalize to a bound for the count of lattice points near (say within a distance ...
3
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1
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244
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Resolution of the ideal of the Abel-Jacobi image of a curve?
Let $C$ be a complex curve of genus $g\ge 2$ and let $a\colon C\to J(C)$ be the Abel-Jacobi map. Is there a finite resolution of the ideal $\mathcal I_{a(C)}$ whose terms are sums line bundles of the ...
2
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1
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185
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Smooth curves in Tangent Developables
Let $C\subset\mathbb{P}^n$ be a smooth curve, and let $Y\subseteq\mathbb{P}^n$ be its tangent developable.
Given two general points $y_1,y_2\in Y$ does there exist a smooth curve $\Gamma\subset Y$ ...
3
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0
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291
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Why define curves over perfect fields?
One may define a curve (e.g. separated scheme of finite type of dim. 1) over an algebraically closed field, as done in Hartshorne's book. A weaker assumption, which is used commonly, is to define a ...
4
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1
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625
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Reference request for general Hurwitz spaces
Let $G$ be a fixed finite group. I'm interested in the structure of the set $\mathcal{H}_{r,g,h,G}$ of tuples $(C,f,\delta)$, where $C$ is a smooth projective genus $g\geq 2$ curve, $\delta:G\to\mbox{...
2
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0
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159
views
Are these moduli problems of curves "well-behaved"?
Let X be a smooth projective surface over $\mathbb C$, and let $d\geq 3$ be an integer. Suppose that all smooth hypersurfaces of degree $d$ are of genus $g\geq 2$.
Let $H_{X,d}$ be the Hilbert scheme ...
1
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0
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425
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Finding the Chern Class of a the pushfoward of a invertible sheaf
I am trying to understand what happens to the Chern Classes of an invertible sheaf $F$ over a complete intersection reduced curve of genus $g$ and degree $d$, when viewed as a invertible sheaf of $\...
1
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2
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215
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Jacobians of twisted coverings
Given a compact Riemann surface $M$ and two double coverings $\hat\pi\colon \hat M\to M$ and $\tilde\pi\colon \tilde M\to M$ which are branched over the same points $p_1,..,p_n\in M.$ As is well-known,...
7
votes
1
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295
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Archimedean fibers "intersecting" curves on arithmetic surfaces
Let's fix a number field $K$ with its ring of integers $O_K$. Moreover consider an arithmetic surface $f:S\to \text{Spec } O_K$. For every archimedean place $\sigma$ in $K$, $K_\sigma$ is the ...
2
votes
0
answers
165
views
Dimension of a sheaf cohomology group on a genus 1 curve
Let $\mathcal{M}_{g,1}$ be the moduli space of genus 1 curves with 1 puncture. For simplicity let's take $g > 1$. As usual, there is a natural fibration $C \rightarrow \mathcal{M}_{g,1} \rightarrow ...
1
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0
answers
87
views
Calculation of cardinality of Jacobians
The problem of calculation the number of rational points on curves over finite fields is $\#P$-complete - "Counting curves and their projections".
Is it true for calculation of number of rational ...
2
votes
1
answer
180
views
Does $\omega_C\simeq N_{C/S}$ always happen on Enriques surfaces?
Let $S$ be an Enriques surface and $C\subset S$ a smooth irreducible curve of genus $g$.
Consider the condition $$\omega_C\simeq N_{C/S}$$
For example, when $g=1$ then $\omega_C=\mathcal{O}_C$ and ...
7
votes
2
answers
488
views
Are ranks of Jacobians over number fields unbounded?
Fix a number field $K$.
Is the rank of $J(K)$ unbounded, where $J$ ranges over the Jacobians of all smooth, projective, geometrically connected curves over $K$?
Does there exist an integer $g$ such ...