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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1answer
81 views

Does de Franchis' theorem hold over any base field

Let $k$ be a field and let $X$ be a hyperbolic curve over $k$. Then, there are only finitely many hyperbolic curves $Y$ over $k$ dominated by $X$. I know this statement holds over $k=\mathbf{C}$. In ...
0
votes
1answer
125 views

Representing a curve as a plane curve in different ways

Let $X$ be a (smooth projective geometrically connected) curve over $k$, where $k$ is a field of characteristic zero. We assume $X$ has genus at least $2$. I know that $X$ has a plane model. More ...
7
votes
1answer
148 views

The number of curves of given genus over a field

Let $k$ be a field. Let $g\geq 0$ be an integer. I have an elementary question. Let $N$ be the "number" of $k$-isomorphism classes of smooth projective geometrically connected curves over $k$ of ...
3
votes
1answer
202 views

the elliptic curves with j-invariant zero

Let $B\in K^\ast$, where $K$ is a number field. Let $y^2=X^3+B$ be the Weierstrass equation for an elliptic curve $E_B$ over $K$. Note that the $j$-invariant of $E$ is zero. When is $E_B$ ...
3
votes
1answer
116 views

How can function fields have different degrees over the projective line

I'm confused. Let $X$ be a curve over a field $k$. Let $K= K(X)$ be its function field. Then, $K(X)$ is a field. Each non-constant morphism $f:X\to \mathbf{P}^1_k$ gives a field extension ...
2
votes
1answer
259 views

writing down the minimal discriminant of an elliptic curve

Let $j$ be an integer. Does there exist an elliptic curve $E_j$ over $\mathbf{Q}$ with $j$-invariant equal to $j$ whose minimal discriminant we can write down in a practical way? For example, can ...
5
votes
1answer
50 views

Testing local freeness on curves

Let $X$ be a smooth variety (over an algebraically closed field, if it makes a difference), and $\mathscr{F}$ a coherent sheaf on $X$. I have heard it claimed that $\mathscr{F}$ is locally free if and ...
5
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1answer
201 views

Intersection number - difficulties with an example calculation

I recently learned about intersection multiplicity and tried to calculate a simple example. Unfortunately, I am having difficulty. Consider the two curves in $\mathbb{C}^2$ given by $y=0$ and ...
5
votes
2answers
177 views

Equivalence of definitions of intersection number

Consider two varieties over an algebraically closed field $k$ given by $f=0$ and $g=0$ that intersect at some point (without loss of generality $(0,0)$) with no common components, and $f$ and $g$ have ...
0
votes
1answer
150 views

Build equation of a curve with set of coordinates

I need to calculate the intersection of two curves. I do not have the equation of the curves, but I will have a finite set of coordinates. Is there a way to build the equation for this curve based ...
1
vote
1answer
75 views

Transforming a Continuous Function

My math is quite limited so please bear with me. I will get to the point: Is there a way to transform a continuous function into a bounded one? In essence I have a normalized Gaussian distribution ...
5
votes
1answer
196 views

Conditions for a curve to be defined over a subfield

I have just finished reading Hartshorne, Chapter 1, Section 6 and have some questions about curves defined over a subfield of an algebraically closed field. For simplicity, let $k$ be a perfect field, ...
2
votes
1answer
558 views

Automorphisms of projective line

how does one find the set of Automorphisms of the complex projective line? PS: no scheme theory is assumed.
3
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1answer
304 views

Localization at a point - an exercise

I'm trying to practice with localization of rings. In particular, I want to determine the generator t of the principal maximal ideal of the localization of the coordinate ring of a curve at a point. ...
3
votes
1answer
83 views

(Non)-Isomorphic Jacobians

I've started reading Hindry-Silverman's Diophantine Geometry and I got a little ahead of myself. Non-isomorphic elliptic curves have non-isomorphic Jacobians, that is because elliptic curves are ...
4
votes
2answers
255 views

software tool for accurate visualization of algebraic curves

First of all, I apologize since this is not strictly speaking a "mathematical" question but I could not find a better place for it. For a work presentation I need a tool for accurate visualization of ...
4
votes
1answer
125 views

Algebraic vs. Analytic curves

I'm familiar with the idea of using algebra to study certain types of plane curves, and my understanding is that there is a whole class of "algebraic curves" that can be studied this way. It would ...
2
votes
2answers
454 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 ...
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vote
0answers
118 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
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2answers
458 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
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0answers
147 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
174 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
234 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
224 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
217 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
681 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
532 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
156 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
191 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
256 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
304 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
233 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
165 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
262 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
160 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
160 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
103 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
253 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
270 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
229 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
224 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
227 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
306 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
481 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 ...