Questions tagged [algebraic-curves]

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 of curves are extensively studied as a basic case in singularity theory. Via algebraic function fields and modular curves they have links to number theory.

619 questions with no upvoted or accepted answers
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12
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551 views

Maximum number of intersection points of two different Bernoulli lemniscates

What is the maximum number of intersection points of two different Bernoulli lemniscates in the real plane? (Of course two identical lemniscates share an infinite number of points.) Here are some of ...
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116 views

Can Yoneda lemma for smooth projective varieties only use curves?

Let $X,Y$ be two projective smooth varieties over an algebraically closed field $k$, if we know functorial isomorphism $Hom(C,X) \cong Hom(C,Y)$ as sets for every smooth projective curve $C$, do we ...
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273 views

Why is the degree of a rational map of projective curves equal to the degree of the homogeneous polynomials?

Let $C_1 \subseteq \mathbb{P}^m$ and $C_2 \subseteq \mathbb{P}^n$ be projective curves, and let $\phi : C_1 \rightarrow C_2$ be a nonconstant rational map given by $\phi = \left[ f_1, \ldots, f_n \...
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102 views

Is there a substitution which transforms every Fermat curve into an elliptic curve?

A Fermat Curve of degree $n$ is the set of solutions to $x^n+y^n=z^n$, $x,y,z\in \mathbb R$. In this question, the OP provides a substitution which relates a Fermat Curve of degree $n=3,4$ to two ...
9
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1answer
157 views

Inhomogeneous polynomial and points at infinity

Let $f=X^2-Y$ be a polynomial in $k[X,Y]$, so $V(Z)$ is a parabola: $V(f)$: According to Bézout theorem the $y$-axis has to intersect the parabola two times. We know the y-axis meets the parabola ...
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224 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 ...
9
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1answer
1k 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 ...
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164 views

A Regular Map Has Finitely Many Ramification Points

Let $C,D$ be nonsingular projective curves, $f \colon C \to D$ nonconstant, $K = k(C), L = k(D)$, $d = \text{deg }f = [K:L]$, and of course $k$ algebraically closed. Furthermore let's suppose that $K:...
8
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403 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 ...
7
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197 views

How to write holomorphic line bundles on compact Riemann surfaces in coordinates

I would like to write down coordinate charts for holomorphic vector bundles on a smooth complex (compact) curve $C$. For example, if $C = \mathbb C \mathbb P^1 = \{ [x_0 : x_1 ] \}$, let \begin{...
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92 views

Given a curve over $\mathbb{P}^1$ how can I determine the monodromy around a ramification point?

Given a smooth curve of the form $$ f(y) - g(x) $$ over $\mathbb{A}^1_x$ with a smooth compactification $C \to \mathbb{P}^1$, how can I determine the monodromy of the points of the fibers? Are there ...
7
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266 views

Relationship between twisting sheaves and divisor sheaves

I'm not really entirely sure how to think about Serre's twisting sheaves $\mathscr{O}(i)$ - on any $\text{Proj}$ construction, really, but let's stick to something like $\mathbb{P}_2$ for now for ...
7
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1answer
405 views

Why do algebraic varieties contain curves passing through two given points

Let $X$ be an algebraic variety over the complex numbers. My definition of an algebraic variety is a finite type separated $\mathbf C$-scheme. Someone told me that such varieties have the following ...
7
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2answers
254 views

A question on Newton's “theorem about ovals”

This is a question about a result from Newton's Principia. It says, roughly, that the if you intersect lines $ax + by + c$ with a smooth, closed, convex curve, then the area of the curve that the line ...
6
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1answer
114 views

Error in Algebraic Curves by Fulton?

The following lemma is from section 3.3 of Fulton's algebraic curves. Notation: $\psi$ is a map from $k[X,Y]/I^n×k[X,Y]/I^m$ to $k[X,Y]/I^{m+n}$ I'm having some difficulty understanding the ...
6
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225 views

Affine plane curves classification

Define an affine plane conic as $\operatorname{Spec}A$ where $A=k[x,y]/(f)$ and $f$ a quadratic polynomial with no multiple factors. Define an equivalence relation on the set of alline plane conics ...
6
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233 views

Katz-Mazur chapter 1 AG questions

I was reading through Katz-Mazur "Arithmetic Moduli of Elliptic Curves", Chapter 1, and ran into some small issues (which might have a lot to do actually with notation). I think most of them are due ...
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152 views

Siegel's Theorem for Genus 0 Curves over Function Fields

Let $V$ be a quasi-projective algebraic curve over a field $k$. Are there finitely many morphisms from $S$ to the triply-punctured sphere $\mathbb{P}^1 -\{0,1,\infty\}$? This is true if $k = \mathbb{C}...
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166 views

For this morphism of integral schemes $X\to Y$, if $X$ is geometrically reduced (irreducible) then is $Y$ also geometrically reduced (irreducible)?

Let $k$ be an arbitrary field. Suppose that $C/k$ is an integral curve which is birationally equivalent to a projective line. Is it true that $C$ is geometrically reduced and irreducible? This is of ...
5
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87 views

Showing that a certain prime is a Weierstrass point

This problem is taken from the exercises in "Number Theory in Function Fields" by M. Rosen, chapter 6, page 76: Suppose that $\omega\in \Omega_K (0)$ and has a zero $P$ of degree 1, and that $...
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0answers
139 views

What is an example of a genus 5 smooth projective curve?

Recall that if $k$ is an algebraically closed field, then any degree $d$ plane curve $X$ will have arithmetic genus $g=(d-1)(d-2)/2$, by a simple calculation of $H^1(X,\mathscr{O}_X)$. This tells us ...
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418 views

How to find the tangent lines to an algebraic curve of a singular point?

I think the question is self-explanatory, but I'd like to put an example of what I am referring to. Let $C = V(f)$ be the curve defined by the polynomial $$f= -X^3 +4X^2Y-3XY^2+Y^3-2XY+Y$$ and $$F =...
5
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1answer
181 views

What is the relation between fractional ideals and divisors on curves?

I've done courses in Algebraic Number Theory and Algebraic Geometry, where I learned the theory of Dedekind domains (in ANT) and Divisors (in AG). Now, the wikipedia article on divisors says the ...
5
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0answers
143 views

Coordinate ring of the complement to the theta divisor

Let $C$ be a smooth projective curve over $\mathbb{C}$ and let $\Theta$ be the theta divisor in $J^{g-1}(C)$. The theta divisor is ample, so $J^{g-1}\setminus \Theta$ is affine. What is coordinate ...
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132 views

Riemann surfaces and folding

I am learning about Riemann surfaces as covering, and was interested to know how Riemann first thought of them. Looking at his collected papers, I found the following definition in his lecture called “...
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1k views

The Picard group of an Elliptic Curve

Let $(E,O)$ be an elliptic curve. Let $\operatorname{Pic}^0(E)$ stand for the divisors that have degree $0$ where : $$D = \sum_{p\in E}n_p(P) \text{ and } \deg D = \sum_{p\in E}n_p.$$ I understand ...
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218 views

Why is the morphism induced by this linear system birational?

I have seen (what seems to be) the following statement used in a few places, but I am not sure why it is true. Any explanation as to why it is (or is not) true would be appreciated. Let $P$ be a ...
5
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1answer
80 views

Book which covers these contents of the same level of Fulton's book.

My question is very specific. I'm studying Chapter 8 of Fulton's algebraic curves book and I would like to find another book (or online sources) which covers these contents: Divisors, the Vector ...
5
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0answers
265 views

Newton's Investigation of Cubics: Generalization?

I recently read about Newton's investigation of cubic curves, and how, like for quadratic curves we can classify them into parabolas, ellipses and hyperbolas, Newton was able to classify cubic curves ...
5
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0answers
376 views

On the bidegree of a curve in $\mathbb{P}^1 \times \mathbb{P}^1$

I was reading Beauville's Complex algebraic surfaces, at page 5 there is an example in which curves in $\mathbb{P}^1 \times \mathbb{P}^1$ are classified by the bidegree up to linear equivalence. I'm ...
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278 views

Homogenous polynomial and partial derivatives

I'm struggling to understand this part in a book I'm reading: Let $F$ be a projective curve of degree $d$ with $P\in F$. Wlog, suppose $P=(a:b:1)$. Let's look the affine chart $(a,b)\mapsto (a,b,...
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212 views

Curves of fixed genus and degree lying on a cubic surface

I would like to prove the following statement: Let $C\subseteq \mathbb{P}^{3}$ be an irreducile nonsingular curve of arithmetic genus $g_{a}(C)=24$ and degree $d(C)=14$. Then there exists an ...
5
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0answers
109 views

Algebraical Fujita cancellation

If there exists some birational equivalence between ruled surfaces $C\times\mathbb{P}^1$ and $C'\times\mathbb{P}^1$ over a sufficiently nice field, then one can cancel the projective line and conclude ...
5
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0answers
153 views

Line bundles over a curve

I keep seeing statements like that and I don't know how they might be established: Let $k \subset \bar{k}$ be fields, $C \rightarrow Spec(\bar{k})$ be a curve and $\mathscr{L}$ a line bundle on $C$. ...
5
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0answers
69 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}, ...
5
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0answers
96 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 $f:X\to \mathbf{P}^1_k-B$. ...
5
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0answers
353 views

intersection multiplicity and partial derivatives of algebraic curves

this will probably be an easy-to-answer and a not-well-posed question, since I'm a total beginner in the field, but here goes: Let $V(F)$ and $V(G)$ be two projective curves in $\mathbb{P}^2$ ($F,G\...
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0answers
469 views

The Milnor Conjecture on the Unknotting Number of a Torus Knot

Let $f \colon (\mathbb{C}^{n},\mathbf{0}) \to (\mathbb{C},0)$ be a complex analytic function with isolated critical point at the origin. Define the singular hypersurface $V_{f, \kappa} = f^{-1}(\kappa)...
4
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0answers
59 views

Computing global monodromy of branched covering of Riemann sphere

Given an irreducible polynomial $F(X,Y)\in\mathbb{C}[X,Y]$, there is an associated Riemann surface $S$ obtained from the zero set of $F$. Projection onto the second coordinate gives a map $\pi:S\to \...
4
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0answers
71 views

Homology of Fermat curve

Let $C(n):X^n+Y^n=Z^n$ be the plane projective Fermat curve of degree $n$ over $\mathbb{C}$. Shorter version of the question: How can I describe explicit representatives for a basis for the singular ...
4
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0answers
57 views

Connected levels and polynomials submersions

Is it true that a polynomial submersion $ p: \mathbb{R}^2 \to \mathbb{R}$ of degree $n$ has at most $n$ connected components on each level? I think I have a proof, can someone point me out any ...
4
votes
1answer
113 views

Does a curve's genus depend on its base field?

Let $C$ be a projective plane curve, defined by a polynomial in $Z[x,y]$, over a field $K$. Does the geometric genus of $C$ depend on the choice of $K$? I think the answer to this question is ...
4
votes
1answer
360 views

When are curves irreducible?

This question might be too vague, but we know lots of things about irreducible curves, but when I'm given a curve I often can't tell if it's irreducible, so I don't know if those things apply to my ...
4
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0answers
79 views

Cardinality of a supersingular hyperelliptic curve of genus $2$ over $\mathbb{F}_q$ (q square)

I am trying to calculate the cardinality of a genus $g=2$ curve under certain hypotheses, but I do not know if the information I have is enough to infer this number or a bound for it (without using ...
4
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0answers
104 views

Is any regular algebraic curve in the plane union of closed curves?

This is related to the attempt I am doing trying to prove that any polynomial lemniscate is the union of regular closed curves. Recall by the way that a polynomial lemniscate is a curve of the form $\{...
4
votes
1answer
106 views

Grobner Basis and Surfaces

Grobner basis are really good at describing polynomial systems of equations with 0-dimensional zero sets. In a sense, Grobner basis yields a better/simplified description of such systems because the ...
4
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0answers
199 views

Why is this cubic polynomial generic for cyclic field extensions?

[EDIT: There doesn't seem to be any interest in answering this question, so could anyone just provide me a reference for understanding (2), and if possible (1)? Hopefully that would be enough to help ...
4
votes
0answers
330 views

(first) cohomology $H^1$ of sheaves on curves

Say we have a reduced, irreducible, complete curve $X$ over a field $k$ and a sheaf $\mathcal F$ on $X$. I am interested in understanding the cohomology groups of $\mathcal F$. By flat base change, ...
4
votes
0answers
150 views

structural (algebraic) sheaf on a Riemann surface “inside” the sheaf of holomorphic functions

This question consists of $3$ points, and each of them deals with the relationship between the "algebraic" structure on a projective curve and its "analytic" structure. I know that this argument has ...
4
votes
0answers
84 views

Group associated with this curve

Consider an Elliptic curve with a singularity (double point) over $\mathbb{F}_p$ $$y^2=x^2(x-a) $$ where $a$ is a quadratic residue in ${\mathbb{F}_p}^*$. We can easily count the number of (non-...