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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42
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677 views

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 ...
21
votes
3answers
1k 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 ...
20
votes
4answers
2k views

intuitive explantions 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 ...
17
votes
3answers
801 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 ...
15
votes
1answer
735 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, ...
15
votes
2answers
575 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 ...
14
votes
1answer
1k 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 ...
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 ...
12
votes
2answers
1k 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 ...
12
votes
2answers
531 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 ...
12
votes
1answer
301 views

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?
12
votes
0answers
179 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 ...
11
votes
3answers
398 views

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 ...
11
votes
3answers
475 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 ...
11
votes
1answer
389 views

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 ...
11
votes
2answers
266 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 ...
11
votes
2answers
195 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 ...
11
votes
1answer
1k 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 ...
11
votes
1answer
149 views

Invariant point (flex) invariant under projective transformations.

Let $C$ be a projective curve in $\mathbb{P}_2$ defined by a homogeneous polynomial $P(x, y, z)$ and let $\alpha$ be a linear transformation of $\mathbb{C}^3$. Let $Q$ be the homogeneous polynomial $Q ...
10
votes
3answers
264 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 ...
10
votes
2answers
415 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 $V$. ...
10
votes
1answer
189 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 ...
10
votes
3answers
216 views

Is there a better way to find the polynomial equation for this curve?

Consider the curve in $\mathbb{R}^2$ defined by the equation $$ x^{1/3} + y^{1/3} + (xy)^{1/3} = 1, $$ where $x^{1/3}$ denotes the real cube root of $x$, etc. Since the equation above involves only ...
10
votes
1answer
496 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 ...
10
votes
1answer
200 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 ...
10
votes
0answers
97 views

Multiplicity of point as a zero of polynomial.

Let $C \subset \mathbb{P}_2$ be a projective curve defined by a homogeneous polynomial $P(x, y, z)$, and let $L \subset \mathbb{P}_2$ be the line $\{z = 0\}$. Assume $L$ is not a component of $C$. If ...
9
votes
4answers
213 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 ...
9
votes
3answers
966 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 ...
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 ...
9
votes
2answers
239 views

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

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 ...
9
votes
2answers
589 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 ...
9
votes
1answer
138 views

$p \in C - D$, inflection point for $C$ iff inflection point for $C \cup D$.

Show that if $C$ and $D$ are projective curves in $\mathbb{P}_2$ and $p \in C - D$ then $p$ is a point of inflection for the curve $C$ if and only if $p$ is a point of inflection for the curve $C \cup ...
9
votes
2answers
89 views

Nonsingular curve $C$ of degree 4, exists rational function $f: C \to \mathbb{CP}^1$ of degree 2?

Suppose $C \subset \mathbb{CP}^2$ is a nonsingular curve of degree $4$. Does there exist a rational function $f: C \to \mathbb{CP}^1$ of degree $2$?
9
votes
1answer
50 views

Irreducible projective cubic, exists continuous surjection?

Let $C$ be an irreducible projective cubic in $\mathbb{P}_2$ with a singular point $p$. So consider $f: \mathbb{P}_1 \to C$ defined as follows. Identify $\mathbb{P}_1$ with the set of lines in ...
8
votes
4answers
285 views

Is there a name for the curve $t \mapsto (t,t^2,t^3)$?

Is there a name for the curve given by the parametrization $\{(t,t^2,t^3); t\in\mathbb R\}$? Here is a plot from WA. An another plot for $t$ from $0$ to $1$. This curve is an example of a ...
8
votes
1answer
286 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 ...
8
votes
2answers
257 views

Equations for double etale covers of the hyperelliptic curve $y^2 = x^5+1$

Let $X$ be the (smooth projective model) of the hyperelliptic curve $y^2=x^5+1$ over $\mathbf C$. Can we "easily" write down equations for all double unramified covers of $X$? Topologically, these ...
8
votes
1answer
266 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 ...
8
votes
2answers
157 views

Principal divisors

How can i calculate the principal divisor $(f)$ where $$f = \frac{(x^{3}-1)}{(x^{4}-1)}$$ with $f\in\mathbb{F}_2(x)$. I am recently reading about the subject, so i am looking for a simple solution ...
8
votes
1answer
256 views

Threefold category equivalence: algebraic curves, Riemann surfaces and fields of transcendence degree 1

According to this article, The category of curves over the complex numbers is equivalent to two other categories: Riemann surfaces and fields of transcendence degree 1 over $\mathbb{C}$. But I ...
8
votes
1answer
247 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 $D_i$ ...
8
votes
1answer
556 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 ...
8
votes
1answer
131 views

Varieties with the property that the cotangent bundle restricted to a complete nonsingular curve is free

Let $X$ be a $d$-dimensional smooth projective connected variety with cotangent sheaf $\Omega^1_X$ over $\mathbb C$. Suppose that for any nonsingular complete curve $C$ and non-constant morphism ...
8
votes
0answers
98 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 ...
8
votes
1answer
110 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 ...
8
votes
0answers
164 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 ...
7
votes
2answers
117 views

If $f(u(x), v(y))=f(x, y)$, can we conclude that either $u(x)=x$ or $v(y)=y$?

Suppose $k$ is an algebraically closed field, and $f\in k[x, y]$ is an irreducible polynomial in two variables. Furthermore, suppose that $f(u(x), v(y))=f(x, y)$ for every $x, y\in k$, where ...
7
votes
1answer
199 views

Is the Fermat scheme $x^p+y^p=z^p$ always normal

Let $K$ be a number field with ring of integers $O_K$. Is the closed subscheme $X$ of $\mathbf{P}^2_{O_K}$ given by the homogeneous equation $x^p+y^p=z^p$ normal? I know that this is true if ...
7
votes
2answers
213 views

Branch points lie in $\mathbb P^1\left(\overline{\mathbb Q}\right)$

Let $X$ be a complex smooth projective curve and suppose moreover that $X$ is defined over $\overline{\mathbb Q}$. Now consider a finite map $f:X\longrightarrow\mathbb P^1(\mathbb C)$ of degree $d$ ...
7
votes
2answers
243 views

endomorphisms of the jacobian of a general hyperelliptic curve

Let $C$ be a curve of genus $g$. If $C$ is very general, we know that the Jacobian $JC$ of $C$ is simple and thus $End(JC)=\mathbb{Z}$. Do we know something about $End(JC)$ if $C$ is a very general ...