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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Euler characteristic singular surface

The setting is the one of algebraic curves over the complex numbers. It is known that in an irreducible nodal curve each node reduces the arithmetic genus by one: if $\tilde{C} \rightarrow C$ is the ...
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33 views

Projective Mapping of a Sphere

Show that the projective completion of the curve $Y=X^2$ is topologically a sphere. Consider the parametrization $X=t, Y=t^2$, where t ranges the sphere $\mathbb C\cup {\infty}$. How do I prove this ...
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1answer
49 views

Maps to $\mathbb{P}^1$ induced by rational functions.

I am reading from Hartshorne, Corollary II.6.10 page 138. Given a nonsingular (is this necessary?) curve $X$ over a field $k$, let $f\in K(X)^*\setminus k$. Then the inclusion of fields ...
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19 views

A question on algebraic curve

Let $A_j \in \mathbb{C}^{n \times n}$, ($j = 0,1,2,\ldots,m$) and $P(z) = A_m z^m + \cdots + A_1 z + A_0$ is a matrix polynomial, and $z $ is a complex variable. $s_1 \ge s_2 \ge \cdots \ge s_n$ ...
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1answer
24 views

Image of morphism between curves

I have this projective curve $C\dots y^2z=x(x+2z)(x-z)$ and I have function $f$ on $C$, ie $f\in k(C)$ given by $f(x:y:z)=(y:z)$. What would be image of this function? Can I see it in some way as ...
2
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1answer
48 views

Order of the pole of projective curve using uniformizer

I need to find divisor of functon $f=y$ i.e. $\frac{y}{z}$ for projective curve $y^2z=x^3-xz^2$ and I have some questions: For example, I have pole: $(0:1:0)$ and zeros $(0:0:1),(1:0:-i),(1:0:1)$. It ...
3
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0answers
87 views

Geometric interpretation of algebraic property

Let $X \subset \mathbb{C}^n$ be an irreducible affine algebraic curve with coordinate ring $$\mathbb{C}[X] = \mathbb{C}[z_1, \ldots, z_n] / (f_1, \ldots, f_m ) $$ with each $f_i \in \mathbb{Z}[z_1, ...
2
votes
1answer
56 views

When is a hypersurface rationally connected?

A projective variety $X$ is said to be "rationally connected" if any two points on it can be connected by a map $\mathbb P^1 \to X$. Let $X$ be a smooth hypersurface in $\mathbb P^n_k$ defined by a ...
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48 views

Smooth completion of algebraic curves

I am having trouble understanding the concept of smooth completions of algebraic curves. I know the definition (smooth, complete curves which contain the curve X as an open subset ) and according to ...
1
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1answer
34 views

local equation of a divisor, localization and local ring at a point

Let $X$ be a regular, proper surface (integral, separated, of fin. type) over a perfect field $k$ and let $Z\subset X$ be an integral irreducible subscheme. Moreover $\eta$ is the generic point of $X$ ...
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32 views

The set of curves of degree $d$ with two singular points or a degenerate singular point is closed?

Suppose $d>0$, $k$ is a field and $L_d$ is the space of projective plane curves over $k$ of degree $d$, namely $L_d=\mathbb P(k[X,Y,Z]_d)$, where $k[X,Y,Z]_d$ is the vector space of homogeneous ...
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37 views

Intersection multiplicity inequality problem

I have a question about an inequality that was stated in my class but never proven. We stated p is simple $I_p(F \cap G+H) \ge \min ( I_p (F \cap G) , I_p(f \cap H))$. Where we defined $I_p(F \cap ...
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1answer
46 views

Question about intersection multiplicity of a curve and it's tangent line

If we have a double point $a$ on some complex curve, call it $C$, defined by some polynomial $f$ and we have only one tangent line at $a$, call it $T_l$, then the intersection multiplicity $I(a,f \cap ...
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46 views

Polar forms of algebraic curves & surfaces

A paper I'm reading says the following ... With homogeneous coordinates $\mathbf{x} = [x,y,z,w]$, let $F(\mathbf{x}) = 0$ be the equation of a surface of degree $n$. The first polar form of ...
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1answer
41 views

Find the generator of the maximal ideal of regular functions on a curve on P

I'm stuck at a probably very simple exercise : Consider the algebraic curve $C: Y^2=X+X^3$ and $P=(0,0)$. Find a generator of the maximal ideal of the local ring of rational functions on $C$ ...
2
votes
1answer
109 views

How can find points such that the tangent fails to exist?

Let $F(x,y)=0$ and $F(x,y)$ be polynomial in $\mathbb R$. How can find points such that the tangent fails to exist ?
3
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1answer
63 views

The independence of the degree of morphisms between two curves on field extension.

Suppose $C_1$ and $C_2$ are two regular projective geometrically irreducible curves over $k$ and $F$ is a surjective morphism between them, then the degree of $F$ is the degree of the field extension ...
2
votes
1answer
35 views

Must a polynomial function of $x$ pass through infinitely many integer lattice points?

I made a mistake in my formulation of this question when I last asked it and got downvoted because the answer was actually trivial. However, I think the intended question is actually an interesting ...
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2answers
34 views

Does every line through the origin in the plane intersect the integer lattice an infinite number of times? [closed]

Question is in the title. What about every algebraic curve through the origin? Does every line through the origin in the complex numbers pass through an infinite number of Gaussian primes? EDIT: Just ...
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45 views

Pushforward on principal divisors

I am having much difficulty proving the following fact. Let $\phi:C_1\to C_2$ be a Galois cover of curves over an algebraically closed field $K$. Let $G=\mathrm{Gal}(K(C_1)/\phi^*K(C_2))$. Let $N: ...
3
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0answers
57 views

Automorphism of elliptic curve, Vakil 19.10.E

There are many other proofs of finding all the possible automorphism groups of elliptic curves, but I am interested in the $Hint$ and the corresponding proof in the following exercise from Vakil's ...
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1answer
29 views

A 3D curve correlation

Forgive me if its too basic, but i am looking to read some materials about a subject in which i don't know its name/field. So what we need to do, is to get a 3 axises curve, with unknown shape, that ...
2
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1answer
37 views

Automorphism group of genus 1 curve

Suppose $E$ is a regular curve of genus 1 over a field $k$ that is not necessarily algebraically closed. The automorphism groups of $E$ forms a one parameter family. If $k=\mathbb{C}$, then $E$ could ...
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1answer
61 views

Picture of elliptic curve double covers Rieman sphere

For simplicity, let us assume we live in the field $\mathbb{C}$. Then all elliptic curves are hyperelliptic, so they admit a double cover of $\mathbb{P}^1$ branched over four points, whose preimage ...
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45 views

Automorphism group of genus 2 curve

Suppose C is a genus 2 curve over a field k such that char k is not 2. Is there an easy way to show that the automorphism group is finite? If we assume k is algebraically closed then C is ...
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0answers
22 views

compute order x/z in Q

Let $x^3+y^2z+yz^2+\alpha^3z^3=0$ be a curve over $F_{16}$ where $\alpha$ is primitive element for $F_{16}$ and $\alpha^4+\alpha+1=0$. The point $Q=(0:1:0)$ is only infinity point this curve. I want ...
2
votes
1answer
90 views

Dimension of the Riemann-Roch Space of a divisor of a cubic (Fulton's Algebraic Curves Exercise 8.10)

Let $C$ be an irreducible projective cubic on $k[X,Y,Z]$, $k$ algebraically closed. Let $x = \frac{X}{Z}$, $y = \frac{Y}{Z}$, $z = x^{-1} = \frac{Z}{X}$. Also define the divisor of zeros of $z$: ...
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0answers
38 views

Definition of Numerical Trivial Invertible Sheaf

I am reading Ravi Vakil's book, Foundations of Algebraic Geometry and in section 18.4.9, the definition of numberically trivial invertible sheaf is, Suppose $X$ is a proper $k-$ variety, and ...
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0answers
47 views

Understanding Localized Rings mod an ideal.

Hi guys I am working with Fulton's book and I am trying to understand for myself the elements of two rings. $O_p(\mathbb{A}^n)/JO_p(\mathbb{A}^n)$ and $O_p(V)/\bar{J}O_p(V)$ Where $I = ...
1
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1answer
61 views

Intersection of n hyperplanes in projective space of dimension n is not empty

I want to prove the following: Let $H_1,\dots,H_n$ be $n$ hyperplanes in $\mathbb{P}^n =\mathbb{P}^n \mathbb{C}$. Then $\cap_{i=1}^n H_i$ is not empty. Please be noted that this is an exercise ...
6
votes
1answer
155 views

Understanding an exercise from Fulton's Book on Algebraic Curves

I am reading Fulton's book Algebraic Curves. Currently I am working on a specific problem (2.43), and I have doubts about my work and would appreciate another opinion(s). Assume $p$ is the origin ...
0
votes
2answers
52 views

Why the image of quadratic Veronese map has the form $v\cdot v$?

It says that the image of quadratic Veronese map $v_2(P^1)$ is the subset of $P(Sym^2V)$ with the form $v\cdot v$. Isn't it has the form $x^2+xy+y^2$? So how can it be some $v\cdot v$?
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0answers
56 views

Etale map from a variety to an elliptic curve

I read this sentence and I can't see why it is true. Let $E$ be an elliptic curve over an algebraically closed field $k$, $f\colon Y\to E$ an etale map; then $Y$ is also a curve over $k$. Can ...
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1answer
46 views

Irreducible quartic curve in projective space

Let $V$ be an irreducible quartic curve in $\mathbb P^2(\mathbb{C})$. Then $V$ cannot have a point of multiplicity greater than or equal to $4$. Furthermore if $V$ has a triple point $(a,b,c)$, ...
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2answers
62 views

Question about projective closure of a curve

I am working on a problem, and I wanted someone to look over my work and comment of I am on the right track. I have a polynomial $f\in K[x]$, where $K$ is an algebraically closed field, and $d=\deg ...
5
votes
1answer
68 views

Isomorphism of the affine circumference over certain fields.

Let us consider the coordinate ring of the circumference $$ A:=K[X,Y]/(X^{2}+Y^{2}-1), $$ and let us suppose that $K$ is infinite but not necesarilly algebraically closed. I wonder if $A$ is ...
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votes
1answer
28 views

Ideal of the hyperbola in a field that is not algebraically closed.

Let $K$ a field not necessarily algebraically closed. I would like to find the coordinate ring of the hyperbola $$ V(XY-1)\subseteq K^{2}. $$ If the field was algebraically closed we could use ...
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38 views

Definition of algebraic cusp

Is not it true that for planar curves, an $\textit{algebraic cusp}$, say at the origin, is the one that can be locally represented by $y^n=x^m$ with $m,n\in\mathbb N$ and $(m,n)=1$? What is a ...
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0answers
26 views

Line through 2 flex passes through a third flex

(in $\mathbb P^2$)Show that a line through two flexes on a cubic passes through a third flex. I've tried to solve this problem using the corollaries of the Max Noether theorem (which talks about ...
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39 views

an exercise of shafarevich (I.1.8)

I have a question related to the following exercise of Shafarevich's Basic Algebraic Geometry: (I.1.8) Prove that for any nonsingular points $P_1,\ldots,P_r$ of an irreducible curve and numbers ...
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1answer
60 views

How to find the equation of the curve defining the intersection of two quadrics.

Let $\mathbb{F}_{q}$ be the finite field of order $q$ where $q$ is the power of an odd prime. Let $u$ be a fixed non-square in $\mathbb{F}_{q}$ and let $\lambda\in\mathbb{F}_{q}$ such that ...
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2answers
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A curve has the equation $y=x^3 - 4x^2 - 3x + 17$. What are the x-coordinates of the point(s) on this curve where the tangent is parallel to 4y=7x-11.

A curve has the equation $$y=x^3 - 4x^2 - 3x + 17$$. What are the x-coordinates of the point(s) on this curve where the tangent is... (a) parallel to $4y=7x-11$ (b) horizontal
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2answers
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Does anyone know of any good sources on the algebraic theory of abelian varieties?

I have a copy of Mumford's book, but as a final year undergraduate I am finding it to be a little too dense as a starting text. Something lighter would be appreciated to get an intuition before ...
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18 views

Finding a cubic function with one real root given its graph.

When given a cubic graph with one real root. I need to find the equation of that graph using the function $$y=a(x-s)(x^2+bx+c)$$ where a, b,and c are unknowns. The y intercept is therefore $t = -asc$ ...
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0answers
36 views

Linearly normal embedding and varietes lying on quadrics

Let $X\subset\Bbb{P}^N$ be a smooth algebraic variety and assume that $X$ is not contained in a hyperplane. The embedding $i\colon X\hookrightarrow\Bbb{P}^N$ is called linearly normal if the linear ...
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23 views

Finding and analyizing the singularities in Affine and Projective space

Hi guys I am working this $F(x,y,z)=xy^4+yx^4+xz^4$ I need to find the singularities in affine and projective space and find the multiplicity of them.I would really appreciate some help tips. So ...
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1answer
20 views

Finding a curve given only its basic form and its tangent line

The basic form is $f(x)=k\sqrt{x}$ and the tangent line is $4x+36$. I've spent over half and hour with wolfram alpha and my notes on this problem: I've tried $(k\sqrt{x})'=4x+36$ and got ...
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1answer
29 views

There exist cuspidal cubic sections in a nonsingular cubic surface in $\mathbb{P}^3$

This is part of Exercise 7.3 in Undergraduate Algebraic Geometry by Reid. Let $S: (f=0) \subset \mathbb{P}^3$ be a nonsingular cubic surface. For $P\in S$ prove that if $P$ is not on a line of $S$ ...
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1answer
29 views

Isomorphism on cubics group law

Let $C$ a non singular cubic projective plane curve with a fix point $O$, we use que chord-tangent method to make $G = (C,+,O)$ an abelian group, if I choose another fix point $O'$ and construct the ...
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1answer
12 views

Set of roots of quadratic form $B(x,y,z,t)$ on the line $z=t=0$ is nonempty.

This is a proof from Section 7.1 of Undergraduate Algebraic Geometry by Reid. Suppose $S\subset \mathbb{P}^3$ is a nonsingular cubic surface, given by a homogeneous cubic $f=f(x,y,z,t)$. Consider ...