An algebraic curve or plane algebraic curve is the set of points on the Euclidean plane whose coordinates are zeros of some polynomial in two variables.

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2
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119 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 ...
3
votes
1answer
81 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
0answers
162 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 ...
2
votes
1answer
98 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{...
20
votes
3answers
801 views

A historical question: Hurwitz, Luroth, Clebsch, and the connectedness of $\mathcal{M}_g$

The connectedness of the moduli space $\mathcal{M}_g$ of complex algebraic curves of genus $g$ can be proven by showing that it is dominated by a Hurwitz space of simply branched d-fold covers of the ...
7
votes
1answer
395 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 ...
2
votes
0answers
129 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 ...
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vote
0answers
104 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
2answers
158 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
1answer
212 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
0answers
96 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
0answers
64 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
1answer
149 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
2answers
380 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 ...
2
votes
1answer
95 views

On some curves of real values of a rational function

For given parameters $a_{1},\dots,a_{k}\in\mathbb{R}$, define the rational function $\phi:\mathbb{C}\to\mathbb{C}$ as $$\phi(z)=\frac{1}{z}-a_{1}z-a_{2}z^{2}-\dots-a_{k}z^{k}.$$ The domain of its real ...
0
votes
0answers
37 views

Field of moduli relative to a different extension

Let $X$ be a Riemann surface of genus $g\geq 2$ defined over $\mathbb{Q}(\sqrt{2})$. Let us consider the automorphism of $\mathbb{Q}(\sqrt{2})$, $$ \tau:\mathbb{Q}(\sqrt{2})\rightarrow\mathbb{Q}(\sqrt{...
5
votes
0answers
100 views

Is this method of finding a “dual curve” correct?

I have a very limited exposure to projective geometry, but I'm having fun exploring the concept of duality. In particular I'd like to know if this naive method of finding a "dual" curve to a given ...
4
votes
1answer
586 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 $(K)$ ...
4
votes
1answer
279 views

Shafarevich conjecture for abelian varieties

In the paper "Arakelov's theorem for abelian varieties" Faltings proves the Shafarevich conjecture for abelian varieties. The statement is the following: Let B be smooth projective a curve, S a ...
8
votes
2answers
344 views

The class of the diagonal in the symmetric product of a smooth curve

Let $C$ be a smooth curve of genus $g$, and let us consider its $d$-th symmetric product $\textrm{Sym}^d(C)$ and its Jacobian $J(C)$. Fixing a point $p_0 \in C,$ there are two maps $$u_d\colon C_d \to ...
6
votes
1answer
195 views

Very stable vector bundles

Let $X$ be a smooth curve, and $E$ a rank $r$ vector bundle over $X$, $E$ is said very stable if every nilpotent map $$u:E\rightarrow E\otimes K_X$$ is zero (nilpotent means that the composition $E\...
4
votes
1answer
208 views

Stable vector bundles in Weil's parametrization

Let $C$ be a smooth projective algebraic curve. Isomorphism classes of vector bundles on $C$ are in bijection with $GL_n(F) \backslash GL_n (\mathbb{A}) / GL_n(\mathcal{O})$, I think (one trivalizes ...
3
votes
2answers
489 views

Recognize this plane curve?

An aspect of my work led to a plane curve with implicit equation $$ x^2+y^2 = 3 (y/2)^{2/3} + 1 $$ Actually, I started with the parametrization below and derived from it the equation above: \begin{...
3
votes
1answer
205 views

Polars of algebraic curves and surfaces

I asked this on Math.StackExchange, but received no response, so trying here ... A paper I'm reading says the following ... With homogeneous coordinates $\mathbf{x} = [x,y,z,w]$, let $F(\mathbf{x}...
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vote
0answers
218 views

About complete residues on curves

Preliminaries: Let $X$ be a projective smooth curve (scheme of finite type, integral and of dimension $1$) over a perfect field $F$. Let $K=K(X)$ be the function field of $X$ and for a closed point $...
3
votes
1answer
148 views

Identifying the canonical principal polarization of a Jacobian

Let $X$ be a curve over an algebraically closed field $k$ (even over $k = \mathbb{C}$ if you want), let $J = Pic^0_{X/k}$ be its Jacobian, let $P \in X(k)$ be a point, and let $i \colon X \...
2
votes
0answers
94 views

Singularities of algebraic curves, and torsion of the pull-back of the differential module by the normalisation

The problem in the following : given an algebraic curve $C$, it's well-known that a smooth projective model of $C$ can be construct as the set of discrete valuations $v$ on it's function field $\...
3
votes
1answer
340 views

Proof of the Belyi's theorem: where is the hypothesis really used?

Consider the Belyi's theorem: If a smooth projective curve $X$ is defined over $\overline{\mathbb Q}$, then there exists a finite morphism $X\longrightarrow\mathbb P^1(\mathbb C)$ with at most $3$...
11
votes
3answers
494 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.
5
votes
1answer
109 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 ...
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vote
0answers
88 views

Fiber of the specialization map of Picard groups

Let $R$ be a Henselian discrete valuation ring with residue field $k$ of positive characteristic and fraction field $K$ of characteristic zero. Let $\pi:X_R \to \mathrm{Spec}(R)$ be flat, projective ...
3
votes
0answers
128 views

Does $C(k)$ necessarily contain a smooth point? [closed]

If $k$ is an infinite perfect field and if $f \in k[x, y]$ is nonconstant irreducible, cutting out the affine plane curve $C$, then does $C(k)$ necessarily contain a smooth point?
6
votes
1answer
242 views

Quotients of curves of genus $4$ by a free $\mathbb{Z}/ 3 \mathbb{Z}$-action

Let $V_2$ and $V_3$ be the two hypersurfaces of $\mathbb P^3$ defined by \begin{equation*} V_2:={x_2x_3 + r(x_0, \, x_1)=0}, \quad V_3:={x_2^3+x_3^3+s(x_0, \, x_1)=0}, \end{equation*} where $r, \, s \...
0
votes
0answers
89 views

Quadrics cutting out a polygon

Let $l_1,l_2,l_3,l_4,l_5\subset \mathbb P^4_k$ be distinct lines such that $|l_i\cap l_{i+1}|=1$ for all $i\ mod\ 5$ and $l_i\cap l_j\neq \emptyset\iff j=i+1\ mod\ 5$ (so that $\cup_{i=1}^5l_i$ is a ...
31
votes
3answers
1k views

Polynomials with the same values set on the unit circle

Assume that $P(z)$, $Q(z)$ are complex polynomials such that $P(S)=Q(S)$, where $S=\{z\colon |z|=1\}$ (equality is understood in the sense of sets, but I do not know the answer even for multisets). ...
2
votes
1answer
199 views

Given a curve $C$, does there exist a rational function on $C$ totally ramified at two given points?

Let $C$ be a smooth projective irreducible curve over $\mathbb C$. Let $x$ and $y$ be distinct points of $C$. We say that $f$ is totally ramified at a point $p$ if the ramification index of $p$ ...
3
votes
1answer
174 views

Is there a covering of Prym variety?

$\mathstrut$Hi, guys! Let $C$, $C^\prime$ be projective smooth irreducible algebraic curves over an algebraically closed field $k$ ($\mathrm{char}(k) \neq 2$), $\phi : C$ $\to$ $C^\prime$ a two-...
4
votes
0answers
169 views

Cohomology and deformations of moduli of vector bundles

I believe that the following is well-known, but I cannot find a reference in the literature... Let $X$ be a smooth variety (in our case $X = M^s(r)$ coarse moduli space of stable rank $r$ vector ...
0
votes
0answers
167 views

Explicit form of certain polynomials and intersection of curves

Let $X$ be a smooth degree $d$ hypersurface in $\mathbb{P}^3$ and $C, D$ two effective divisors on $X$ intersecting at finitely many points. Is it true that if $C$ and $D$ intersect in ''low'' number ...
2
votes
2answers
374 views

Rational points on towers of curves

Let $\ldots \to X_n \to X_{n-1} \to \ldots \to X_0$ be etale maps between smooth projective curves of genera $g(X_n)>1$, all defined over a fixed number field $K$. By Faltings' Theorem, we know ...
10
votes
3answers
477 views

Quotient of a smooth curve by a finite group and differentials

Let $X$ be a proper smooth connected curve over an algebraically closed field $k$ of characteristic $0$, and suppose that $X$ is equipped with a $k$-linear action of a finite group $G$. It makes sense ...
3
votes
1answer
221 views

Existence of pencils on some special curves of genus 10

Everything over $\Bbb{C}$. Say we have a smooth curve $C$ of genus $10$ which is a double cover of a smooth plane cubic curve. Therefore $C$ admits a 1-dimensional family of pencils of degree 4 (...
3
votes
2answers
181 views

Are curves with maximal Clifford index Brill-Noether general?

By the Brill-Noether Theorem, a general curve $C$ of genus $g\geq2$ has maximal Clifford index $\lfloor \frac{g-1}{2}\rfloor$. Hence a very naive question is: (Q1) Is a curve with maximal Clifford ...
5
votes
2answers
303 views

The gonality of smooth plane curves

I have often seen the assertion that for a smooth plane curve $C$ of degree $d$ the gonality of $C$ is $d-1$ and each gonality pencil is obtained by projection from a point of $C$ onto a line. (let ...
0
votes
1answer
162 views

Rational maps between elliptic curves [closed]

I am studying Silverman's "The Arithmetic of Elliptic Curves" and I got the following question: In the first chapters he defines rational between projective varieties (see the first definition in I.3)...
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vote
0answers
66 views

Cubic, divisor of rational function $x/z$? [closed]

Let $k$ be a field, and let $a \neq 0$, $1 \in k$. Let $C = V(y^2z - x(x-z)(x - az))$. What is the divisor of the rational function $\psi([x, y, z]) = x/z \in k(C)$?
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votes
0answers
117 views

How to compute the arithmetic genus of a nonreduced curve?

Let $X$ be a nonreduced curve over an algebraically closed field $k$. Suppose that the reduced scheme $X^{\rm red}$ associated to $X$ is a smooth projective curve with genus $g$, and $J \subset O_{X}$ ...
1
vote
1answer
298 views

Negative degree line bundles over a singular projective curve have no sections?

Let $C$ be a local complete intersection projective curve in $\mathbb{P}^3$. Assume that $C$ is integral. Let $\mathcal{L}$ be a line bundle on $C$ of negative degree. We know that if $C$ is smooth ...
10
votes
3answers
580 views

what is the maximum number of rational points of a curve of genus 2 over the rationals

Conjecturally, there exists an integer $n$ such that the number of rational points of a genus $2$ curve over $\mathbf{Q}$ is at most $n$. (This follows from the Bombieri-Lang conjecture.) We are ...
4
votes
1answer
164 views

Non canonical singularities of moduli spaces of curves

Is it true that for any $g\geq 1$ and $n$ such that $\overline{M}_{g,n}$ has dimension at least two the locus in $\overline{M}_{g,n}$ parametrizing reducible curves which are union of an elliptic ...