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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3
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2answers
481 views

When to read of the degree of a variety from its defining polynomials

The question concerns algebraic varieties. I just read the question The degree of an algebraic curve in higher dimensions and great answer by user M P. One of the thing he says is that if a curve in ...
1
vote
0answers
120 views

Dimension of Secant variety of an irreducible projective curve not contained in a plane

I'm trying to show that this dimension is three, but I'm stuck. Could anyone give me a hint?
2
votes
2answers
470 views

Characterization of Rational Normal Curve

Given a curve $C\subset P^n$ in projective space such that any $n+1$ points on $C$ are linearly independent. I've heard from multiple people that this implies that $C$ is a rational normal curve, some ...
8
votes
0answers
151 views

Tropical-like redefinitions of addition and multiplication?

I've been impressed at the rich structure of tropical mathematics, all consequences of the seemingly mundane starting point of replacing classical $a+b$ by $\min(a,b)$ (or max), and replacing ...
4
votes
3answers
176 views

How do I write down a curve with exactly one rational point

Let $g\geq 1$. I would like to write down (for all $g$) a smooth projective geometrically connected curve $X$ over $\mathbf{Q}$ of genus $g$ with precisely one rational point. Is this possible? For ...
4
votes
2answers
246 views

Genus of curves embedded into some projective space

The Plucker formula allows one to calculate the genus of a curve embedded into $\mathbf{P}^2$. Does there exist a "higher-dimensional" analogue of the Plucker formula? More precisely, let ...
2
votes
2answers
227 views

Rational points on singular curves and their normalization

Let $X$ be a curve over a field $k$. Assume that $X$ is geometrically connected, geometrically reduced and stable. Let $Y\to X$ be the normalization. Is $Y(k) = X(k)$?
3
votes
2answers
227 views

genus of normalization of stable curve

Let $X$ be a stable curve of genus $g$ over the field $k$, i.e., a $k$-rational point of the Deligne-Mumford stack $M_g$. What is the genus of the normalization of $X$? Does it depend on the number ...
15
votes
1answer
700 views

How to properly use GAGA correspondence

currently studying algebraic surfaces over the complex numbers. Before i did some algebraic geometry (I,II,start of III of Hartshorne) and a course on Riemann surfaces. Now i understood that by GAGA, ...
9
votes
2answers
557 views

Elementary proof of the Hurwitz formula

I am aware of two forms of the Hurwitz formula. The first is more common, and deals only with the degrees. So if $f:X \rightarrow Y$ is a non-constant map of degree $n$ between two projective ...
0
votes
2answers
157 views

Why are these curves not defined over a smaller field

Let $K$ be a number field and let $\pi$ be an element in $K$. Assume that $\pi$ is not contained in a subfield of $K$. Consider the curve $y^2 = x^{2g+1}+\pi$. This defines (after homogenization and ...
6
votes
1answer
194 views

Examples of stable curves $g\geq 2$?

I'd like to get my hands on some simple examples of families of stable curves. Ideally these would come in the form of a projective curve $C$ over a 1 dimensional base $B$, say $B = \mathbb{A}^1$. ...
3
votes
1answer
84 views

When is this quotient by an action on the product of a variety with itself non-singular

Let $X$ be a smooth projective geometrically connected variety over a field $k$. Let the cyclic group $G=\{e,a\}$ with two elements act on $X \times X$ via $a\cdot (x_1,x_2) = (x_2,x_1)$. When is ...
3
votes
3answers
112 views

Show that if the curve $y^2 = p(x)$ has a double point, then it must be of the form $(r,0)$ where $r$ is a double root of $p(x)$.

Let $p(x) = ax^3 + bx^2 + cx + d$ where $a,b,c,d \in\mathbb{R}$. Show that if the curve $y^2 = p(x)$ has a double point, then it must be of the form $(r,0)$ where $r$ is a double root of $p(x)$. ...
10
votes
3answers
259 views

Is every algebraic curve birational to a planar curve

Let $X$ be an algebraic curve over an algebraically closed field $k$. Does there exist a polynomial $f\in k[x,y]$ such that $X$ is birational to the curve $\{f(x,y)=0\}$? I think I can prove this ...
3
votes
1answer
311 views

Two notions of uniformizer

Let $X$ be a projective algebraic curve and consider a `uniformizing' map $h:X \rightarrow \mathbb{P}^1$. Is there any connection between this notion of uniformizer and a uniformizer of the maximal ...
2
votes
1answer
237 views

Hyper-elliptic curves in positive characteristic

I have been looking at hyperelliptic curves in characteristic two, in particular using Algebraic Geometry and Arithmetic Curves by Qing Liu, which gives a description in all characteristics. For the ...
0
votes
1answer
172 views

Smallest genus example of a non planar curve

A curve is a smooth projective connected curve over an algebraically closed field. Every curve of genus 2 is planar. Also, every curve of genus 3 is planar. But what about curves of genus 4? What ...
6
votes
4answers
266 views

Irreducible polynomial not attaining squares over finite field

Is it possible to construct an irreducible polynomial $f$ over $\mathbb{F}_{q}$ such that $f(x)$ is a non-square for any $x \in \mathbb{F}_{q}$? I can prove the existence of irreducible polynomials ...
2
votes
1answer
166 views

Divisor of degree 2 on a smooth plane curve

Let $X$ be a smooth plane curve of genus $3$ (assume a smooth plane quartic) and $D$ a divisor of degree $2$ on this curve. Assume that $\mathcal{l}(D)>0$. It means that there exists a rational ...
4
votes
2answers
164 views

zeroes of forms on Riemann surfaces

Let $P$ be a point on a Riemann surface. Does there exist a non-trivial differential form $\omega$ on $X$ such that $\omega$ vanishes at $P$? Does there exist a non-constant rational function $f$ on ...
3
votes
1answer
104 views

Homogenous polynomials

In section 3.1 (3rd paragraph on page 4) in this paper, I cannot understand why $Q$ and $R$ are homogeneous: (Given $A$, $B$ are homogenous. Capital letters denote homogenous polynmials.)
3
votes
1answer
265 views

A theorem in Fulton's Algebraic Curves

In 5.23., Fulton proves the theorem that if $P$ is an ordinary flex of a plane curve $C=V(F)$, then $C$ and it's Hessian $H$ intersect with multiplicity one, that is $I(P, C \cap H) = 1$. After a bit ...
1
vote
1answer
274 views

Affine change of coordinates

Let $P,U$ be points in $K^2$ ($K$ is a field). Let $L(1)$, $L(2)$ be two lines through $P$, and $L(3)$, $L(4)$ be two distinct lines through $U$. How to show that there is an affine change of ...
8
votes
1answer
239 views

Tangent sheaf of a (specific) nodal curve

Given a nodal (= reduced, connected, projective, having only ordinary double points as singularities) curve $C$ consisting of 5 $\mathbb P^1$ labeled $C_0, D_1, D_2, D_3, D_4$ such that $D_i$ ...
1
vote
0answers
98 views

Deformations preserving dual graph

Let $C$ be a nodal curve over an algebraically closed field $k$. A deformation of $C$ over an Artinian ring $A$ over $k$ consists of a flat scheme $C'$ over $A$ and a closed immersion $i: C \to C'$ ...
5
votes
1answer
230 views

A question about modular curves and base change

Let $X$ be a smooth projective geometrically connected curve over a number field $K$. Suppose that the curve $X\times_{K,\sigma} \mathbf{C}$ is a modular curve for some $\sigma:K\to \mathbf{C}$. Can ...
7
votes
0answers
232 views

Do Neron models of hyperbolic curves exist

Let $X$ be a smooth projective geometrically connected curve over a number field $K$. Assume that $g\geq 2$. Does there exist a Neron model $\mathcal X$ for $X$ over $O_K$? By a Neron model, I mean ...
2
votes
1answer
326 views

Showing $\sin(1/x)$ is not a rectifiable curve

Intuitively it looks like near $0$, $\sin(1/x)$ oscillates wildly so that two points will be very far apart, but how can I properly formulate this?
5
votes
1answer
133 views

Very special rational points on curves over number fields

For some reason, I'm convinced the answer to the following question should be (obviously) negative, but I can't come up with a good reason. Does there exist a number field $K$, a smooth projective ...
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
503 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
203 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
115 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
711 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
87 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
401 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
128 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
542 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
361 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
289 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
213 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
103 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 ...