An algebraic curve is an algebraic variety of dimension one. An affine algebraic curve can be described as the zero-locus of $n-1$ independent polynomials of $n$ variables in affine $n$-space over a field. Examples include conic sections, compact Riemann surfaces and elliptic curves. Singularities ...

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9
votes
2answers
3k views

How does one calculate genus of an algebraic curve?

I've been reading about parametrization of algebraic curves recently and the idea of the "genus of a curve" appears quite often (my impression is that a curve is parametrizable exactly when it has ...
5
votes
1answer
483 views

algebraic curves with negative (arithmetic) genus?

By an algebraic curve I mean a projective reduced connected scheme of pure dimension 1 over a field. My question is: Is there a lower bound for the arithmetic genus of such curves? If the answer is ...
7
votes
1answer
202 views

Does every curve over a number field have infinitely many rational functions of fixed degree

Let $X$ be a curve over a number field $K$ of genus $g\geq 2$. Does there exist an integer $d$ such that $X$ has infinitely many rational functions (i.e., finite morphisms $f:X\to \mathbf{P}^1_K$) of ...
3
votes
0answers
94 views

Is this a map $X \to \text{Sym}^n Y$?

Let $X$ and $Y$ be smooth projective curves over an algebraically closed field $k$ and let $D$ be a divisor on $X \times Y$ meeting every vertical fiber ($X$ is the horizontal axis). Let's assume that ...
1
vote
1answer
114 views

Proving the existence of a particular linear subvariety

I'm trying to prove that if $V$ is a non-empty linear subvariety then there is an affine change of coordinates $T$ of $ \Bbb A^n $ such that $V^T = V(X_{m+1}, \ldots, X_n) $. A set V in $ \Bbb A^n(k) ...
13
votes
5answers
1k views

Trefoil knot as an algebraic curve

Is the trefoil knot with its usual embedding into affine $3$-space an algebraic curve (maybe after extending scalars to $\mathbb{C}$)? Is there even some thickening to some algebraic surface? If ...
3
votes
2answers
703 views

Canonical divisor on the symmetric product of a hyperelliptic curve

Let $C$ be a hyperelliptic curve of genus $g$ and let $S = C^{(2)}$ denote the symmetric square of $C$. Let $\nabla$ be the divisor on $C^2$ defined by $\{(P, \overline{P}) \mid P \in C\}$ where ...
5
votes
1answer
117 views

projective cubic

I have some difficulties to prove that the image of the function $f:\,\mathbb P^1\longrightarrow\mathbb P^3$ such that $$(u,v)\longmapsto (u^3,\,u^2v,\,uv^2,\,v^3)$$ is the algebraic projective set ...
4
votes
1answer
86 views

The universal cover of the multiplicative group over the field of algebraic numbers

Let $X=\mathbf{A}^1_{\overline{\mathbf{Q}}}-\{0\} = \mathbf{G}_{m,\overline{\mathbf{Q}}}$ be the multiplicative over the field of algebraic numbers. Each finite etale cover $Y\to X$ (with $Y$ ...
1
vote
1answer
379 views

Family of curves (in algebraic geometry)

How can I view, in algebraic geometry, a family of curves over a base curve? For instance, can the family $y^2 = x(x-1)(x-t)$ be viewed as a family over $\mathbb{P}^1$ ? How can I understand this ...
1
vote
1answer
93 views

Is the study of algebraic curve is techniquely equal to the advanced division of analytic geometry, if not, what is the difference?

Is the study of algebraic curve is techniquely equal to the advanced division of analytic geometry, if not, what is the difference? And what is other branch of advanced analytic geometry called? in ...
1
vote
1answer
67 views

Defining invariants of varieties over fields

Let $f$ be a real-valued function on the set of curves over $\overline{\mathbf{Q}}$. Assume that isomorphism curves give rise to the same value for $f$. Let $K$ be a number field and let $X$ be a ...
1
vote
1answer
126 views

Gaps in the Genera of Space Curves

We learned the following relationship between the degree and genus of plane curves in my algebraic geometry course: \begin{array} a \text{degree} &d &1 &2 &3 &4 &5 &6 ...
8
votes
0answers
526 views

A problem about the twisted cubic

I have some difficulty with the following problem: Let $f : k → k^3$ be the map which associates $(t, t^2, t^3)$ to $t$ and let $C$ be the image of $f$ (the twisted cubic). Show that $C$ is an ...
4
votes
1answer
350 views

Ramification on hyperelliptic curves

I am using Rick Miranda's book "Algebraic curves and Riemann Surfaces" to try and check some things about hyperelliptic curves. I have completed almost all of one of the exercises, but there is one ...
4
votes
0answers
67 views

Automorphism of $L|K$ mapping 3 distinct rational points of $S_{L|K}$ to other 3 distinct ones

Let $K$ be a field and consider $L = K(x)$ the field of rational functions. Let $v_{1}, v_{2}, v_{3}$ rational points in the abstract Riemann surface $S_{L|K}$, distinct from each other, and $w_{1}, ...
1
vote
0answers
285 views

Artin-Schreier extensions over characteristic two fields

I have been looking at hyperelliptic curves over an algebraically closed field $k$ of characteristic two, with a view towards finding the basis for the vector space of holomorphic differentials. To do ...
3
votes
1answer
82 views

Twists of rational points

Let $X$ be a "nice" algebraic curve over some field $K$, say characteristic zero. The twists of $X$, i.e., the curves $Y$ over $K$ such that $X_{\overline K} \cong Y_{\overline K}$, are in bijection ...
4
votes
1answer
210 views

Jacobian of a curve

Let $C$ be a curve and $J$ be its Jacobian. What is the relation between $H^1(C,\mathcal{O}_C)$ and $H^1(J,\mathcal{O}_J)$ ? Can someone point me to an easy reference for this subject?
1
vote
0answers
102 views

Rational curve cover/Transcendental Galois field extension

Suppose the rational curve $C$ is a finite cover for the rational curve $D$ and the field of rational functions of $C$ is the purely transcendental extension $k(x)$ and that of $D$ is the subfield ...
2
votes
1answer
316 views

Is there a fundamental domain for $\Gamma(2)$ contained in the following strip

Let $\Gamma(2)$ be the subgroup of $\mathrm{SL}_2(\mathbf{Z})$ satisfying the usual congruence conditions. It acts on the complex upper half-plane. Does it have a fundamental domain contained in the ...
1
vote
1answer
133 views

Can a non-proper variety contain a proper curve

Let $f:X\to S$ be a finite type, separated but non-proper morphism of schemes. Can there be a projective curve $g:C\to S$ and a closed immersion $C\to X$ over $S$? Just to be clear: A projective ...
0
votes
1answer
137 views

Are elliptic curves also Galois covers of degree 3

Let E be an elliptic curve with equation $y^2=x^3+Ax+B$. The projection onto the $x$-coordinate is a Galois morphism of degree $2$. But what about the projection onto the $y$-coordinate? Is it ...
6
votes
1answer
269 views

Why is Klein's quartic curve not hyperelliptic

Let $X$ be Klein's quartic curve given by $x^3y + y^3z+z^3x=0$ in $\mathbf{P}^2$. It is isomorphic to $X(7)$. How do I easily show that $X$ is not hyperelliptic? I can see that $X$ is of genus $3$ ...
2
votes
1answer
456 views

Parameterizing a rational curve

I'm having trouble finding a parameterization for the following curve: $x^4 - 2x^2yz + y^2z^2 - y^3z = 0$ taken to be a curve in $\mathbb{C}\mathbb{P}^2$. I followed the example on Wikipedia where ...
7
votes
1answer
195 views

Is the Fermat scheme $x^p+y^p=z^p$ always normal

Let $K$ be a number field with ring of integers $O_K$. Is the closed subscheme $X$ of $\mathbf{P}^2_{O_K}$ given by the homogeneous equation $x^p+y^p=z^p$ normal? I know that this is true if ...
3
votes
1answer
151 views

self-intersections in a product of two curves

Let $X$ be a smooth projective curve of genus $g$ over an algebraically closed field and consider the intersection pairing on the surface $X \times X$. I remember hearing that $\Delta^2 = 2-2g$: how ...
2
votes
1answer
100 views

Residue map of a place

The following definition for a residue map is given in "Algebraic Curves over a Finite Field" by Hirschfeld, Korchmáros and Torres (page 265): "Let $\Sigma$ be a field of transcendence degree 1 over ...
3
votes
1answer
137 views

A field isomorphism related to polynomial rings and their field of fractions

There are 2 ways to approach function fields: the algebraic approach, i.e. looking at finite extensions of $K(s)$, where $s$ is transcendental. The other is geometric, i.e. considering functions over ...
0
votes
0answers
368 views

What is a global section, and why do we use cohomology?

I have a few questions about the use of cohomology. Firstly,we use cohomology to measure the obstruction of a section from a global section, so what can we do about a global section? I got very ...
3
votes
1answer
132 views

Can an algebraic group only have trivial elements over $k$

Let $G$ be an algebraic group over $k$ such that $G(k) = \{e\}$ is the trivial group. Does this imply that $G_{\overline{k}}$ is trivial? I think the answer is no. I think you can just take an ...
2
votes
0answers
136 views

Intersection of a cone $x^2+y^2-z^2$ and a generic plane in $\mathbb{RP}^3$

If we take the zero locus of $x^2+y^2-z^2$ to be our cone, I'd like to know how to go about finding the intersection of the cone and a generic plane $Ax+By+Cz+Dw=0$. The result will be a conic, but ...
1
vote
0answers
298 views

What is the equation for a cone in $\mathbb{RP}^3$?

The zero locus of $x^2+y^2-z^2$ is a cone in $\mathbb{R}^3$. What is the projective version of this cone? That is, what is the homogeneous polynomial whose zero locus is a cone in $\mathbb{RP}^3$? ...
3
votes
0answers
64 views

What applications does the theory of fibered surfaces have

Let $C$ be a smooth projective connected curve over $\mathbf{C}$. Let $X$ be a curve over the function field of $C$. Arakelov and Parshin proved the Mordell conjecture by considering a model for $X$ ...
5
votes
0answers
79 views

Can we make topological covers of $\mathbf{P}^1$ minus three points into schemes

Let $k=\overline{\mathbf{Q}}$. Fix a finite closed subset $B\subset \mathbf{P}^1_k$. Let $X$ be a "nice" topological space and suppose that there is a continuous morphism $f:X\to \mathbf{P}^1_k-B$. ...
6
votes
1answer
354 views

Intersection number & non-singular curves

I was looking at a proof of the following theorem... Let $S/\mathbb{C}$ be a smooth projective surface, let $C$ be a non-singular irreducible curve on $S$. Then for all $L \in \text{Pic}\,S$, the ...
2
votes
0answers
79 views

Is the degree of a Galois morphism bounded by $84(g-1)$

Let $X\to Y$ be a finite morphism which is Galois in the category of smooth projective connected curves over $\mathbf{C}$. Assume $g=g(X) \geq 2$. Is the degree of $X\to Y$ bounded by $84(g-1)$? I ...
0
votes
1answer
150 views

Does the absolute Galois group act on the moduli space of curves

Do elements of the absolute Galois group $G=\mathrm{Gal}(\overline{\mathbf{Q}}/\mathbf{Q})$ of $\mathbf{Q}$ induce automorphisms of $\mathbf{C}$ if we extend the morphisms trivially to the rest of ...
3
votes
1answer
105 views

For curves, is being defined over a number field invariant under birational equivalence

Suppose that a (connected) Riemann surface $X$ is birational to a Riemann surface $Y$ which can be defined (algebraically) over the field of algebraic numbers. Does this imply that $X$ itself can be ...
3
votes
2answers
176 views

Does there exist a finite morphism of algebraic curves such that…

Let $K\subset L$ be a finite field extension. Let $X$ and $Y$ be (smooth projective geometrically connected) curves over $L$. Let $f:X\to Y$ be a finite morphism of curves over $L$. Assume that ...
3
votes
2answers
246 views

Are fundamental groups of Riemann surfaces always finitely generated

For any finite subset $B\subset \mathbf{P}^1$, the fundamental group of the Riemann surface $\mathbf{P}^1-B$ is finitely generated. Is this true if we replace $\mathbf P^1 $ by a higher genus compact ...
4
votes
1answer
126 views

Ramification of a prime in a Dedekind ring and curves

Let $\phi:C_1\to C_2$ be a nonconstant map of two smooth curves over some algebraically closed field $K$ and let $P\in C_1$. $\phi$ gives us an induced map of fields $\phi^*:K(C_2)\to K(C_1)$, ...
5
votes
1answer
186 views

Relationship between two distinct notions for divisors on curves

I've seen divisors on curves before, a few years ago in a course in algebraic geometry. Now I've come across them again, but they're somewhat more generalized. I was hoping someone could explain the ...
3
votes
1answer
85 views

Do gonal morphisms have non-trivial automorphisms

Let $X$ be a compact connected Riemann surface. Let $\pi:X\to \mathbf{P}^1$ be a gonal morphism, i.e., a morphism of minimal degree. Can $\pi$ have non-trivial automorphisms? (An automorphism of ...
1
vote
1answer
171 views

Galois covers of Riemann surfaces

Let $G$ be a finite abelian group, and $C$ a compact Riemann surface (algebraic curve) of genus $g$. I am interested in topological Galois $G$-covers $X \to C$, aka \'etale $G$-principal bundles over ...
11
votes
3answers
461 views

Is a cover Galois if and only if it is geometrically Galois

Let $K$ be a number field and let $\pi:X\to \mathbf{P}^1_K$ be a finite morphism, where $X$ is a smooth projective geometrically connected curve. Is $\pi$ a Galois cover if and only if the base ...
2
votes
1answer
71 views

Covers without automorphisms

Let $X\to \mathbf{P}^1$ be a branched cover of the complex projective line, where $X$ is a compact connected Riemann surface. Let $G=\mathrm{Aut}(Y/\mathbf{P}^1)$. Question 1. Could somebody provide ...
2
votes
1answer
67 views

What is the length of the following local ring

Let $f:Y\to X$ be a finite etale cover of smooth projective connected varieties. (Or, just a finite degree connected topological cover of connected Riemann surfaces.) Let $y\in Y$ and let $x=f(y)$. ...
2
votes
0answers
53 views

Why does a non-constant endomorphism of curves have isolated fixed points

Let $f:X\to Y$ be a non-constant morphism of smooth projective connected curves over $\mathbf{C}$ (or compact connected Riemann surfaces). Suppose that $X=Y$ and that $f$ is not the identity. Why ...
2
votes
1answer
117 views

Is the intersection of the diagonal with a graph always transverse in characteristic zero

Let X be a projective smooth connected curve over $\mathbf{C}$. Let $f:X\to X$ be a non-constant morphism. Is the intersection of the diagonal $\Delta_X$ and the graph $\Gamma_f$ on $X\times X$ ...