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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19
votes
3answers
932 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
1answer
119 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 ...
24
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 ...
11
votes
3answers
498 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 ...
6
votes
5answers
419 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 $x+y$.
5
votes
1answer
131 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 ...
2
votes
1answer
80 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
0answers
394 views

Cardinality of the Fiber of a Finite Morphism Vs. Degree (via Vakil)

I would like to show the following (note: it is not an assigned problem, so it may be false) (EDIT: Indeed it is false, see end of post): Suppose $f:X \rightarrow Y$ is a finite, surjective morphism ...
4
votes
3answers
274 views

Compactness of Algebraic Curves over $\mathbb C^2$

I was reading through Kirwan's Complex Algebraic Curves and I've been stuck on the following exercise: Given a (non-constant) polynomial $P(x,y)$, show that the curve in $\mathbb C^2$ defined by ...
3
votes
1answer
218 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 ...
6
votes
1answer
586 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.
3
votes
1answer
196 views

General surface no lines

I've been studying surfaces recently and I came across the following statement: A general surface $S \subset \mathbb{P}^{3}$ of degree $m \geq4$ contains no lines. Does anyone have any idea how to ...
1
vote
2answers
289 views

Number of solutions of an arithmetic function's equation

Say, an equation is given below \begin{equation} 2\pi(x) - \pi(2x)=\omega(x) \end{equation} where $x$ is a positive integer, $\pi(x)$ is the prime-counting function, and $\omega(x)$ is the number of ...
12
votes
2answers
4k 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 ...
5
votes
2answers
518 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 ...
21
votes
3answers
2k 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 ...
13
votes
1answer
383 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?
10
votes
2answers
309 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 ...
6
votes
1answer
559 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 $f \colon M \to \mathbb{P}^1_{/\mathbb{C}}$, ...
13
votes
1answer
2k 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 ...
12
votes
2answers
288 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
490 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
265 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 ...
7
votes
0answers
616 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
160 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 ...
4
votes
0answers
136 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
2answers
146 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 ...
3
votes
1answer
112 views

Divisor on curve of genus $2$

I suffer from lack of concrete examples in Algebraic Geometry, so I will appreciate it if somebody can help me in understanding a bit better this one: Let $\mathcal{C}$ be a genus $2$ curve ...
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 ...
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 ...
11
votes
2answers
197 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 ...
9
votes
3answers
187 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 ...
6
votes
1answer
146 views

Conic by three points and two tangent lines

With the exception of degenerate situations, a conic is uniquely determined by five points lying on it. Likewise, five lines tangent to a conic uniquely define that conic. With four points and one ...
6
votes
1answer
324 views

Are varieties of Kodaira dimension zero precisely the varieties with torsion canonical sheaf

Let $B$ a smooth projective connected variety over $\mathbf C$. Suppose that $K_B$ is torsion. Then, clearly, the Kodaira dimension of $B$ is zero. Does the converse hold? That is, suppose that $B$ ...
6
votes
1answer
778 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 $$D=kP$$ ...
4
votes
2answers
274 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) ...
1
vote
2answers
294 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
210 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 ...
-4
votes
1answer
124 views

How can I get smooth curve at the sigmoid function?

I'm trying to implement the sigmoid curve by using the following function. A is 3.2505508013 B is 1.5223545069 and K is 0.56. ...
9
votes
1answer
195 views

Weil does not imply Cartier on variety $X$.

Show that the divisor $D$ defined by $a = b = 0$ in the variety $X \subset \mathbb{A}^4$ defined by $ad - bc = 0$ $($the cone on a smooth quadric surface$)$ is not locally principal. My attempt ...
8
votes
2answers
284 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
265 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
309 views

Why is Klein's quartic curve not hyperelliptic

Let $X$ be Klein's quartic curve given by $x^3y + y^3z+z^3x=0$ in $\mathbf{P}^2$. It is isomorphic to $X(7)$. How do I easily show that $X$ is not hyperelliptic? I can see that $X$ is of genus $3$ ...
5
votes
2answers
127 views

Normal bundle of twisted cubic.

Let $C$ be a twisted cubic in $\mathbb P^3$. I'd like to compute the splitting type of normal bundle $N_{C/\mathbb P^3}$? I understood that $T_{\mathbb P^3}|_C=\mathcal O(4)^{\oplus 3}.$ So we have an ...
5
votes
2answers
118 views

Divisors of differentials.

Let $\mathbb{C}$ be the base field. Suppose $C \subset \mathbb{P}_2$ is a nonsingular projective curve of degree $d > 3$. Must it be the case that $D \in \text{Div}(C)$ is the divisor of a ...
3
votes
1answer
55 views

Smooth affine plane curve with non-trivial cotangent sheaf?

Question: Let $A = \mathbb C[x,y]/(f)$ be a non-singular plane curve. Under what conditions is the module of Kahler differentials $\Omega_A^1$ (over $\mathbb C$) a free module? I am not sure what ...
3
votes
0answers
102 views

Question about cusp cubic example in Hartshorne

In Hartshorne's Algebraic Geometry, in Chapter II.6 on Divisors he computes the Cartier class group (denoted $\operatorname{CaCl}$) of the cuspidal cubic cut out by $y^2z=x^3$ in $\mathbb{P}^2$. He ...
3
votes
1answer
59 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
2answers
254 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
2answers
65 views

Generator for Kahler differentials of an affine elliptic curve

Consider the affine (nonsingular) elliptic curve $A = \mathbb C[x,y]/(y^2-x^3+x)$. Since the cotangent bundle is trivial, $\Omega_A^1 = A\,dx\oplus A\,dy /(2y\,dy - (3x^2-1)\,dx)$ is a free ...