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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18
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 ...
16
votes
3answers
709 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 ...
6
votes
5answers
377 views

For $(x+\sqrt{x^2+3})(y+\sqrt{y^2+3})=3$, compute $x+y$ .

If $(x+\sqrt{x^2+3})(y+\sqrt{y^2+3})=3$, compute $x+y$.
4
votes
1answer
72 views

Why is the extension $k(x,\sqrt{1-x^2})/k$ purely transcendental?

Consider the function field $k(x,\sqrt{1-x^2})$ of the circle over an algebraically closed field $k$. Is $k(x,\sqrt{1-x^2})$ a purely transcendental extension over $k$? I'm curious because I was ...
3
votes
1answer
136 views

What is a field of definition of a morphism?

An algebraic variety $X$ over a field $K$ is a $K$-scheme integral separated and of finite type over $K$. Definition: If $L$ is a subfield of $K$ we say that $X$ is defined over $L$ if there ...
2
votes
1answer
70 views

the diagonal $\Delta (Y) =\{(y,y)\in Y \times Y\}$ is closed in $\Bbb P^m \times \Bbb P^m $

Asumme that $\Pi : \Bbb P^n \times \Bbb P^m \rightarrow \Bbb P^m $is a closed map. $X\subset \Bbb P^n$ And $Y\subset P^m$ $f: X\rightarrow Y$ be a morphism of projective varieties. ...
4
votes
1answer
414 views

Set that is not algebraic

I'd like some hints for the problem: Show that the following set is not algebraic: $ \{ (\cos(t),\sin(t),t) \in \mathbb{A}^3 : t \in \mathbb{R} \} $ thanks.
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 ...
5
votes
2answers
191 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 ...
6
votes
1answer
393 views

Computing the monodromy for a cover of the Riemann sphere (and Puiseux expansions)

Given an irreducible polynomial $P(X,Y) \in \mathbb{C}[X,Y]$, one obtains by analytic continuation a Riemann surface $M$ with a branched covering $f \colon M \to \mathbb{P}^1_{/\mathbb{C}}$, ...
11
votes
2answers
256 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 ...
10
votes
2answers
318 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 $V$. ...
6
votes
2answers
191 views

Is the circle a rational curve and what is its function field?

It does seem like the circle ($S^1=\{X^2+Y^2=1\}\subseteq k^2$ for $k$ a field) is a rational curve: it has parameterization $X=2T/(T^2+1)$ and $Y=(T^2-1)/(T^2+1)$. On the other hand, we have a ...
0
votes
2answers
2k views

equation of a curve given 3 points and additional (constant) requirements

Given 3 pairs of coordinates, $x_1, y_1, x_2, y_2, x_3, y_3$, I need a function $y(x)$ that will return the $y$ coordinate of any $x$ coordinate between $x_1$ and $x_3$ (it can be assumed that $x_1 ...
11
votes
1answer
229 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?
11
votes
2answers
852 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 ...
11
votes
2answers
192 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 ...
7
votes
0answers
475 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 ...
4
votes
0answers
78 views

Computing Riemann surfaces of a given algebraic function

I've never seen written in a book a way or an algorithm for computing Riemann surfaces of a given algebraic function. I would like to know how to construct such Riemann surface using intuitive cutting ...
4
votes
1answer
104 views

Duality in algebraic de Rham cohomology

I am trying to prove that the following is a short exact sequence $$ 0 \rightarrow H^0(X,\Omega_X) \rightarrow H^1_{\text {dR}}(X/k) \rightarrow H^1(X,\mathcal O_X) \rightarrow 0, $$ where $k$ is an ...
8
votes
2answers
225 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
213 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 $D_i$ ...
6
votes
1answer
499 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 $$D=kP$$ ...
2
votes
1answer
37 views

When is an holomorphy ring a PID?

I posted this question on mathstackexchange but I realized it is probably more suitable for mathoverflow. I will use the notation and language of Stichtenoth, Algebraic Function Fields and Codes. ...
2
votes
0answers
84 views

When do two integral superellipses have 'nice' intersections?

A recent question posed the nonlinear system \begin{cases} 3x^3+4y^3=7\\ 4x^4+3y^4=16 \end{cases} for real $(x,y)$ and asked for the sum $x+y$. As noted by commentary in the question, this regrettably ...
2
votes
1answer
301 views

what is genus of complete intersection for: $F_1 = x_0 x_3 - x_1 x_2 , F_2 = x_0^2 + x_1^2 + x_2^2 + x_3^2$

In the below problem I want to answer second part: Show that the curve in $\mathbb{P}^3$ defined by the two equations $x_0 x_3 = x_1 x_2$ and $x_0^2 + x_1^2 + x_2^2 + x_3^2 = 0$ is a smooth complete ...
1
vote
2answers
137 views

Does the projective Fermat curve have singular points?

I want to check if the projective Fermat curve, $$X^n+Y^n+Z^n=0, n \geq 1$$ has singular points. Could you give me a hint how we could do this? Do we have to check maybe if the curve is ...
1
vote
1answer
89 views

through two points passes an unique line and generalizations

I would like to prove for each integer $d\ge 1$, there are $\frac{d(d+3)}{2}$ points in the plane for with passes exactly a curve of degree $d$. For $d=1$, according to the statement there are 2 ...
0
votes
0answers
44 views

How to find the tangent vector to a curve at a special point

The two points $A=(x_{0},y_{0},z_{0})$ and $B=(x_{1},y_{1},z_{1})$ are given. The line segments $AC$ and $BC$ make equal angle $\alpha$ with the horizontal plane through $C$. The angle ...
0
votes
0answers
69 views

$V(f)$ is irreducible iff $f=g^k$, $g$ irreducible

I'm trying to prove this theorem $V(f)$ is irreducible iff $f=g^k$, $g$ irreducible. To prove the converse, we have $V(f)=V(g^k)=V(g)$, since $g$ is irreducible $V(g)$ is irreducible, then ...
6
votes
1answer
137 views

explict form of the equation of elliptic curve

Let $E(\mathbb{F}_{q^2})$ is elliptic curve with #$E(\mathbb{F}_{q^2}) =q^2 + q + 1$. Can we write equation of this curve in the explicit form?
5
votes
1answer
82 views

Normal curve after base change (p > 0)

Suppose that $C$ is a normal projective curve over some base field $k$, possibly of positive characteristic. I am wondering to what extent one can modify the base field $k$ while not braking the ...
4
votes
1answer
83 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 ...
3
votes
2answers
86 views

Definition of intersection multiplicity in Hartshorne VS Fulton for plane curves

In exercise I.5.2 in Harshorne there is the following definition of intersection multiplicity for two curves in $\mathbb{A}^2$: \begin{equation} \mathrm{length}_{\mathcal{O}_P}\mathcal{O}_P / (f,g) ...
3
votes
1answer
41 views

Valuation rings and total order

Let $K$ be a field and $\mathcal{O}$ be a valuation ring of $K$ (So $\mathcal{O}\subset K$ is an integral domain with the property that either $x$ or $x^{-1}$ is in $\mathcal{O}$ for all $x\in K$). ...
3
votes
4answers
213 views

how to find the equation of one curve in XY plan which satisfies such functions?

I want to find the equation of one curve in $X-Y$ plan which satisfies the functions as follows: 1) $A(x_1,y_1)$, $B(x_2,y_2)$ are two known points and $f(x,y)$ (or $y(x)$) is the equation of the ...
3
votes
1answer
87 views

The intersection numbers in Fermat curve

I'm a beginner in this subject and I think this "easy" exercise could help me to have more practice in basic algebraic curves. Let $F=X^{p+1}+Y^{p+1}+Z^{p+1}$ be a Fermat curve in the field $k$, with ...
3
votes
2answers
106 views

$Im(\phi)$ is closed subset of $\mathbb{A}^2$

let $\alpha(t)$ and $\beta(t)$ $\in$ $K[t]$ , $\phi(t)=(\alpha(t),\beta(t))$ is a morphism from $\mathbb{A}^1$ to $\mathbb{A}^2$ show that $Im(\phi)$ is closed subset of $\mathbb{A}^2$. it seems ...
3
votes
1answer
100 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.)
2
votes
1answer
57 views

Computing these multiplicities

I'm trying to use some Algebraic Geometry techniques to check my understanding on them. I'm using the most stupid of all the examples: trying to compute the multiplicities of the intersections of the ...
2
votes
1answer
59 views

intersection multiplicity from Shafarevich

In Basic Algebraic Geometry, Shafaravich proves the following theorem: Theorem. If $X$ is an irreducible affine curve, and $P \in X$ is a nonsingular point, then there is a function $t$, regular at ...
2
votes
2answers
176 views

Pole set of rational function on $V(WZ-XY)$

Let $V = V(WZ - XY)\subset \mathbb{A}(k)^4$ (k is algebraically closed). This is an irreducible algebraic set so the coordinate ring is an integral domain which allows us to form a field of fractions, ...
2
votes
1answer
223 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 ...
2
votes
1answer
109 views

Is the intersection of the diagonal with a graph always transverse in characteristic zero

Let X be a projective smooth connected curve over $\mathbf{C}$. Let $f:X\to X$ be a non-constant morphism. Is the intersection of the diagonal $\Delta_X$ and the graph $\Gamma_f$ on $X\times X$ ...
1
vote
1answer
36 views

Intersection Number of $B = Y^2 - X^3 + X$ and $F = (X^2 + Y^2)^3 - 4X^2Y^2$ using the fact $I(P,F \cap B) = ord_P^B(F) $.

Firstly, I apologise that this question is specific to two polynomials. I understand that this post will not help a lot of people, and for that I am sorry. I will make it up to you all by anwsering ...
1
vote
2answers
62 views

Why aren't those Cartier Divisors equivalent?

Please refer to Gathmann's notes http://www.mathematik.uni-kl.de/~gathmann/class/alggeom-2002/main.pdf at Example 9.3.6 for context. It's trying to give an example that the map between $Div(X)$ and ...
1
vote
1answer
80 views

Merge two or more cubic Bézier curves for optimization

I am looking for an algorithm which can merge several cubic Bezier curves. For instance, I have a lot of cubic Bezier that are joined to form a poly-Bezier curve. The idea is to merge dynamically some ...
1
vote
1answer
107 views

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

I'm working on the following problem for several days without finding any solution: Let $X$ be a complex smooth projective curve and suppose moreover that $X$ is defined over $\overline{\mathbb ...
1
vote
1answer
72 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 ...
0
votes
1answer
117 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 ...