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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Group actions on Čech cohomology

Suppose we have a curve $X$ and a group $G$ acting on $X$. Then one has an induced action of $G$ on the sheaf cohomology of $\mathcal O_X$. I wondered what one can say about the group action on the ...
2
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0answers
131 views

Genus of a function field

There is a one-to-one correspondence between isomorphism classes of smooth absolutely irreducible curves $X/\mathbb{k}$ and isomorphism classes of fields $\mathbb{K}$ of transcendence degree $1$ over ...
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123 views

Analogy between Picard group and Ideal class group

Can you give a reference where the conformity between Picard group and Ideal class group is explained? What is analogy of Picard group of elliptic curve over finite field?
2
votes
0answers
101 views

Computing algebraic de-Rham cohomology via Čech cohomology

I have been reading this paper about de-Rham cohomology of hyperelliptic curves, and I have been trying to recompute some of what has been done in section 3. In particular, I am trying to see why ...
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46 views

Curve of genus $0$ is rationally parametrizable .

I have recently come across a statement that an algebraic curve of genus $0$ exhibits rational parametrization . Can someone tell me where i can read the proof or give me a proof . Thanks .
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0answers
47 views

Are there Galois covers of curves branched at 1 point?

Let $G$ be a finite group, not necessarily abelian. Is there any smooth algebraic curve $C$, with an action of $G$ on $C$, such that the natural quotient map $C \to C/G$ is branched at precisely one ...
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0answers
76 views

Galois actions on extensions of algebraic function fields

Let $k$ be an algebraically closed field, $C/k$ and $C'/k$ be smooth projective curves, and $C'/k \rightarrow C/k$ be a $k$-morphism which is corresponding to the field extension $k(C) \hookrightarrow ...
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votes
1answer
52 views

Existence of a holomorphic map from Riemann Surface to an algebraic curve .

Let $C$ be an algebraic curve in $\mathbb P^2( \mathbb C)$ with singular points $p_i : \{1 \le i \le n \}$ . Then there exists a holomorphic map $\Phi : S \to C$ , where $S$ is a Riemann surface. ...
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100 views

number of integral points on an ellipse

Let $Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0$ be an ellipse. How can the number of integral points lying on such an ellipse be calculated ($A,B,C,D,E,F$ are, of course, integers) ?
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99 views

Are these rational curves?

I have to find the singular points of the following curves and tell if they are rational. The curves are $C=Z(x^2+y^2+x^2y^2)$ and $C=Z(x^3+y^3-1)$, and the base field is the complex one. I think I ...
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votes
1answer
93 views

Embedding of curves in projective spaces… typo?

I'm reading from the book "Geometry of algebraic curves", by Griffiths, Harris, Arbarello and Cornalba. In the middle of page 5 they define the map $\phi_{\mathscr{D}}:C\to \mathbb{P}V^*$, from a ...
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1answer
149 views

Every $n$-dimensional variety is birationally equivalent to a hypersurface in $\mathbb{A}^{n+1}.$

Problem: Show every $n$-dimensional variety is birationally equivalent to a hypersurface in $\mathbb{A}^{n+1}.$ Thoughts: For a (quasi-projective) variety $X,$ the function field $k(X)$ is a finitely ...
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vote
0answers
28 views

Bézier curves as portions of algebraic curves

Can every Bézier curve of any degree be defined as the algebraic (polynomial) curve of which it is a part and it's endpoints? If some Bézier's (such as those of degree $n$ or greater) cannot be ...
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53 views

$H^0 (\psi_i)$ on $\mathcal M_{0;n}$

Thinking about moduli spaces of tropical curves, I obtained the following result: On the moduli space of tropical marked rational curves $\mathcal M^{trop}_{0;n}$ for any $i$ we have: $\dim H^0 ...
3
votes
1answer
90 views

Genus over finite fields

Is there a way of computing the genus of a parametrized curve over a finite field? For instance I am interested in the genus of the following space curve in the m-dimensional space over $F_{q^k}$ ...
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30 views

Is the following variety reducible? [duplicate]

$\{y^2z-x^2z-x^3=0\}$ in $\mathcal{O}_{\mathbb{C}^3,(0,0,z)}$ while $z\neq 0$.
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73 views

A problem of irreducibility

Consider the cone in $\mathbb{C}^3$ over the curve $\{y^2-x^2-x^3=0\}\subset\mathbb{C}^2$. Show that $g=zy^2-zx^2-x^3$ with $g = 0$ giving the cone, is irreducible in $\mathcal{O}_{\mathbb{C}^3,~0}$, ...
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2answers
142 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, ...
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1answer
153 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 ...
3
votes
1answer
99 views

Intersections of tangents with cubic are colinear

I am trying to do Exercise 5.33 on page 64 of Fulton's book on algebraic curves. 5.33 Let $C$ be an irreducible cubic, $L$ a line such that $L\bullet C = P_1+ P_2 + P_3,$ $P_i$ distinct. Let $L_i$ ...
3
votes
1answer
66 views

Smallest projective subspace containing a degree $d$ curve

Is it true that the smallest projective subspace containing a degree $d$ curve inside $\mathbb{P}^n$ has dimension at most $d$? If not, is there any bound on the dimension? Generalization to ...
4
votes
1answer
85 views

curves and surfaces. curvature of a regular curve

Let $\gamma(t)$ be a regular curve lies on a sphere $S^2$ with center $(0, 0, 0)$ (origin) and radius $r$. Show that the curvature of $\gamma$ is non-zero, i.e., $κ \ne 0$. Furthermore, if the ...
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0answers
79 views

Finding singular points , its multiplicity and tangents .

I am looking for an example to know How could I possibly find all the singular points with multiplicity and tangents? I have a particular curve in interest $$G:=2X^4-3X^2YZ + ...
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36 views

Determining the equivalence class of local parametrization of affine algebraic curve with center $(0,0)$

I would like to know how i can determine the equivalence class of local parametrization of affine algebraic curves ? Lets say for example a curve $C=V((X^2+y^2)^2+3X^2Y-Y^3)$ Can someone help me ? ...
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votes
1answer
40 views

Nonsingular affine curve which is not unmixed

Let $C$ be any nonsingular curve in $A^3_{\mathbb C}$. Can a point be an irreducible component of $C$? I am not able to find an example of such $C$.
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110 views

Why do algebraic varieties contain curves passing through two given points

Let $X$ be an algebraic variety over the complex numbers. My definition of an algebraic variety is a finite type separated $\mathbf C$-scheme. Someone told me that such varieties have the following ...
2
votes
1answer
148 views

Definition of simple spectrum

From the book "Spinning Tops" by Audin, given Lax equation $[A_{\lambda},B_{\lambda}]$ where $\lambda$ is a parameter (so called spectral parameter), he claims that we have spectral curve ...
2
votes
0answers
71 views

Puiseux series and Resolution of Singularities

I have a very basic knowledge of algebraic geometry(no schemes!), and am trying to study the resolution of singularities. So the Newton's method gives us a Puiseux series parametrizing the branches of ...
6
votes
5answers
326 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
3answers
78 views

Good source to learn about surface singularities?

I am looking for something that treats singularities on algebraic surfaces and curves over $\mathbb{C}$, starting from the very basics but not stopping there. I checked out Miles Reid his lectures on ...
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1answer
52 views

Existence of a variety with prescribed properties

In these notes that give a proof of the Weil conjectures for curves, the author writes on page 17 that given a smooth projective curve $X$ over a finite field $k = \mathbb{F}_q$ for a fixed prime $q$, ...
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89 views

When is an intersection of varieties finite

Consider the general Bezout's theorem: If $p_1 \ldots p_n$ are polynomials with degrees $d_1,\ldots, d_n$ in $\mathbb{R}[x_1,\ldots,x_n]$, with $V = \{a=(a_1,\ldots, a_n) | p_i(a) = 0, \forall i\}$ ...
5
votes
1answer
50 views

deg functions and maps

For any map $f$ between curves $C_1$ and $C_2$, one defines $\mathrm{deg}(f) = [K(C_1) : f^*K(C_2)]$ as given in "The Arithmetic of Elliptic Curves" by Silverman. For algebraic functions on elliptic ...
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votes
2answers
235 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 $V$. ...
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50 views

How to find the dimension of linear system of curves of degree $d$

Consider two curves $C_1$ and $C_2$ in $\mathbb P^2 (\mathbb C)$ . How can i find the expected and real dimension of the linear system of cuves of degree $d$ passing through points lying on the both ...
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0answers
38 views

Solving the algebraic equations .

I am working with an equation to find the singular points in $\mathbb P^2 (\mathbb C)$ . Basically after taking the partial derivatives and doing some manipulations it reduces to $$y^2 + (2-k)xz ...
2
votes
1answer
62 views

Extension of prime ideal in $k[V]$ to $\mathcal{O}_P(V)$ is prime?

Let $k$ be an algebraically closed field, $I\subset k[X_1,\cdots, X_n]$ be a prime ideal, $V=V(I) \subset \mathbb{A}^n$ a variety and $P=(a_1,\cdots, a_n)\in V.$ Recall that $\mathcal{O}_P(V)$ is the ...
3
votes
1answer
83 views

Computing a rational function at a point in terms of a uniformising parameter

I am not quite sure how to ask this precisely, but vaguely I would like to know how difficult it is to write a function on an algebraic curve at a point $P$ as a power series of a uniformising ...
2
votes
1answer
64 views

Number of intersection multiplicity points .

I need help for the following problem : Consider $C_1 = V(F_1)$ and $C_2=V(F_2)$ be algebraic curves in $\mathbb P (\bar K )$ (where $K$ is a field,) without a common component and $F_1, F_2 \in ...
11
votes
2answers
408 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 ...
3
votes
2answers
166 views

Intersection of smooth projective plane curves

I need to calculate the number of intersections of the smooth projective plane curves defined by the zero locus of the homogeneous polynomials $$ F(x,y,z)=xy^3+yz^3+zx^3\text{ (its zero locus is ...
5
votes
1answer
228 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 ...
2
votes
2answers
126 views

Large Intersection Multiplicity

A cubic curve, say, $x^3+y^3=1$ and some quadratic curve $f(x,y)=0$ generally have six intersection points in $\mathbb{CP}^2$. Question: If all the intersection points coincide, what will be the ...
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votes
1answer
88 views

What's the sense in a Hyperelliptic Riemann Surface?

Can someone explain me, possibly using some very intuitive ideas, of what kind of object a hyperelliptic Riemann Surface is? What's the goal of constructing it (my lecture on is was based in Miranda's ...
2
votes
1answer
43 views

Creating and using calibration factors

Perhaps simple question, but I (the simple) need some guidance. The following applies to a project ongoing and is a challenge in that I am not a math whiz! As example, I wish to measure temperature ...
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vote
1answer
59 views

Degree of morphism of quotient of upper half-plane

Recall that SL$_2(\mathbf R)$ acts on the complex upper half-plane $\mathbf H$. Let $\Gamma$ be a finite index subgroup of SL$_2(\mathbf Z)$. Then there is the quotient $Y_\Gamma = \Gamma \backslash ...
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votes
1answer
329 views

Circumference of a superellipse?

Could someone help me formulate the circumference of a superellipse? $$\frac{x^n}{a^n} + \frac{y^n}{b^n} = 1$$ If it makes things easier, I'm considering only the cases $n>2$, and ...
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vote
0answers
87 views

A funny condition for ampleness on a curve

Let $E$ be a locally free sheaf on a complete nonsingular curve $C$ over $\mathbb C$. Suppose that, for all points $P$ in $C$, the locally free sheaf $$ E\otimes \mathcal{O}_C(P)$$ is an ample ...
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1answer
104 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 ...
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30 views

Homogeneity of translated polynomial

I am currently trying to understand the very basics of complex algebraic curves and I came across the following statement in the book by F. Kirwan (Definition 2.9): The multiplicity of the complex ...