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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229 views

Problem related to tangents and normals of a curve.

I 've been trying the both sums, while first one I've no clue how to start about in the second one I a getting stuck. [1] The equation of the normal at any point $\theta$ on the curve $x=a ...
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52 views

Algebraic curve over rings $\mathbb{Z}/n\mathbb{Z}$

Can you give a reference about algebraic curve over rings $\mathbb{Z}/n\mathbb{Z}$? I'm very interested in analogy Hasse-Weil theorem, R-R theorem... Thank you.
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261 views

Twist of elliptic curve

It is continuation of this question: explict form of the equation of elliptic curve Let $p$ is prime and $p = 3 ($mod $4)$. $q = p^n$. It is easy to see that $E: y^2 = x^3 + x$ has $1 \pm 2q + q^2$ ...
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176 views

Function that is identically zero

Is it true that: Any rational function $f$ on $\mathbb{C}^2$ that vanishes on $S=\{(x,y)\in\mathbb{C}^2 : x=ny \text{ for some } n \in \mathbb{Z}\}$ must be identically zero. I have a theorem that ...
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1answer
71 views

Is this set an algebraic set

Is the set $\{z\in\mathbb{C}:|z|=1\}$ and algebraic set? Intuitively, I think the answer is no because it is not possible to use a polynomial to split an arbitrary complex number into its real and ...
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155 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?
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189 views

Polynomial curve fitting with a set of points plus an angle constraint

I'm trying to find the polynomial equation of degree 3 passing through a set of given points as explained in this wikipedia article. However, instead of providing 4 points, I'd like to provide 3 ...
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3answers
215 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 ...
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2answers
128 views

Space of global sections for a smooth projective curve of genus $g$

Let $X$ be a smooth projective curve of genus $g$, $T_{X}$ its tangent bundle and $H^{0}(X,T_{X})$ the space of global sections for $X$. What is $\dim H^{0}(X,T_{X})$ and why?
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255 views

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 ...
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133 views

Line bundles over a curve

I keep seeing statements like that and I don't know how they might be established: Let $k \subset \bar{k}$ be fields, $C \rightarrow Spec(\bar{k})$ be a curve and $\mathscr{L}$ a line bundle on $C$. ...
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90 views

Reading multiplicity of cusps , singularity etc from initial polynomial.

Here I have an example which I found. Can someone help me to understand what's happening here? The following are my concerns: 1) What do we need to do co-ordinate transformation? 2) How does the ...
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81 views

Is there an algebraically normal function from $\mathbb{Z}^{2}$ to $\{ 0 , 1\}$?

Let $\gamma : \mathbb{R} \to \mathbb{R}^{2}$ be a real algebraic curve. Let $r \geq 0$ and $I \subset \mathbb{R} $ then $\gamma_{r} (I)= \{a \in \mathbb{R}^{2} : \exists b \in \gamma(I), d(a,b)\leq r ...
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0answers
21 views

automorphism groups of hyerelliptic curves in positive charactersitic

It appears that the automorphism groups of hyperelliptic curves are at least well studied, if not understood, in the characteristic zero case. I would imagine that most of these results would carry to ...
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88 views

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 ...
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319 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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193 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 ...
2
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56 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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95 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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1answer
64 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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152 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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119 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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143 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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301 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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34 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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55 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 ...
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1answer
122 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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2answers
226 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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190 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 ...
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142 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 $L_i$ ...
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1answer
77 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
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1answer
105 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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1answer
61 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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159 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 ...
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1answer
345 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 ...
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111 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 ...
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5answers
408 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 $x+y$.
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3answers
105 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 ...
6
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1answer
58 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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124 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 $V = \{a=(a_1,\ldots, a_n) | p_i(a) = 0, \forall i\}$ ...
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1answer
53 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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2answers
417 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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108 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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52 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 ...
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1answer
75 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
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1answer
101 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
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1answer
76 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 ...
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3answers
808 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
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2answers
329 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 ...
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1answer
564 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 ...