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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 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.

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Geometric interpretation of the Riemann-Roch for curves

Let $X$ be a smooth projective curve of genus $g\geq2$ over an algebraically closed field $k$ and denote by $K$ a canonical divisor. I have some clues about the geometrical interpretation of the ...
Abramo's user avatar
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48 votes
4 answers
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Intuitive explanations for the concepts of divisor and genus

When trying to explain AG-codes to computer scientists, the major points of contention I am faced with are the concepts of divisors, Riemann-Roch space and the genus of a function field. Are there any ...
yohay kaplan's user avatar
38 votes
2 answers
4k views

The genus of the Fermat curve $x^d+y^d+z^d=0$

I need to calculate the genus of the Fermat Curve, and I'd like to be reviewed on what I have done so far; I'm not secure of my argumentation. Such curve is defined as the zero locus $$X=\{[x:y:z]\in\...
Marra's user avatar
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34 votes
2 answers
20k 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 ...
smackcrane's user avatar
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30 votes
1 answer
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What is the best way to see that the dimension of the moduli space of curves of genus $g>1$ is $3g-3$?

This fact was apparently known to Riemann. How did Riemann think about this?
Bruno Joyal's user avatar
30 votes
1 answer
4k 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, ...
Joachim's user avatar
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30 votes
3 answers
8k views

Explicit Derivation of Weierstrass Normal Form for Cubic Curve

In page 22-23 of Rational Points on Elliptic Curves by Silverman and Tate, authors explain why is it possible to put every cubic curve into Weierstrass Normal Form. Here are relevant pages: (My ...
Prism's user avatar
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28 votes
4 answers
6k views

What is a local parameter in algebraic geometry?

Shafarevich offers the following theorem-definition: "At any nonsingular point $P$ of an irreducible algebraic curve, there exists a regular function $t$ that vanishes at $P$ and such that every ...
Zach Conn's user avatar
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28 votes
1 answer
802 views

Intersection of twisted cubics in $\mathbb{P}^3$

Suppose we have two twisted cubics $C_1$, $C_2$ in $\mathbb{P}^3$ such that both of them lie in some cubic surface, which means that $h^0(\mathbb{P}^3, I_{C_1\cup C_2}(3))>0$. I want to show that ...
guest31's user avatar
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25 votes
1 answer
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Tate's Thesis: in what sense is Tate's Theorem 4.2.1 the Riemann-Roch theorem for curves?

I am reading Tate's Thesis. Tate derives a theorem which he calls "the number-theoretic analogue of the Riemann-Roch theorem" from an abstract Poisson summation formula. I am accustomed to thinking of ...
Ben Blum-Smith's user avatar
24 votes
2 answers
10k views

Geometric interpretation and computation of the Normal bundle

My question concern the definition, geometric meaning, and usage of the normal bundle in algebraic geometry. Let $X$ be a nonsingular variety over an algebraically closed field $k$, and $Y\subseteq X$ ...
Fq00's user avatar
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23 votes
2 answers
2k views

Geometric interpretation of ramification of prime ideals.

I am trying to understand geometrically the ramification of primes in a finite separable field extension. Let $A$ be a Dedekind domain with fraction field $K$ and $L/K$ a finite separable field ...
Pedro's user avatar
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23 votes
4 answers
4k views

When do equations represent the same curve?

Suppose we have two sets of parametric equations $\mathbf c_1(u) = (x_1(u), y_1(u))$ and $\mathbf c_2(v) = (x_2(v), y_2(v))$ representing two 2D planar curves. When I say "2D planar curves" I mean ...
bubba's user avatar
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21 votes
6 answers
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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 not, ...
Martin Brandenburg's user avatar
21 votes
1 answer
4k views

Proving that the genus of a nonsingular plane curve is $\frac{(d-1)(d-2)}{2}$

I'm studying from Joseph Silverman's book The Arithmetic Of Elliptic Curves and I'm trying to do as many exercises as I can. Right now I'm trying to do Exercise 2.7 from chapter II which reads as ...
Adrián Barquero's user avatar
19 votes
2 answers
8k views

Irreducible components of the variety $V(X^2+Y^2-1,X^2-Z^2-1)\subset \mathbb{C}^3.$

I want to find the irreducible components of the variety $V(X^2+Y^2-1, \ X^2-Z^2-1)\subset \mathbb{C}^3$ but I am completely stuck on how to do this. I have some useful results that can help me ...
Katie Dobbs's user avatar
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18 votes
2 answers
3k views

Why is it so difficult to find beginner books in Algebraic Geometry? [duplicate]

I don't understand why it is so difficult to find a really beginner book in Algebraic Geometry. Let's take for example Fulton's algebraic curves: An introduction to Algebraic Geometry. I've already ...
user42912's user avatar
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17 votes
3 answers
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Applications of Belyi's theorem

Belyi's theorem (1979) is stated as follows: A smooth projective curve over $\mathbb C$ is defined over a number field if and only if there exists a finite morphism (of varieties over $\mathbb C$...
Dubious's user avatar
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17 votes
2 answers
3k views

Good books/expository papers in moduli theory

I have been studying mathematics for 4 years and I know schemes (I studied chapters II, III and IV of Hartshorne). I would like to learn some moduli theory, especially moduli of curves. I began ...
Andrea's user avatar
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17 votes
1 answer
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Link between Riemann surfaces and Galois theory

In my notes for a Geometry of Surfaces course that I'm studying, there is the following quote: (For those of you who like algebra and Galois theory) Studying compact connected Riemann surfaces is ...
Tim's user avatar
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16 votes
3 answers
3k views

When is a Morphism between Curves a Galois Extension of Function Fields

My apologies if this question has already been answered somewhere on this site: when I searched, I could only find specific examples rather than the general question. It is known that the category of ...
Dorebell's user avatar
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16 votes
2 answers
3k views

Is a divisor in the hyperplane class necessarily a hyperplane divisor?

Let $V$ be a smooth irreducible projective curve over an algebraically closed field $k$, embedded in some projective space $\mathbb{P}^n$, and let $[H]$ be the induced hyperplane divisor class on $V$. ...
Zhen Lin's user avatar
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16 votes
1 answer
2k views

Weil and Cartier divisors on a curve

I'm trying to understand the relationship between Weil divisors and Cartier divisors, and I would like to see why these are the same in the simple case where $X$ is a nonsingular projective curve over ...
Justin Campbell's user avatar
16 votes
1 answer
940 views

Exercise 1.24 from Joe Harris' Algebraic Geometry: First Course

I have a question about solvability of Exercise 1.24 (p. 14) from Joe Harris' Algebraic Geometry: A First Course and correctness of following 'synthetic' construction which according to the book (or ...
user267839's user avatar
  • 7,201
15 votes
3 answers
5k views

rational points of an algebraic variety

In http://en.wikipedia.org/wiki/Rational_point we read : a $K$-rational point is a point on an algebraic variety where each coordinate of the point >belongs to the field $K$. This means that, if ...
palio's user avatar
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15 votes
1 answer
2k views

Reasons for defining sheaves of holomorphic and meromorphic functions on complex manifolds

I am hoping this question is sensible and non-trivial. I am learning algebraic geometry at the moment, and have taken a strong liking to it. Unfortunately my complex analysis is weaker and I only know ...
Joe's user avatar
  • 609
14 votes
3 answers
2k views

Prove $X^2+Y^2-1$ is irreducible using geometrical tools.

I'm trying to understand what is meant in this paragraph: of "Conics and Cubics. A Concrete Introduction to Algebraic Curves (by Robert Byx)": He wants to prove that the polynomial $X^2+Y^2-1$ is ...
user42912's user avatar
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14 votes
2 answers
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On the automorphisms of the Klein Quartic

I am trying to solve a problem from Miranda's book, Algebraic Curves and Riemann Surfaces. On page 84, problem K gives the Klein curve $X$ as a smooth projective plane curve defined by the equation $...
FGerard's user avatar
  • 447
14 votes
2 answers
987 views

What is Riemann-Roch in arithmetic all about?

I learn number theory recently and I could not understand what Riemann-Roch was all about in arithmetic; could someone give me a bit hint? What is the advantage of viewing all this stuff geometrically ...
Yoshinobu Osawa's user avatar
14 votes
0 answers
798 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 ...
Mostafa's user avatar
  • 1,674
13 votes
3 answers
1k 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 ...
Harry's user avatar
  • 3,063
13 votes
3 answers
961 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 ...
Hoedan's user avatar
  • 359
13 votes
1 answer
1k 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 \...
kless135's user avatar
  • 164
12 votes
4 answers
1k views

What are some applications of the Weil conjectures for algebraic curves?

I have been interested in the Weil conjectures for some time, and the easiest place to start has been in studying them for elliptic curves. I've been able to see some of their applications and ...
Supersingularity's user avatar
12 votes
1 answer
364 views

Why does Mumford want to avoid "reduction to Jacobians"?

In the introduction to his Abelian Varieties book, David Mumford writes: I don't believe the word "Jacobian" is ever used in this book. Rather stubbornly I wanted to prove that the theory of ...
Miles Lake's user avatar
12 votes
1 answer
3k views

Basis for the Riemann-Roch space $L(kP)$ on a curve

Let $C$ be a smooth algebraic plane complex curve of degree $d$ defined as the zero locus of a homogeneous polynomial $F$. Let $P\in C$. For a positive integer $k$ consider the divisor $$D=kP$$ ...
Heitor Fontana's user avatar
12 votes
1 answer
2k views

How to compute the order $\text{ord}_P (f)$ for $f \in K(C)$

First lets fix some notation. Let $C$ be a projective curve (i.e. projective variety of dimension 1) defined over a field $K$. Suppose that $P \in C$ and that $P$ is a smooth point. It is known that ...
Adrián Barquero's user avatar
12 votes
1 answer
896 views

Why do varieties with torsion canonical sheaf have finite etale covers with trivial canonical sheaf

Let $B$ be a variety with torsion canonical sheaf, i.e., $\omega^{\otimes n}_B \cong \mathcal O_B$ for some $n>0$. Then, why does there exist a finite (etale?) morphism $X\to B$ such that $K_X$ is ...
Tom's user avatar
  • 647
12 votes
2 answers
4k views

Canonical divisor on algebraic curve

Can someone help me with this problem? Let $D$ be a divisor on an algebraic curve $X$ of genus $g$, such that $\deg D = 2g-2$ and $\dim L(D) = g$. Then $D$ must be a canonical divisor. By Riemann-...
Tony's user avatar
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12 votes
1 answer
4k views

What is normal crossing?

I could not find any reference for normal crossings. The definition here is not so clear to me. In some texts, they sometimes said that two varieties have normal-crossing (non-normal crossing) with ...
9999's user avatar
  • 959
11 votes
2 answers
865 views

Precise definition of an "algebraic function"

Remark. I'd like to avoid the "ring of formal expressions" viewpoint for this question. I know we can avoid these kinds of questions by working "purely algebraically" and in particular by taking the ...
goblin GONE's user avatar
11 votes
1 answer
661 views

Nonsingular projective variety of degree $d$

For each $d>0$ and $p=0$ or $p$ prime find a nonsingular curve in $\mathbb{P}^{2}$ of degree $d$. I'm very close just stuck on one small case. If $p\nmid d$ then $x^{d}+y^{d}+z^{d}$ works. If $p\...
TheNumber23's user avatar
  • 3,304
11 votes
2 answers
462 views

Can there be a point on a Riemann surface such that every rational function is ramified at this point?

Let $X$ be a compact connected Riemann surface, and let $S\subset X$ be a finite subset. Does there exist a morphism $f:X\to \mathbf{P}^1(\mathbf{C})$ which is unramified at the points of $S$? I'm ...
Ohdur's user avatar
  • 255
11 votes
2 answers
2k views

Abstract Nonsingular Curves

In section I.6 of Algebraic Geometry, Hartshorne establishes a that every curve is birationally equivalent to a nonsingular projective curve. To do this, he defines for any given curve $C$ with ...
Lalit Jain's user avatar
11 votes
2 answers
245 views

Families of curves over number fields

Is there known a number field $K$ and a curve $F(x, y) \in K(t)[x, y]$ such that $F(x, y)$ does not have points over the field of rational functions $K(t)$ but for all but finitely many positive ...
streetcar277's user avatar
11 votes
1 answer
790 views

Looking for elementary proof that irreducible/smooth curve in $\mathbb C^2$ is connected in Euclidean topology of $\mathbb C^2$

Let $f(X,Y)\in \mathbb C[X,Y]$ be an irreducible polynomial. I know that the zero set of $f$ , $V(f):=\{(a,b)\in \mathbb C^2 : f(a,b)=0\}$ is connected in the usual Euclidean topology of $\mathbb C^2$ ...
user's user avatar
  • 4,454
11 votes
1 answer
2k views

Intuitive/geometric way of thinking about effective divisors?

What is the motivation/intuition/geometric way of thinking about an effective divisor? I know that a divisor is effective if all its coefficients are non-negative. We write $D \ge 0$ for effective ...
user avatar
11 votes
1 answer
311 views

What is the minimum background required to understand moduli of curves?

Recently I've coincidentally run into various relatives of the moduli stack $\mathcal{M}_g$ in several unrelated contexts. I tried reading Harris and Morrison's "Moduli of Curves," but it seems to ...
Daniel McLaury's user avatar
10 votes
2 answers
3k 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 $...
Joachim's user avatar
  • 5,345
10 votes
2 answers
304 views

Finding an example of a degree 5 rational curve in $\mathbb{P}^3$ which is not in a quadric.

This is exercise IV.6.2 in Hartshorne, which asks us to find an example of a nonsingular rational degree $5$ curve in $\mathbb{P}^3$ which is not contained in a quadric. I'm somewhat at a loss for how ...
Daniel's user avatar
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