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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 ...

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Is this plane curve irreducible?

I want to define a plane curve in $\mathbb{A}^2(\mathbb{C})$ by the polynomial $f(x,y)=x(x-1)^2-(y-1)^2=0$ where $(x,y)\in\mathbb{A}^2(\mathbb{C})$, but my goal is for the plane curve to be ...
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Plot projective curves

Do you know any website, app or program that can plot curves in the projective plane? I know the projective plane $\mathbb{P}^2 \mathbb{R}$ can be visualized as a sphere with the antipodal points ...
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Using resultants to show extension of function fields of curves is algebraic.

We are given two irreducible nonsingular plane curves, the zero sets of $f,g\in \bar{k}[x,y]$, with $\bar{k}$ algebraically closed. We have an injective map given on algebras as $\phi^*:C_f\rightarrow ...
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Example for intersection, multiplicity and tangents for projective plane curves

We recently got introduced to projective plane curves in our class, however we just defined it and never really talked about examples. And when I am trying to come up with a just any curve, ...
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Why is this partial derivative zero? (Algebraic functions)

Why is $F'(a,z_i) \ne 0$? An algebraic function $y=f(x)$ is defined by the algebraic equation $$ F(x,y) := g_n(x)y^n + g_{n-1}y^{n-1} + \cdots + g_0(x) = 0 $$ where $g_j$ are polynomials. In ...
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Smooth curves of odd genus

Let $C$ be a smooth curve of genus $g$ and $J_C$ its intermediate Jacobian. Recall that $J_C$ is a ppav of dimension $g$. Fixing a point $p\in C$, one can define the Abel-Jacobi map $$a\colon C\...
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What is the relation between torsion elements of the class group and covering spaces of curves?

For a Dedekind domain $A$ we have the following result relating torsion elements of the class group to (mostly) unramified extensions: If $a\in A$ is such that there exists an ideal $I$ with $I^n=(a)$...
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When is it the case that all closed immersions of all irreducible components are flat?

Let $X$ be a projective, reduced scheme of pure dimension one (irreducible components have dimension one) over some field $k$. Let $X_1,\ldots,X_r$ be the irreducible components of $X$ and let $f_i: ...
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Show that $f(x,y,z)=x^2-y^2z$ is irreducible in $\mathbb{C}[x,y,z]$.

Let $p\in\mathbb{C}[x,y,z]$ be defined by $p(x,y,z)=x^2-y^2z$. Goal: Prove that $p$ is irreducible. Let $I\subset\mathbb{C}[x,y,z]$ be the ideal defined by $$I:=(p).$$ My approach is to show that ...
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Does $\deg_k F(-D)/F = \deg_k D$ hold for effective divisors $D$ and coherent, torsion-free $\mathcal{O}_X$-modules $F$?

$\DeclareMathOperator{\F}{\mathcal{F}}\DeclareMathOperator{\o}{\mathcal{O}}$Let $X$ be a reduced, pure dimensional, projective curve over some field $k$. Let $\F$ be a coherent and torsion-free $\o_X$...
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Number of branch points for a projection to $\Bbb{CP}^1$

Based on a helpful response to my previous post which offered advice on how to split the image of a map into affine and non-affine components, I've come up with a solution to the following problem. ...
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Connection between definitions in function fields and on curves

(Sorry for the weird title, I really don't know how to describe this question in a line) I am reading the book "Number Theory in Function Fields" by Rosen and it has an algebraic perspective on all ...
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Branch points of a projection to $[X:Y] = \Bbb{CP}^1$ using homogeneous coordinates

I have a question that I've been stuck on for some time now, and although I'm able to understand similar questions I'm stuck on this particular type of projection to the complex projective line. ...
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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 ...
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Hartshorne algebraic geometry IV Proposition 3.8.

In Hartshorne's book for Algebraic Geometry, I am a little confused about the proof in Chapter IV Proposition 3.8. In particular, why does (a) imply (b)? If $\psi$ is inseparable, why does every ...
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Type I pencils on Riemann surfaces (complex curves)

I was reading the paper https://www.researchgate.net/publication/265872523_On_the_number_of_pencils_of_minimal_degree_on_curves_with_small_gonality and found the statement in the second paragraph to ...
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Hyperelliptic curves express by branch points

In a book of algebraic geometry, the author says that equation $$ y^2=x(x^4-1) $$ defined a curve $S$ of genus $2$. My problem is this: Because the curve has genus 2, it can be expressed as follows $...
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Smooth projective curves of genus 0

Is there a classification of smooth projective curves of genus $0$ over $\mathbb{Q}$? I know that if the curve has a rational point, then it is isomorphic to $\mathbb{P}^1$. The curve must embed as ...
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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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Exercise 8.3 Fulton [closed]

Let $C=X$ be a nonsingular cubic. a) Let P,Q $\in{C}$. Show that $P \equiv Q$ if and only if $P=Q$. (Hint: Lines are adjoints of degree 1) Where $P \equiv Q$ if and only if $P=Q+div(z)$ Please give ...
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Is it possible to (naturally) integrate an algebraic curve into quantum computers?

A projective algebraic curve is defined over an projective space as zeros of some homogeneous polynomial. While quantum computers almost canonically holds their states in a "ket," i.e. a vector in the ...
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Non-trivial Extensions of Sheaves of $\mathcal{O}$-Modules

I am working on Problem XI.4. E of Miranda’s book “Algebraic Curves and Riemann Surfaces”. Let $X=\mathbb{P}^1, p = \infty, \mathcal{O}[n]=\mathcal{O}[n\cdot p]$. Write down a nontrivial extension ...
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Singular points on complex projective-algebraic curve vs affine curves

I am trying to understand singular points on a complex projective -algebraic curve. I remember that singular points on a affine algebraic curve are determined by taking the partial derivatives and ...
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A question on Poincaré-Hopf Theorem for meromorphic forms.

In Griffiths' Introduction to Algebraic Curves. In the proof of the following statement, Let $\omega$ be a meromorphic 1-form on a compact Riemann surface $C$, then $\sum_{p\in C} \...
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Draw the curve $8x^2+6xy-\frac x{\sqrt{10}}+3\frac y{\sqrt{10}}=1$

Draw the curve Draw the curve $8x^2+6xy-\frac x{\sqrt{10}}+3\frac y{\sqrt{10}}=1$. let $v = (x,y)\in \mathbb{R}^2$ and $B=\{(\frac 3{\sqrt{10}},\frac 1{\sqrt{10}}),(\frac {-1}{\sqrt{10}},\frac 3{\...
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Global sections of a Vojta divisor: a lemma for Faltings' theorem and Vojta's inequality

This is E.5 of Hindry, Silverman's Deophantine Geometry. I want a global section of a Vojta divisor from some polynomials. Let $K$ be a number field, $C$ a smooth projective $K$-curve of genus $g \ge ...
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63 views

Hartshorne IV 2.2

This is exercise 2.2 in Hartshorne's AG: What I have done: a):Apply Hurwitz theorem and observe that ramification index is no greater than degree. b):There is $ k[x]\rightarrow k[x,z]/(z^2-\...
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Gap Numbers for the Canonical Linear Systems

I am working on an exercise in Miranda’s book “Algebraic Curves and Riemann Surfaces” [Chapter VII.4.S]. Let $X$ be a nonhyperelliptic curves of genus $g\ge 3$. Denote by $G_p(|K|)=\{n_1<\dots<...
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A polynomial vanishing on a set having a limit point.

Let $C\subset\Bbb{R}^2$ be an algebraic curve defined by an irreducible polynomial $P$. Take $Q\in\Bbb{R}[x,y]$ an another polynomial such that $Q(x,y)=0$ on a set $A\subset C$ such that $A$ ...
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Irreducibility of polynomial $x^3-y^2$

I was told that $x^3-y^2$ is irreducible in $\Bbb C[x,y]$. But I cannot really give a sounding argument. I supposed that it may be factored as $g_1, g_2$, and considered those monomials ...
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Level curves of exponential

I have the following two-variable function: f(x,y) = exp(-x^2-(y-1)^2) And I need to compute/sketch the level curves for ...
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Reference book/ Notes/ articles for p rank

I am a graduate student. I want to learn about $p$-rank and $p$-torsion points on curves over finite fields. As the base, I have read Stictenoth's book ''Algebraic Function Fields and Codes'' but he ...
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Uniformizer of a projective curve

Let us consider a smooth projective curve over a algebraically closed field. I would like to know if there is any algorithm to compute the uniformizer at any point of the curve ? at the page 3 of this ...
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Hartshorne II Prop. 6.8 (last step).

I am having trouble understanding the last line in the proof of this proposition. I have seen this question once on Math.SE and another time in MO, but I couldn't find in either case a clear solution ...
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Local Behaviour of a Node of a Plane Curve

In Miranda’s book “Algebraic Curves and Riemann Surfaces”, in page 70, it says At a node, we locally have a curve of the form $xy=0$; the nearby curve looks locally like $xy=t$ for some small ...
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Example of family of integral curves with constant gonality and increasing genus

Are there examples of a family of integral curves over some field $k$ which have constant gonality but increasing genus? A related question: If I give you two non-negative integers $n$ and $g$, can ...
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How to find the center of an arithmetic spiral? Tangent to two lines

attempt at finding center of arithmetic spiral to be tangent to two lines Given the two orange lines in the picture, how can I calculate the center for an arithmetic spiral that will join the two ...
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Asymptotes' intersection with curve at infinity

I read in my course book that asymptotes have at least two intersection with algebraic curve at infinity. How can I take this fact in my head in a visualized way? And what does that at least means? ...
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Moduli space of stable curves of genus $g$ does not admit a universal family

I am trying to understand why the (coarse) moduli space $\overline{M}_g$ of stable curves of genus $g$ does not admit a universal family. I am following the proof in p.267 of this book. A key step is ...
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Quadratic forms with common real zeros

Is it possible to find three quadratic forms $f_1,f_2,f_3$ in $K[x,y,z]$ where $K$ is a real number field such that $f_1,f_2$ and $f_3$ have exactly 2 common real zeros $f_1^2+f_2^2+f_3^2$ can be ...
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Logistical Curve Tansformations

Mostly I am having a brain fart. Working on a game and wanted to use logistical curve for the experiance gain of skill levels. Idea is to create a system the reflects natural learning; early on you ...
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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?
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Pushforward under map of curves

If $C$ and $D$ are smooth projective curves and $f:C\to D$ is a map, how do we compute $c_1(f_{*}\mathscr{O}_C)$? I think applying Grothendieck Riemann-Roch from wikipedia yields the answer, but I'm ...
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Why are these called “meromorphic” differential forms?

On p.30 of Silverman's The Arithmetic of Elliptic curves, the notion of a meromorphic differential form on a curve $C$ is defined as follows: Definition. Let $C$ be a curve. The space of (...
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Does a curve which is homeomorphic to interval always have a finite length independent of the parametrization in general metric spaces?

Let $(X,\rho)$ be a metric space and $C\subseteq X$ be a subset which is homeomorphic to interval $[a,b]$ for some $a,b\in\mathbb{R},a<b$. Each homeomorphism $\gamma:[a,b]\rightarrow C$ is called a ...
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Rational point and its residue field

How is the relation between the rational point $P$ of a non-singular projective curve over field $k$ and its residue field $F_P=\mathcal{O}_P/\mathcal{M}_P$? Do we have the relation $F_P\cong k$? If ...
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Meaning of expression : “uniquely determined”

I am having difficulty understanding (also because of my native language) the expression that is widely used in mathematics: "Uniquely Determined". For example: $1)$ Cat theorem asserts that color ...
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What part of algebra can I use to describe linked curves?

Can anybody tell me what is the algebraic structure of linked curves in 3D space? Can their linking number be described using a vector space? Or does it require some other structure such as a group, ...
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The function field of an affine plane curve has transcendence degree one

I am looking for a book which shows that the function field of an affine plane curve has transcendence degree 1. Sincerely, Hypertrooper
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Longest function for the not each other dominating points

I stumbled over this little exercise on StackOverflow where the poster asks for an algorithm to find the biggest subset of a point cloud which contains only points that do not dominate each other. $...