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Questions tagged [algebraic-curves]

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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 of curves are extensively studied as a basic case in singularity theory. Via algebraic function fields and modular curves they have links to number theory.

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For every function field $L$ there is a smooth projective curve $C$ with $L\simeq k(C)$

Let $k$ be an algebraically closed field. It is a well-known result that: The category of smooth (i.e., non-singular) projective curves with dominant morphisms is equivalent to the category of ...
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Proof of $Z(I(X))=X$ for $X\subset \mathbb{P}^n$ a projective algebraic set.

Let $X\subset \mathbb{P}^n$ be a projective algebraic set. That $X \subset Z(I(X))$ is clear. For the other inclusion, let's take $x\in Z(I(X))$ and we want to see that $x\in X$. Since $x\in Z(I(X))$,...
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Smooth special fiber implies smooth generic fiber when proper and flat

Let $f : X \rightarrow S$ be a morphism between schemes, with $f$ locally of finite presentation, proper and flat, and $S$ the spectrum of a discrete valuation ring with closed point $s$. Assume that ...
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22 views

Is it true that $P_f(k)\simeq C_{~^df}(k)\sqcup C_{~^d f_p}(k)$?

Let $f\in k[x_1, \ldots, x_{n+1}]$ be an homogeneous degree $p$ polynomial and let $~^d\! f\in k[x_1, \ldots, x_n]$ be defined by: $$~^d\! f(x_1, \ldots, x_n):=f(x_1, \ldots, x_n, 1).$$ Let $P_f(k)$ ...
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26 views

Topological Euler characteristic of a cycle of algebraic curves

The topological Euler characteristic of a smooth algebraic curve of genus $g$ equals $2-2g$. Let $C=\bigcup_{i=1}^n C_i$ be a union of smooth algebraic curves. I am wondering if there is a way to ...
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21 views

Characterize the values of A and B for which variety is singular

Let $A, B ∈ \overline{K}$ with $K$ a field. Characterize the values of A and B for which each of the following variety is singular: $V : Y^2 Z + AXY Z + BY Z^2 = X^3$ Looking at the point where ...
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1answer
49 views

Affine vs Projective curves?

Let $\mathbb A^n_k$ be affine $n$-space and let $\mathbb P^n_k$ be the projective $n$-space. An affine curve is a set of the form $$C_f(k):=\{(a_1, \ldots, a_n)\in\mathbb A^n_k: f(a_1, \ldots, a_n)=...
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Why is $xy=yz=xz=0$ not a complete intersection?

In a class on algebraic curves we were given the problem to show, that the union of the three axes $$C=\lbrace(x,y,z)\in\mathbb{C}^3\vert xy=yz=xz=0\rbrace$$ is not the intersection of two surfaces, ...
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Collinear Points on a Curve

I encountered a problem recently wherein I had to find the value of the coefficient of the second degree term in a 4th degree equation (other coefficients were given) such that there are $4$ ...
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56 views

Let $I=(y^2-x^2,y^2+x^2)$. Find $V(I)$ and $\text{dim}_{\mathbb{C}}(\mathbb{C}[x,y]/I)$

To find $V(I)$ we note that the only point $(x,y)\in \mathbb{C}$ satisfying both $y^2-x^2=0$ and $y^2+x^2=0$ is $(0,0)$, so $V(I)=(0,0)$. Now, to find $\text{dim}_{\mathbb{C}}(\mathbb{C}[x,y]/I)$, we ...
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55 views

Equivalence between function fields and curves

Sometime ago I overheard a conversation in which someone said "studying functions fields is the same thing as studying algebraic curves". After looking it up, I've found these two results (I'll ...
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64 views

Questions about 1-dimensional scheme

Suppose $X$ is a regular, Noetherian, separated, connected, one-dimensional scheme over a field $F$. Questions: (1) Does there always exist a point $P\in A$ such that $X-P$ is an open affine ...
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$X \to C$ is Finite Etale Map and $C$ Smooth Curve then $X$ also Smooth

My question refers to the proof of lemma 4.11 (page 9) from A. Kundu's "THE ETALE FUNDAMENTAL GROUP OF AN ELLIPTIC CURVE" ( math.uchicago.edu/~may/REU2017/REUPapers/Kundu.pdf) The statement is: ...
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58 views

A subbundle of a direct sum of two nonisomorphic line bundles

Let $0 \to N \to L_1 \oplus L_2 \to O_x \to 0$ be a short exact sequence, where $L_1, L_2$ are nonisomorphic line bundles on a smooth projective algebraic curve $X$ and $x$ is a point of $X$. Is it ...
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Montgomery form elliptic curve with given j-invariant

For a given $j\in K$, I would like to find a Montgomery form elliptic curve $M_{A,B}: By^2=x^3+Ax^2+x$ such that j-invariant of $M_{A,B}$ is j (if it is possible). In the form $y^2=x^3+Ax^2+x$ is also ...
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27 views

Dimension of $L(D)$ space

Let $X$ be a compact Riemann surface such that $g(X)=1$, and $p \in X$. Let's consider the divisor $D = n[p]$, where $n$ is n is natural. What's the dimension of $L(D)$ ? Thanks in advance for your ...
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On the definition of Edwards curves group addition

I'm working with Hales The Group Law of Edwards Curves where he defines the addition $z_1 \oplus z_2 = (\frac{x_1x_2 - c y_1y_2}{1 - d x_1x_2y_1y_2}, \frac{x_1y_2+y_1x_2}{1+dx_1x_2y_1y_2})$ and he ...
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Change of coordinates for $X^3+Y^3+60Z^3=0$

I have to demonstrate that the curve $$ X^3+Y^3+60Z^3=0$$ is birationally equivalent to $$Y^2=X^3-2^43^360^2$$ or to $$Y^2=X^3-3^330^2.$$ I can't find a proper change of coordinates for this purpose. ...
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80 views

non constant morphism of integral curves

Let $f: C_1 \to C_2$ be a non constant morphism of integral curves. by a curve I mean a proper one dimensional scheme over fixed base field $k$. I saw often the usage of the fact that under ...
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$[u:v]\mapsto[x(u^2v^3):y(u^2v^3):z(u^2v^3)]$ gives a rational curve

Given the map $\mathbb P^1\to\mathbb P^2$ with $[u:v]\mapsto[x(u^2v^3):y(u^2v^3):z(u^2v^3)]$, s.t. $[x:y:z]$ is on $C$(curve) how shall I deduce that the curve is rational ? I think I must show the ...
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Proving that the Quotient of an Algebraic Curve $X/G$ is an Algebraic Curve.

I am self-studying Miranda's book Algebraic Curves and Riemann Surfaces and I am looking for a hint for a problem. For those with the book, it is on page 178, and is Problem VI.$1$.L. The problem is ...
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Picard group of curves

I suppose the Picard group of a complete curve $C$ in $\mathbb P^n$ of degree $d>2$ is complicated. So, if we remove a general point $x\in C$ and denote $C'=C-x$, I think we have $\mathcal O_{C'}(1)...
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What goes wrong in the proof that $Spec \mathbb{R}[x,y]/(x^2+y^2)$ is a principal divisor in $Proj \mathbb{R}[x,y,z]/(x^2+y^2+z^2)$

Let $C=\text{Proj}\mathbb{R}[x,y,z]/(x^2+y^2+z^2)$. Consider a closed subset $Z=\{z\neq 0\}$. I claim that $Z=(f)$ with $f=x^2+y^2\in K(C)$. Indeed, if $z=0$ then the we get exactly $Z$. If $...
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26 views

Riemann Roch theorem for surfaces

Hi am a student of Maths at university; I am studying the theorem of Riemann-Roch for curves. I am interested in understanding what happens in the case of surfaces. I do not want to look for the whole ...
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1answer
35 views

Is the unit circle always an infinite set if the field is infinite?

Let $k$ be a field. Consider the unit circle $X=V(x^2+y^2-1):=\{(x,y) \in k^2 \mid x^2+y^2-1=0\}$. Question: Show that $V(x^2+y^2-1) \cong k$ as affine varieties if and only if char $(k)=2$. My ...
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Compute genus of compact Riemann surface $\{[x_0,x_1,x_2]\in\Bbb P^2|x_2^2x_0=\prod_{i=1}^3(x_1-\lambda_ix_0),\lambda_i\in\Bbb C\}$

Compute genus of compact Riemann surface $X=\{[x_0,x_1,x_2]\in\Bbb P^2|x_2^2x_0=\prod_{i=1}^3(x_1-\lambda_ix_0),\lambda_i\in\Bbb C\}, \lambda_i$ are distince numbers. I know the degree-genus formula, ...
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40 views

Show that $V(4xy+8xz+4yz-3y^2) \equiv V(x^2+y^2-z^2)$.

Definition An affine map from $F^n$ to $F^m$ is a function $f : F^n \to F^m$ given by $f(x)=Ax+b$ for some $m \times n$ matrix $A$ with entries in $F$ and some vector $b \in F^m$. Definition An ...
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The variety $X=V(x^2+y^2-1) \subseteq k^2$ is isomorphic to k if and only if $k$ has characteristic 2

Question: The variety $X=V(x^2+y^2-1) \subseteq k^2$ is isomorphic to k if and only if $k$ has characteristic 2. I don't even know where to start....Can anyone give me a hint?
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If $X$ is an irreducible variety in $\mathbb{C}^n$, then $\mathbb{C}^n \setminus X$ is path-connected.

My question: Prove that if $X$ is an irreducible variety in $\mathbb{C}^n$, then $\mathbb{C}^n \setminus X$ is path-connected in the standard metric topology. For every two distinct points $p,q$ in $\...
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Cubic curves and their lattices

Let $C_{\lambda}$ denote the non-singular cubic curve $y^2z = x(x-z)(x-\lambda z)$ in $\mathbb{C}\mathbb{P}^2$ for $\lambda \neq 0, 1$. a) Find projective transformations $\sigma, \tau, \nu$ which ...
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1answer
41 views

Recursive formula for points of algebraic curves over finite fields

Given $f$ an algebraic curve, let $N_f(p,k)$ the points of $f$ over $\mathbb{F}_{p^k}$ I need to prove that there exists a recursive formula of order $2g+2$ where $g$ is the genus of $f$. I know that ...
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Intersection between projective curves

They ask me to calculate the points of intersection of the projective curves $ C_1 = Z (x^2 + y^2-z^2) $ and $ C_2 = Z (x^2 + y^2 -2z^2) $ What I have done: I tried to solve the system $ x^2 + y^2 =...
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29 views

Isomorphism and birrationality between projective varieties

a) Let $ X $ projective variety and $ S (X) $ your homogeneous coordinate ring Let $ \phi: \mathbb{P}^1 \rightarrow \mathbb{P}^2 $ given by $ \phi([a: b]) = [a^2: ab: b^2] $ be $ C = Im \phi $. How ...
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Irreducibility of Lemniscates

I am reading S.Yu.Orevkov's new paper named "Irreducibility of Lemniscates", but I am stuck in the conclusion of this paper. The main result of this paper: Any lemniscate is an irreducible real ...
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Uniformizer $x$ on a curve then $x - x(p)$ uniformizer at all but fin. many points $p$?

Is it true that on a projective curve $C$, we have that if $x$ is a uniformizer at some point, then $x - x(p)$ is a uniformizer at $p$ for all but finitely many $p \in C$? If so can u give a proof? ...
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Help with example about divisors of differentials

I'm currently studying Silverman's Arithmetic in Elliptic Curves book. I hope someone can help me with example 2.4.6. Let $C$ be the curve $$C:y^2=(x-e_1)(x-e_2)(x-e_3)$$ with char$(K)\neq 2$, $e_1,...
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Coverings map the degree $2$ of the $\mathbb{P}^1$

Let $z$ be an affine coordinate on $\mathbb{P}^1$. Consider the curve $Y$ defined by the equation $$ y^2=\prod_{i=1}^{n}(z-z_i), $$ where $z_i \in \mathbb{C}$ are distinct. $1)$ Then, the curve Y ...
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42 views

Extending map between curves

Consider two integral curves $X,Y$ over an arbitrary base field $K$. Assume that $X$ is regular, $Y$ proper and there exists an open subset $U \subset X$ such that there exists a morphism $f_U: U \to ...
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Every plane projective conic is normal

I'm trying to solve exercise I.$3.17$ a) from Hartshorne's Algebraic Geometry, which states that every conic $X\subset\mathbb{P}^2$ is normal. By definition $X$ is normal $\Leftrightarrow \mathcal{O}...
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Smooth Affine Curve arises from a Field Extension

My question arises from following former thread of mine:Category Equivalence between Curves and Fields of Transcendence degree $1$ and refers to a comment of @reuns: Let $K/k(t)$ a finite ...
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33 views

Ideal of polynomials vanishing on algebraic set containing a polynomial with roots only in the set.

Lets say I have an algebraically closed field $\mathbb{K}$. I also have an algebraic set $Y \subset A^n(\mathbb{K})$. Is it true that the ideal of polynomials vanishing on $Y$ denoted as $I(Y)$ ...
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Riemann-Hurwitz formula for curves without schemes?

Is it possible to learn the proof and statement of the Riemann-Hurwitz formula for curves in Algebraic geometry without knowing schemes? If so: any good references? Thanks
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42 views

Redundant reference to the Nullstellensatz?

I just started with Kirwan's "Complex Algebraic Curves", and on page $29$ the notion of a complex algebraic curve is defined: Let $P(x,y)$ be a non-constant polynomial in two variables with complex ...
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29 views

Computing the intersection number of a line and a curve (Fulton Algebraic Curves 3.21.)

The problem is stated as follows: Let $F$ be an affine plane curve. Let $L$ be a line that is not a component of $F$. Suppose $L = \{(a+tb, c +td)\, | \, t \in k \}$. Define $G(T) = F(a+Tb, c+Td)$. ...
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58 views

Category Equivalence between Curves and Fields of Transcendence degree $1$

It is a well known fact that there exist a so called threefold category equivalence between following tree categories: 1) the category of finitely generated field extensions $K/k$ of transcendence ...
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1answer
50 views

Blow-Up in p not singular in p

I came upon a Lemma, which stated that the Blow-Up of an algebraic Curve $C \in \mathbb{C}[x,y]$ in a singular point $p$ of $C$ is non singular in $p$. For the proof the author referred to the ...
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34 views

Intersection multiplicity and contact order of plane curves

A standard fact about intersection multiplicities of algebraic plane curves is that, for curves $X=V(F(x,y))$ and $Y=V(G(x,y))$, if $p\in X\cap Y$ is a non-singular point of $X$ and $Y$ then $$ I_p(X,...
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29 views

Notation for Divisors in Liu's AG & AC

I have a question about a notation in Liu's "Algebraic Geometry and Arithmetic Curves" in the excerpt below (or look up at page 276): What is here means by $(f)_0$ and $(f)_{\infty}$? Does anybody ...
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2answers
122 views

Oblique Asymptote to a Curve

For a given curve: $$C: \frac {ax^2+bx+c}{dx+e} $$ where $a,b,c,d,e$ are integers. Let $f(x)=ax^2+bx+c$ . Oblique asymptote can be found by long division of numerator by denominator.Here oblique ...
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101 views

Dual of an elliptic curve?

Since an elliptic curve given by the homogenized polynomial $$y^2z=x^3+axz^2+bz^3$$ is a plane projective curve, we can get its dual. From this Wikipedia link, eliminating $p$, $q$, $r$, and $λ$ from ...

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