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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1 vote
1 answer
615 views

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 vote
0 answers
626 views

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 votes
2 answers
516 views

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 vote
1 answer
201 views

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 vote
0 answers
142 views

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 votes
0 answers
342 views

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 answer
805 views

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 vote
0 answers
103 views

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 votes
1 answer
247 views

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 votes
2 answers
5k views

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 ...
9 votes
3 answers
1k views

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 votes
1 answer
891 views

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 votes
1 answer
178 views

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 votes
2 answers
341 views

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 votes
2 answers
770 views

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 votes
0 answers
251 views

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 votes
1 answer
98 views

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 answer
382 views

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 votes
2 answers
704 views

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 votes
0 answers
374 views

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 vote
0 answers
207 views

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 votes
0 answers
157 views

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 votes
0 answers
108 views

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 votes
0 answers
262 views

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 votes
2 answers
317 views

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 votes
0 answers
137 views

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 votes
1 answer
350 views

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 votes
2 answers
976 views

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 vote
0 answers
104 views

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 votes
0 answers
244 views

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 votes
1 answer
137 views

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 votes
1 answer
605 views

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 votes
1 answer
1k views

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 votes
1 answer
350 views

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 votes
1 answer
1k views

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 votes
0 answers
207 views

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 votes
0 answers
185 views

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 votes
1 answer
818 views

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 votes
1 answer
244 views

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 votes
1 answer
185 views

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 votes
0 answers
291 views

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 votes
1 answer
625 views

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 votes
0 answers
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 vote
0 answers
425 views

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 vote
2 answers
215 views

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 answer
295 views

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 vote
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 ...

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